diff --git "a/49AyT4oBgHgl3EQfpPiQ/content/tmp_files/load_file.txt" "b/49AyT4oBgHgl3EQfpPiQ/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/49AyT4oBgHgl3EQfpPiQ/content/tmp_files/load_file.txt" @@ -0,0 +1,859 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf,len=858 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='00522v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='RT] 2 Jan 2023 Irreducible module decompositions of rank 2 symmetric hyperbolic Kac-Moody Lie algebras by sl2 subalgebras which are generalizations of principal sl2 subalgebras TSURUSAKI Hisanori∗ Abstract There exist principal sl2 subalgebras for hyperbolic Kac-Moody Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In the case of rank 2 symmetric hyperbolic Kac-Moody Lie algebras, certain sl2 subalgebras are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' These subalgebras are generalizations of principal sl2 subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show that the rank 2 symmetric hyperbolic Kac-Moody Lie algebras themselves are irreducibly decomposed under the action of this sl2 subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Furthermore, we classify irreducible components of the decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In particular, we obtain multiplicities of unitary principal series and complementary series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 1 Introduction A nilpotent orbit in a finite dimensional simple Lie algebra g0 is an orbit ob- tained by acting on the nilpotent element x of g0 by inner automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In [Dyn57], these are classified by weighted Dynkin diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the Jacobson- Morozov theorem, for a nilpotent element x of g0, we can construct a sl2-triple with x as a nilpositive element ([CM93, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This makes it equiva- lent to classify nilpotent orbits of g0 and to classify sl2 triples in g0 up to inner automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Among the nilpotent orbits of a finite dimensional simple Lie algebra, the one whose dimension as an algebraic variety is maximal is called the principal nilpotent orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Correspondingly, we can construct a principal SO(3) subalgebra that is compatible with compact involution ([Kos59]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Kac-Moody Lie algebras are generalizations of finite-dimensional simple Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' They are classified into three types: finite type, affine type, and indefinite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The finite type Kac-Moody Lie algebras are finite dimensional simple Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Within indefinite Kac-Moody Lie algebra, there is a class called hyperbolic Kac-Moody Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' A hyperbolic Kac-Moody Lie algebra ∗Graduate School of Mathematical Sciences, University of Tokyo , htsu- rusaki1929@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='com 1 is an indefinite type Kac-Moody Lie algebra such that any true subdiagram of its Dynkin diagram is of finite or affine type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' By analogy with the above theory, in [NO01], for a hyperbolic Kac-Moody Lie algebra, its principal SO(1, 2) subalgebra was constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Note that [GOW02] shows that it is possible to construct a principal SO(1, 2) subalgebra for certain indefinite Kac-Moody Lie algebra that is not hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Corresponding to this principal SO(1, 2) subalgebra, we can construct a principal sl2-subalgebra in a hyperbolic Kac-Moody Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In [Tsu], for the rank 2 symmetric hyperbolic Kac-Moody Lie algebras g, the following re- sult is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let the space that the positive real root vectors span be Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we consider sl2 subalgebras whose nilpositive element exists in Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then we can construct certain sl2 subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' These subalgebras are generalizations of principal sl2 subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In this paper, for an sl2 subalgebra of rank 2 symmetric hyperbolic Kac- Moody Lie algebra g constructed in [Tsu], we show g is decomposed into irre- ducible sl2-modules by its action on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We are going to more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let s be an sl2 subalgebra constructed in [Tsu].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let H, X, Y be an sl2 triple and assume that s is spanned by H, X, Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let the Chevalley generators of g be ei, fi, hi, (i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' , n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let hR be the R-span of Chevalley generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [Kac90, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2], g has a C-valued nondegenerate invariant symmetric bilinear form (· | ·) called the standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' An antilinear automorphism ω0 of g, called compact involution, is determined by ω0(ei) = −fi, ω0(fi) = −ei (i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' , n − 1), ω0(h) = −h (h ∈ hR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [Kac90, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='7], we can determine a nondegenerate Hermitian form (· | ·)0 on g with (x | y)0 = −(ω0(x) | y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' s-module V is called unitarizable if the following conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) (· | ·)0 on V is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) for v1, v2 ∈ V , the conditions as follows are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' ([X, v1], v2)0 = −(v1, [Y, v2])0, ([H, v1], v2)0 = −(v1, [H, v2])0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' g can be decomposed into a direct sum of irre- ducible s-modules such that s itself is one of the direct summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' All of these modules except for s are unitarizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Also, we classify how many highest weight modules, lowest weight modules, and modules that are neither highest weight module nor lowest weight module appear in this decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We regard a root sα1 + tα2 as a point (s, t) in xy-plane, and We define a region L, −L in xy-plane in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If a root α satisfies α(H) ∈ (0, 2), α ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If a root α satisfies α(H) ∈ (−2, 0), α ∈ −L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 2 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2 (Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider an irreducible decomposition of g by the action of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) Let M is an irreducible component of decomposition of g, which contain a root space for a real root in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then, M is an unitary principal or complementary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' [Tsu, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3]) There is an unitary principal series represen- tation containing an 1-dimensional space in h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (3) g is decomposed into a direct sum of s-submodules described in (1) and (2) above, s itself, irreducible lowest weight modules, and irreducible highest weight modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We also discuss how to calculate multiplicities of irreducible highest or low- est modules (§7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Furthermore, we classified irreducible components which are neither highest weight modules nor lowest weight modules, as either unitary principal or complementary series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3 (Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider irreducible components which are neither highest weight modules nor lowest weight modules and contain root vectors about real roots in L, obtained by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The irreducible com- ponents are complementary series representations, except those described in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For the exceptions, the irreducible components are unitary principal series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 2 General theory of Kac-Moody Lie algebras Let g be a symmetrizable Kac-Moody Lie algebra on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let A be the Cartan matrix of g and let A be an n × n matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let h be a Cartan subalgebra of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let the Chevalley generators of g be ei, fi, hi, (i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' , n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let hR be the R-span of Chevalley generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [Kac90, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2], g has a C-valued nondegenerate invariant sym- metric bilinear form (· | ·) called the standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' An antilinear automorphism ω0 of g, called compact involution, is deter- mined by ω0(ei) = −fi, ω0(fi) = −ei (i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' , n − 1), ω0(h) = −h (h ∈ hR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [Kac90, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='7], we can determine a nondegenerate Hermitian form (· | ·)0 on g with (x | y)0 = −(ω0(x) | y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Write n+ for a subalgebra of g generated by ei’s and n− for a subalgebra of g generated by fi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We can construct a 3-dimensional subalgebra of g which is spanned by three non-zero elements J+ ∈ n+, J− ∈ n−, J3 ∈ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' J+, J− and J3 satisfy [J3, J±] = ±J±, 3 [J+, J−] = −J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This subalgebra is called SO(1, 2) subalgebra of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' A representation of SO(1, 2) subalgebra is called unitary if the representation space V has a Hermitian scalar product (·, ·) and the following two conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) The actions of J+ and J− are adjoint each other, and the action of J3 is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' That is, for x, y ∈ V , we have ([J+, x], y) = (x, [J−, y]), ([J3, x], y) = (x, [J3, y]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) Hermitian scalar product (·, ·) is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When considering the adjoint action of an SO(1, 2) subalgebra of g to g, from [Tsu, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2], we can see that the adjoint action satisfying the condition (1) to be unitary and J− = −ω0(J+) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In [NO01], prin- cipal SO(1, 2) subalgebras for hyperbolic Kac-Moody Lie algebras are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Principal SO(1, 2) subalgebra satisfies that J− = −ω0(J+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When three non-zero elements X ∈ n+, Y ∈ n−, H ∈ h of g satisfy [H, X] = 2X, [H, Y ] = −2Y, [X, Y ] = H, these three elements are called sl2-triple of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' A g-subalgebra that these elements span is called sl2 subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The SO(1, 2) subalgebras and the sl2 subalgebras can be converted by J+ = 1 √ 2X, J− = − 1 √ 2Y, J3 = 1 2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The condition J− = −ω0(J+) in SO(1, 2) subalgebra is converted to Y = ω0(X) in sl2 subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In the following paper, we consider sl2 subalgebra that satisfies Y = ω0(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 4 3 sl2-triples of rank 2 hyperbolic symmetric Lie algebra that is compatible to compact involu- tion Let a be an integer that satisfies a ≥ 3, and let g be a hyperbolic Kac-Moody Lie algebra on C such that the Cartan matrix of g is � 2 −a −a 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let α0, α1 be the simple roots of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let {Fn} be the sequence of numbers determined by F0 = 0, F1 = 1, Fk+2 = aFk+1 − Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1 ([KM95, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The real positive roots of g are of the form α = Fk+1α0 + Fkα1 or β = Fkα0 + Fk+1α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We distinguish these roots as type α and type β, and we also distinguish root vectors belonging to each root as type α and type β (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' [Tsu, §4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let X be an element of the space which real positive root vectors span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then X can be written as X = � k ckEk, (k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=', nX − 1}, ck ∈ C, ck ̸= 0, Ek ∈ gβk, Ek ̸= 0) where βk (k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' , nX − 1}) are distinct real roots and nX is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We call this nX the length of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2 ([Tsu, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let X be an element in the space which real positive root vectors span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) When the length of X is 1 or more than 3, X, Y = ω0(X), H = [X, Y ] do not form sl2-triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) Suppose the length of X is 2 and E0, E1 are real positive root vectors of different types (in the sense of α-type and β-type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then, taking the appropriate c0, c1 ∈ C, X = c0E0 + c1E1, Y = ω0(X), and H = [X, Y ] form sl2-triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In particular, c0, c1 can be chosen so that c0, c1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3 ([Tsu, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Take ⟨H, X, Y ⟩ in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let X = c0E0 +c1E1, where E0 is type α and E1 is type β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, using integers i, j ≥ 0, we can write E0 ∈ gFi+1α0+Fiα1, E1 ∈ gFjα0+Fj+1α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If and only if i = j − 1, j, j + 1, H is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 5 4 Irreducible decomposition of g as an sl2 mod- ule In this section, we consider an sl2-subalgebra s = ⟨H, X, Y ⟩ of g, which satisfies the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) H ∈ h and H is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) X is in the space which is spanned by positive root vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (3) Y = ω0(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show that g is decomposed to irreducible modules by the action of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' s-module V is called unitarizable if the following conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) (· | ·)0 on V is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) for v1, v2 ∈ V , the conditions as follows are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' ([X, v1], v2)0 = −(v1, [Y, v2])0, ([H, v1], v2)0 = −(v1, [H, v2])0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When Y = ω0(X), the condition (2) are automatically satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, (· | ·)0 is positive definite on V if and only if V is unitarizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, we put U = {x ∈ g | ∀y ∈ s (x | y)0 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' U is closed under the action of s, and g = s ⊕ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (· | ·)0 is positive definite on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [Kac90, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='7], (· | ·)0 is positive definite on n+ ⊕ n−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The sign of (· | ·)0 on h is (n − 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since s itself is not unitarizable, when we write h = s ⊕ h′, (· | ·)0 is not positive definite on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, (· | ·)0 is positive definite on h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since U = h′ ⊕ n+ ⊕ n−, (· | ·)0 is positive definite on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Consider a subspace V of U that is closed under the action of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let V ⊥ be the subspace of U orthogonal to V with respect to the Hermitian form (· | ·)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then U = V ⊕ V ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider the eigenspace decomposition of U by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let Uλ be the eigenspace for λ and write U = � λ∈C Uλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since H is a Hermitian operator on (· | ·)0, Uλ and Uµ are orthogonal with respect to this inner product if λ ̸= µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since H is dominant, Uλ is finite- dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For each λ, V also inherits the eigenspace decomposition of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 6 Let Vλ be an eigenspace of V for λ, and V can be written as a direct sum of Vλ’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let V ′ λ = {v ∈ Vλ | ∀x ∈ Vλ (v | x)0 = 0}, and V ′ = � λ∈C V ′ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Vλ is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, (· | ·)0 is positive definite on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Thus we have Uλ = Vλ ⊕ V ′ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have U = V ⊕ V ′ and V ′ = V ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In the following, we show that U can be decomposed into irreducible modules by the action of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Any non-zero sl2-submodule V of U includes an irreducible sub- module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Take the eigenspace decomposition of U by the action of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' V is also decomposed into eigenspaces with this decomposition, and each eigenspace of V is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We regard H as a linear transform on V and take some eigenvalue λ of H on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let U(sl2) be an universal enveloping algebra of sl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Considering the Casimir element C of U(sl2), it preserves Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since Vλ is finite- dimensional, there exists an eigenvector of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let v denote this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Consider the sl2-submodule generated by v, which includes an irreducible submodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' U can be decomposed into direct sum of irreducible s-modules, and all of these modules are unitarizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider a set of irreducible submodules of U such that these sub- modules are orthogonal to each other with respect to (· | ·)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let T be the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We order the elements of T by inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then T is non-empty and in- ductively ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, from Zorn’s lemma, T has a maximal element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Take a maximal element of T and denote it by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Consider the direct sum of all submodules belonging to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let M denote this sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Suppose U ̸= M, we derive the contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since M is a subspace of U which is closed by the action of H, from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, we have U = M ⊕ M ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since M ⊥ is non-zero sl2 submodule of U, from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3, M ⊥ includes an irreducible submodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let W denote this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we have M ∪ {W} ∈ T , that is contradict the maximality of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have U = M, and U can be decomposed into direct sum of irreducible submodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Combining this with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, we can also see the unitarizability of the modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' g can be decomposed into direct sum of irreducible s-modules, which consists s itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' All of these modules except for s are unitarizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 7 5 sl2 modules in g In the following, we consider what kind of modules appear when g is decom- posed into irreducible s-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In particular, we consider how many unitary principal or complementary series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For a lie algebra a, Let U(a) be the universal enveloping algebra of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let V be an irreducible s-module which is an irreducible component of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The Casimir element C of U(s) acts on V by constant multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let µ be this constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [HT92, Chapter II, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='11], for an eigenvalue λ0 ∈ C of H on V , some interval I ⊂ Z exists, and V can be expressed as a direct sum of 1-dimensional eigenspaces such that the eigenvalues of H are λk = λ0 + 2k (k ∈ I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [HT92, Chapter II, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='13], for an eigenvalue λ of H on V , we define s1(k) for an integer k as s1(k) = 8µ − (λ + 2k − 1)2 + 1 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (A) We take an element vk of the eigenspace of V with respect to an eigenvalue λ + 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then we have X(Y vk) = s1(k)vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If k ∈ Z such that s1(k) = 0 does not exist, then V is an irreducible module that is neither highest weight module nor lowest weight module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If there exists a k ∈ Z such that s1(k) = 0, V is a highest weight module or a lowest weight module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let W be the Weyl group of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, we may write H, X, Y in s as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' X = c0w0(ep) + c1w1(eq) (c0, c1 ∈ R, w0, w1 ∈ W, (p, q) ∈ {(0, 1), (0, 0), (1, 1)}), Y = −c0w0(fp) − c1w1(fq), H = −c0w0(hp) − c1w1(hq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let ks, ls, ms, ns be real numbers such that c0w0(ep) ∈ gksα0+lsα1, c1w1(eq) ∈ gmsα0+nsα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3, we can write ks = Fi+1, ls = Fi, ms = Fj, ns = Fj+1 with integers i, j ≥ 0, and furthermore, i ∈ {j − 1, j, j + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When we take the root vector E ∈ gsα0+tα1 with s, t ∈ Z, we want to find out which of the three types of modules E generates under the action of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We define L in the xy-plane as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' L is a region satisfying x ≥ 0, y ≥ 0, (x, y) ̸= (0, 0), x2 − axy + y2 ≤ 1 and the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' x < ks = Fi+1, (when i = j − 1) x + y < ks + ls = Fi + Fi+1, (when i = j) y < ls = Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (when i = j + 1) If we take the root sα0 + tα1 with s, t ∈ Z, then from [KM95, Cor 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3], the point in the xy-plane given by (s, t) is in the interior or on the boundary of the hyperbola x2 − axy + y2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let hC be this hyperbola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let λ ∈ R as the value for which HE = λE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We have λ = (sα0 + tα1)(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' λ ∈ (0, 2) if and only if (s, t) ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In the following, we regard a root sα0 + tα1 as a point (s, t) in the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 8 Figure 1: Imaginary roots and real roots in L, a = 3, X = c0r0r1(e0) + c1r1r0(e1) 0 2 4 6 8 0 2 4 6 8 x y imaginary roots real roots Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider the hyperbola hC on the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The hC was represented by x2 − axy + y2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let lb be a line represented by the function y = −x+ b with some real number b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' There are two intersections of hC and lb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let p1 and p2 be these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let db be a distance between p1 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' db is strictly monotonically increasing with respect to b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The same result holds when lb is a line represented by y = b or x = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, we consider the case where lb is represented by y = −x + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Cal- culating the y-coordinates of p1, p2 gives y = (a + 2)b ± � (a + 2)(a − 2)b2 + 4(a + 2) 2(a + 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have db = √ 2 · � (a + 2)(a − 2)b2 + 4(a + 2) a + 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This db is strictly monotonically increasing with respect to b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Next, we consider the case where lb is represented by y = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Calculating the x-coordinates of p1, p2 gives x = ab ± � (a2 − 4)b2 + 4 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have db = � (a2 − 4)b2 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This db is strictly monotonically increasing with respect to b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The same argument is presented when lb is a line represented by x = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 9 Let R be the interior of hC and hC itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For s, t ∈ Z, (s, t) is a root if and only if (s, t) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (x, y) ∈ L ∪ −L, then neither (x + ks − ms, y + ls − ns) nor (x − ks + ms, y − ls + ns) are roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First we assume (x, y) ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The points (ks, ls) and (ms, ns) are on the hyperbola hC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let l1 be the line connecting these two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using some real number b > 0, l1 is represented by y = −x+ b when i = j, y = b when i = j − 1, and x = b when i = j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let l2 be a line parallel to l1 and passing through (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using some real number 0 < b′ < b, l2 is represented by y = −x + b′ when i = j, y = b′ when i = j − 1, and x = b′ when i = j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let p11, p12 be intersections of hC and l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let d1 be the distance between p11 and p12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let p21, p22 be intersections of hC and l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let d2 be the distance between p21 and p22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, we have d1 > d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The distance between (ks, ls) and (ms, ns) is d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The distance between (x, y) and (x+ks −ms, y+ls −ns) is also d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' These two points are on l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The length of the part of l2 that is inside the hyperbola is d2 < d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the fact that (x, y) is inside hC, (x + ks − ms, y + ls − ns) is outside the hyperbola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, (x + ks − ms, y + ls − ns) is not in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The same argument for (x − ks + ms, y − ls + ns) shows that it is not in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From symmetry, the case when (x, y) ∈ −L is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For a point (s, t) ∈ L corresponding to the root, we consider the root vector E ∈ gsα0+tα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then [X, [Y, E]] ∈ gsα0+tα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We have X = c0w0(ep) + c1w1(eq), Y = −c0w0(fp) − c1w1(fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Also we have c0w0(ep) ∈ gksα0+lsα1, c1w1(eq) ∈ gmsα0+nsα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then we have [X, [Y, E]] ∈ gsα0+tα1 + g(s−ks+ms)α0+(t−ls+ns)α1 + g(s−ms+ks)α0+(t−ns+ls)α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since (s, t) is a root, from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, (s − ks + ms, t − ls + ns) and (s − ms + ks, t − ns + ls) are not roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have g(s−ks+ms)α0+(t−ls+ns)α1 + g(s−ms+ks)α0+(t−ns+ls)α1 = 0, and [X, [Y, E]] ∈ gsα0+tα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider the Casimir element C of U(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We can write C = 1 8H2 − 1 4H + 1 2XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' C acts on a root space as endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The action is diago- nalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3, C acts on the root spaces as endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since g is completely reducible as an s-modules, the action on the root space is diago- nalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For a point (s, t) ∈ L corresponding to the root, we can take the root vector E ∈ gsα0+tα1 such that E is an eigenvector of the Casimir element C, and E generates an irreducible s-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4, we have the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 10 From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5, if we decompose g by the action of s, the decomposition is compatible with the root space decomposition in the root in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider how many unitary principal or complementary series represen- tations appear in the decomposition of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since the set of eigenvalues of unitary principal or complementary series representations is {λ + 2k | k ∈ Z} for some λ, such a module must contain an eigenspace such that its eigenvalue lie on [0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we consider the root vector of H such that the eigenvalue λ of H satisfies λ ∈ [0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If λ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=', s = t = 0, Since the dimension of h is 2, there are two irreducible components of V which have 0-eigenspace (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' [Tsu, §7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since one is sl2 itself, we consider the other module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The casimir element C acts on this module by a constant multiple (let µ times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If k satisfies s1(k) = 0, we get 8µ+ 1 = (2k − 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since µ < −1 from [Tsu, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3], the left hand side is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, there is no integral solution to s1(k) = 0, and this is an irreducible module that is neither highest weight module nor lowest weight module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In particular, this module is an unitary principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In the following, we consider the case of λ ∈ (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In this case, (s, t) is a root in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We compute [X, [Y, E]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since Y = −c0w0(fp) − c1w1(fq), we have [Y, E] = [−c0w0(fp), E] + [−c1w1(fq), E].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We have also [−c0w0(fp), E] ∈ g(s−ks)α0+(t−ls)α1, [−c1w1(fq), E] ∈ g(s−ms)α0+(t−ns)α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If [−c0w0(fp), E] and [−c1w1(fq), E] are not 0, then the eigenvalue of H for them must be in the (−2, 0) interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we consider root vectors which the eigenvalue of H are in the (−2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since R = −R, the roots with respect to these root vectors are −L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, if we take two points such that the difference is (ks − ms, ls − ns) and one of which is a root in −L, then the other is not a root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Now we have ((s − ms) − (s − ks), (t − ns) − (t − ls)) = (ks − ms, ls − ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we know that at least one of [−c0w0(fp), E], [−c1w1(fq), E] is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When both of these are 0, we have [Y, E] = 0 and from the fact that C = 1 8H2 − 1 4H + 1 2XY , we can write 8µ = λ2 − 2λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When [−c0w0(fp), E] ̸= 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=', (s − ks, t − ls) ∈ R, we have [X, [Y, E]] = [c0w0(ep), [−c0w0(fp), E]] = [E, [−c0w0(fp), c0w0(ep)]] + [−c0w0(fp), [c0w0(ep), E]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We define ps ∈ C by [−c0w0(fp), [c0w0(ep), E]] = psE, then we have [X, [Y, E]] = [E, c2 0w0(hp)] + psE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When ps = 0, we have [X, [Y, E]] = −[c2 0w0(hp), E].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 11 Therefore in this case, if we let −[c2 0w0(hp), E] = k0E, then we have 8µ = λ2 − 2λ + 4k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When [c0w0(ep), E] = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=', (s + ks, t + ls) ̸∈ R, we have ps = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' To summarize the above, we take an irreducible decomposition of g by s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' let sα0 +tα1 be a root in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let E ∈ gsα0+tα1 such that E generates an irreducible component of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let C be the Casimir element of U(s), and Let µ be a complex number such that CE = µE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let k0 and ps be complex numbers satisfying [−c2 0w0(hp), E] = k0E, [−c0w0(fp), [c0w0(ep), E]] = psE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s − ms, t − ns) ̸∈ R, we have 8µ = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 λ2 − 2λ ((s − ks, t − ls) ̸∈ R) , λ2 − 2λ + 4k0 ((s − ks, t − ls) ∈ R and (s + ks, t + ls) ̸∈ R) , λ2 − 2λ + 4k0 + ps ((s − ks, t − ls) ∈ R and (s + ks, t + ls) ∈ R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s − ks, t − ls) ̸∈ R and not necessarily (s − ms, t − ns) ̸∈ R, we have 8µ = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 λ2 − 2λ ((s − ms, t − ns) ̸∈ R) , λ2 − 2λ + 4k0 ((s − ms, t − ns) ∈ R and (s + ms, t + ns) ̸∈ R) , λ2 − 2λ + 4k0 + ps ((s − ms, t − ns) ∈ R and (s + ms, t + ns) ∈ R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Solving s1(k) = 8µ − (λ + 2k − 1)2 + 1 4 = 0 for k on R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we obtain that k = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 1 − λ ((s − ks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t − ls) ̸∈ R and (s − ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t − ns) ̸∈ R) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 1 − λ ± � (λ − 1)2 + 4k0 2 \uf8eb \uf8ec \uf8ed (s − ks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t − ls) ∈ R and (s + ks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t + ls) ̸∈ R or (s − ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t − ns) ∈ R and (s + ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t + ns) ̸∈ R \uf8f6 \uf8f7 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 1 − λ ± � (λ − 1)2 + 4k0 + ps 2 \uf8eb \uf8ec \uf8ed (s − ks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t − ls) ∈ R and (s + ks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t + ls) ∈ R or (s − ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t − ns) ∈ R and (s + ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' t + ns) ∈ R \uf8f6 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When (s − ks, t − ls) ̸∈ R and (s − ms, t − ns) ̸∈ R, since (s, t) ∈ L, we have 1 − λ ∈ (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we know that the only integral solution of s1(k) = 0 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In this case E belongs to an irreducible lowest weight module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 6 Classification by roots Based on the previous section, we classify the root (s, t) in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We define the types of roots as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 12 (1) We say that (s, t) is of type A when (s−ks, t−ls) ̸∈ R and (s−ms, t−ns) ̸∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) We say that (s, t) is of type B when \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 (s − ks, t − ls) ∈ R and (s + ks, t + ls) ̸∈ R or (s − ms, t − ns) ∈ R and (s + ms, t + ns) ̸∈ R \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (3) We say that (s, t) is of type C when \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 (s − ks, t − ls) ∈ R and (s + ks, t + ls) ∈ R or (s − ms, t − ns) ∈ R and (s + ms, t + ns) ∈ R \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' All roots belong to one of the above types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We put f(x, y) = x2 − axy + y2 for x, y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [KM95, Cor 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3], for s, t ∈ Z, (s, t) ̸= (0, 0), (s, t) is a real root if and only if f(s, t) = 1, and (s, t) is an imaginary root if and only if f(s, t) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For x, y, x′, y′ ∈ R, if there exists w ∈ W such that (x′, y′) = w(x, y), then f(x′, y′) = f(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' It is sufficient to check the case w = r0 and the case w = r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the symmetry, it is sufficient to check the case w = r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In this case, from the fact that x′ = ay − x and y′ = y, we have f(x′, y′) = f(ay − x, y) = (ay − x)2 − ay(ay − x) + y2 = x2 − axy + y2 = f(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, we know the following results on real roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s, t) is a real root in L and s > t, then f(s − ks, t − ls) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Also, If (s, t) is a real root in L and s < t, then f(s − ms, t − ns) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From symmetry, it is sufficient to show f(s − ks, t − ls) ≤ 0 when s > t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We can write s = Fc+1, t = Fc with c ≥ 0 being an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since ks = Fi+1 and ls = Fi, we have c < i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let dic = i − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, by acting r0 and r1 on (s − ks, t − ls), we know that f(s − ks, t − ls) = f(Fc+1 − Fi+1, Fc − Fi) = f(r0(Fc+1 − Fi+1, Fc − Fi)) = f(Fc−1 − Fi−1, Fc − Fi) = f(r1(Fc−1 − Fi−1, Fc − Fi)) = f(Fc−1 − Fi−1, Fc−2 − Fi−2) = · · · 13 = � f(F1 − Fdic+1, F0 − Fdic) (when c is even) f(F0 − Fdic, F1 − Fdic+1) (when c is odd) = f(F1 − Fdic+1, F0 − Fdic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since F1 = 1, F0 = 0, we have f(s − ks, t − ls) = f(1 − Fdic+1, −Fdic) = 2 − aFdic + 2Fdic−1 < 2 − 2(Fdic − Fdic−1) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s, t) is a real root in L, then (s, t) is of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First we show that (s, t) is not of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the fact that (s, t) is a real root and from symmetry, we can write s = Fc+1, t = Fc with c ≥ 0 being an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From ks = Fi+1, ls = Fi, we have c < i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, f(s − ks, t − ls) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, (s − ks, t − ls) ∈ R and so we know that (s, t) is not of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Next, we show that (s, t) is of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' To show this, we need to show that (s + ks, t + ls) ̸∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show f(s + ks, t + ls) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let dic = i − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, by acting r0 and r1 on (s + ks, t + ls), we have f(s + ks, t + ls) = f(Fc+1 + Fi+1, Fc + Fi) = f(r0(Fc+1 + Fi+1, Fc + Fi)) = f(Fc−1 + Fi−1, Fc + Fi) = f(r1(Fc−1 + Fi−1, Fc + Fi)) = f(Fc−1 + Fi−1, Fc−2 + Fi−2) = · · · = � f(F1 + Fdic+1, F0 + Fdic) (when c is even) f(F0 + Fdic, F1 + Fdic+1) (when c is odd) = f(F1 + Fdic+1, F0 + Fdic) = f(1 + Fdic+1, Fdic) = 2 + aFdic − 2Fdic−1 > 2 + 2(Fdic − Fdic−1) > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This shows that (s, t) is of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We classify also for imaginary roots in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s, t), (s′, t′) are imaginary roots, then (s + s′, t + t′) is also imaginary root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since f(s, t) ≤ 0, for any r ∈ R, we have f(rs, rt) = r2f(s, t) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' It shows that the line connecting the origin and (s, t) is inside the asymptotes of the hyperbola x2 − axy + y2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Similarly, the line connecting the origin and (s′, t′) is also inside the asymptotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since (s+ s′, t+ t′) is the midpoint of (2s, 2t) and (2s′, 2t′), this point is also inside the asymptotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, (s + s′, t + t′) is an imaginary root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let (u, v) ∈ L (u > v) be a real root such that (uα0+vα1)(H) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Put (s, t) = (ks−u, ls−v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then (s, t) is a type C imaginary root in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Similarly, let (u′, v′) ∈ L (u′ < v′) be a real root such that (u′α0 + v′α1)(H) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Put (s′, t′) = (ms − u′, ns − v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then (s′, t′) ∈ L and (s′, t′) is the imaginary root of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The other imaginary roots in L are of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, f(−s, −t) = f(s, t) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' It shows that (s, t) is a imaginary root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We also see that the eigenvalue of H for (s, t) is in the range (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, (s, t) ∈ L is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show that (s, t) is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' To show this, we show that f(s + ks, t + ls) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using c ∈ Z, we can write (u, v) = (Fc+1, Fc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Together this with s + ks = 2ks − u, t + ls = 2ls − v, we have f(s + ks, t + ls) = f(2Fi+1 − Fc+1, 2Fi − Fc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let dic = i−c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1, acting r0, r1 on (s+ks, t+ls), i−c = λ ≥ 1, we have f(2Fi+1 − Fc+1, 2Fi − Fc) = f(r0(2Fi+1 − Fc+1, 2Fi − Fc)) = f(2Fi−1 − Fc−1, 2Fi − Fc) = f(r1(2Fi−1 − Fc−1, 2Fi − Fc)) = f(2Fi−1 − Fc−1, 2Fi−2 − Fc−2) = · · · = � f(2Fdic+1 − F1, 2Fdic − F0) (when c is even) f(2Fdic − F0, 2Fdic+1 − F1) (when c is odd) = f(2Fdic+1 − F1, 2Fdic − F0) = f(2Fdic+1 − 1, 2Fdic) = −2aFdic + 4Fdic−1 + 5 < −6Fdic + 4Fdic−1 + 5 = (−4Fdic + 4Fdic−1) + (−2Fdic + 5) < −4 − 2Fdic + 5 ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This shows that f(s+ ks, t+ ls) ≤ −1 and that (s, t) is type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From symmetry, we also know that (s′, t′) is in L and is the imaginary root of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Finally, we show the other imaginary roots in L are of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let (s′′, t′′) ∈ L be such an imaginary root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show (s′′−ms, t′′−ns) ̸∈ R and (s′′−ks, t′′−ls) ̸∈ 15 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s′′ − ms, t′′ − ns) ∈ R or (s′′ − ks, t′′ − ls) ∈ R, (s′′ − ms, t′′ − ns) ∈ −L or (s′′−ks, t′′−ls) ∈ −L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since (s′′−ms, t′′−ns)−(s′′−ks, t′′−ls) = (ks −ms, ls− ns), from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2, we know (s′′ − ms, t′′ − ns) ̸∈ R or (s′′ − ks, t′′ − ls) ̸∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From symmetry, it is sufficient to consider when (s′′ − ms, t′′ − ns) ̸∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Under this assumption, (s′′ − ks, t′′ − ls) is an imaginary root or not a root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s′′ − ks, t′′ − ls) is imaginary root, then (ks − s′′, ls − t′′) is also imaginary root from the symmetry of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider that (ks, ls) = (s′′, t′′) + (ks − s′′, ls − t′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The left hand side is real root and the right hand side is the sum of imaginary roots, which contradicts Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, (s′′ − ks, t′′ − ls) is not a root and (s′′, t′′) is of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The contents of this section can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) A real roots in L is of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) We consider an imaginary root that can be written as (ks − s, ls − t) or (ms − s, ns − t) where (s, t) is a real root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Such an imaginary root is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (3) The other imaginary roots are of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We now summarize the irreducible s-modules through type A and type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For s-modules through type A, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' An irreducible s-module containing a root vector about a root of type A in L is a lowest weight module which the root vector is the lowest weight element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since (s − ks, t − ls) ̸∈ R and (s − ms, t − ns) ̸∈ R for the root (s, t) of type A, we know that acting Y on the type A root vector will result in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This shows the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let M be an irreducible s-module containing a root vector (say v) with respect to type C root in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then one of the following conditions (1), (2), or (3) is hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) M is a lowest weight module such that v is a lowest element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) M is a highest weight module such that v is a highest element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (3) M contains a real root vector with respect to a real root in −L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The type C root (s, t) can be written with some real root (sr, tr) that (ks − sr, ls − tr) or (ms − sr, ns − tr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, the root vector E of type C becomes either zero or a real root vector when Y act on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If E becomes 0 under the action of Y , then E generates an irreducible lowest weight module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If E becomes a real root vector, then the real root for this vector is in −L, and this lemma is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 16 We also give the type A, B, C distinction to the root of −L by defining Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then, if there is a unitary principal or complementary series representation that passes through a root vector of type C in L, −L, it will also pass through the root vector of type B in −L, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, We have only to classify the modules that contains a type B root space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Figure 2: a = 3, X = c0r0(e1) + c1r1(e0) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 3 0 1 2 3 x y type A type B type C roots of X Figure 3: a = 3, X = c0r0(e1) + c1r1r0(e1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 3 0 2 4 6 8 x y type A type B type C roots of X 17 7 Irreducible modules which contains a root space with respect to a type B root We consider an irreducible decomposition of g by s, and we consider an irre- ducible component M containing a type B root space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The multiplicity of a real root space is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We can take 0 < λ < 2 such that {λ + 2k′ | k′ ∈ Z} is the set of the eigenvalues of H in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider the H eigenspace of M such that the eigenvalue is λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We assume this eigenspace is gsα0+tα1 such that (s, t) ∈ L, and (s, t) is real root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider k such that s1(k) = 0 in (A) in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show that it is not an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let e0, e1, f0, f1, h0 and h1 be Chevalley generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using some c0, c1 ∈ R, w0, w1 ∈ W, and (p, q) ∈ {(0, 1), (0, 0), (1, 1)}, let X = c0w0(ep) + c1w1(eq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Suppose s > t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We take the root vector E with respect to the root sα0+tα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We define λ by HE = λE, and define k0 by [−c2 0w0(hp), E] = k0E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Thus s1(k) = 0 implies k = 1 − λ ± � (λ − 1)2 + 4k0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We put k+ = 1 − λ + � (λ − 1)2 + 4k0 2 , k− = 1 − λ − � (λ − 1)2 + 4k0 2 and we show that k± ̸∈ R or 0 < k± < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When (λ − 1)2 + 4k0 < 0 or (λ − 1)2 + 4k0 ̸∈ R, k± are imaginary numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore we can assume (λ − 1)2 + 4k0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From 0 < λ < 1, it is clear that k+ > 0 and k− < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' To show k+ < 1, we need to show 1 − λ + � (λ − 1)2 + 4k0 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we can easily show that it is reduced to k0 < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Also, to show that k− > 0, we need to show 1 − λ − � (λ − 1)2 + 4k0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we can easily show that it is reduced to k0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In summary, we have only to show that k0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, consider the case (s, t) = (1, 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=', E ∈ gα0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In this case, from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3, we have c0w0(ep) ∈ gFi+1α0+Fiα1 and i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since k0E = [−c2 0r0r1r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' r1−p(hp), E] = [−c2 0(Fi+1h0 + Fih1), E] = −c2 0(2Fi+1 − aFi)E = −c2 0(Fi+1 + Fi−1)E, 18 we have k0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When (s, t) = (0, 1), we can show that k0 < 0 by replacing i with j, p with q and making the same argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If (s, t) is general and s > t, we can write (s, t) = (Fi′+1, Fi′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let p′ be 0 or 1, we can write E = r0r1r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' r1−p′(ep′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From this, we have [−c2 0w0(hp), E] = −c2 0[r0r1r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' r1−p(hp), r0r1r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' r1−p′(ep′)] = −c2 0r0r1r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' r1−p′[rp′r1−p′rp′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' r1−p(hp), ep′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider k0 and c0 when i is replaced by i−i′, and rewrite them as k′ 0 and c′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Considering (s, t) = (1, 0) or (0, 1) cases, we have [rp′r1−p′rp′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' r1−p(hp), ep′] = − k′ 0 c′ 0 ep′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' That is, k0 = c2 0 c′2 0 k′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since k′ 0 < 0, we have k0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When s < t, we can show that k0 < 0 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the above, it can be shown that k0 < 0 in any case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=', k is not an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From this, we can see the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider an irreducible decomposition of g by the action of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (1) Let M is an irreducible component of decomposition of g, which contain a root space for a type B root sα0 + tα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then, M is an unitary principal or complementary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (2) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' [Tsu, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3]) There is an unitary principal series represen- tation containing an 1-dimensional space in h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (3) g is decomposed into a direct sum of s-submodules described in (1) and (2) above, s itself, irreducible lowest weight modules, and irreducible highest weight modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [KM95, §3], the multiplicity of each root of g is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using this, we can find how many modules appear such that the following condition is satisfied: the modules are highest or lowest modules, and eigenvalues of H for root vectors with the highest or the lowest roots are certain value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, the modules which contain root spaces in L and −L can be seen from previous contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Among the positive root spaces not in L, those with the smallest eigenvalue in H are considered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let λH be their eigenvalue and dH be their dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Suppose pH modules which contain space with eigenvalue λH that also contain the root spaces already obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then there are dH−pH lowest weight modules with the root with eigenvalue λH as the lowest root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The multiplicities of modules can be obtained inductively by replacing λH with the next smallest eigenvalue of H and performing the same calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Negative root spaces can be classified by the same calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 8 Unitary principal series representation and com- plementary series representation In this section, we consider a module (say M) that is neither highest weight module nor lowest weight module containing a root vector about the root of type 19 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We compute whether the module is a unitary principal series representation or a complementary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, we state the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If 8µ ≤ −1, then M is a unitary principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If 8µ > −1, then M is a complementary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [HT92, §II 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2], M is isomorphic to U(ν+, ν−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' U(ν+, ν−) is a sl2-module with H eigenvectors {vn | n ∈ Z} as a basis of linear space, such that Hvn = (ν+ − ν− + 2j)vn (n ∈ Z), e+vn = (ν+ + n)vn+1, e−vn = (ν− − n)vn−1, 8µ = (ν+ + ν− − 1)2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [HT92, §III Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3], if ν+ + ν− = 1, U(ν+, ν−) is a unitary principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When 8µ ≤ −1, from λ = ν+ − ν− ∈ R, 8µ = (ν+ + ν− − 1)2 − 1 < −1, using b ∈ R we can write ν+ − ν− = λ, ν+ + ν− = 1 + bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (i = √ −1) In this case, we have ν+ + ν− = λ + 1 2 + b 2i + −λ + 1 2 − b 2i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, M is a unitary principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Consider the case when 8µ > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From [HT92, §III Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3], if ν± ∈ R and if ν− − 1 and −ν+ are both contained in the interval (l − 1, l) with some l ∈ Z, then U(ν+, ν−) is a complementary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From 8µ > −1, we have ν+ + ν− = 1 ± � 8µ + 1, ν+ − ν− = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have −ν+, ν− − 1 = −λ − 1 ± √8µ + 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show that they are in (−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 20 We show first that 0 < λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let n, m be integers such that n > m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We can write λ = 2(Fm+1 + Fm) Fn+1 + Fn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' It is clear that λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From a ≥ 3, for integer z ≥ 0, we have Fz+2 = aFz+1 − Fz > (a − 1)Fz+1 ≥ 2Fz+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Hence we have Fm+1 + Fm Fn+1 + Fn < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show that −1 < −λ − 1 + √8µ + 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From λ < 1, we have −1 < −λ−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, this inequality is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Next we show −λ − 1 + √8µ + 1 2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We have 8µ = λ(λ − 2) + 4k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From 0 < λ < 1, we have λ(λ − 2) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Also, since k0 < 0, we have 8µ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we have √8µ + 1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using 0 < λ again, we know that −λ − 1 + √8µ + 1 2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For −λ − 1 − √8µ + 1 2 < 0, this is clear from λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Finally, we show −1 < −λ − 1 − √8µ + 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From k0 < 0 and 8µ = λ2 − 2λ + 4k0, we have λ2 − 2λ > 8µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From this and λ < 1 we get 1 − λ > √8µ + 1, which can be transformed to −1 < −λ − 1 − √8µ + 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the above, −λ−1±√8µ+1 2 are both in (−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, M is a comple- mentary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 21 Hereafter, we want to determine when M is complementary series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, we consider the case where i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we have 8µ = λ2 − 2λ + 4k0, λ = 2(Fn+1 + Fn) Fi+1 + Fi , k0 = −2(2Fi+1 − aFi) a(F 2 i + F 2 i+1) − 4FiFi+1 − 2, (*) where n is an integer such that i > n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' That is, 8µ is determined by i, n, and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We show that 8µ is greater than −1 with finite exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We assume i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If we consider 8µ to be a function of n by (*), 8µ is monotonically decreasing with respect to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' k0 is independent on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' λ is monotonically increasing with respect to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since 8µ = λ(λ − 2) + 4k0 and 0 < λ < 1, we know that 8µ is monotonically decreasing with respect to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' To show that 8µ is greater than −1 with finite exceptions, we need to examine when n is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We assume i = j, n = i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If we consider 8µ to be a function of i by (*), 8µ is monotonically increasing with respect to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First we write {Fi} explicitly as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The real solutions of x2 − ax + 1 = 0 are x = a± √ a2−4 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' As α = a− √ a2−4 2 , β = a+ √ a2−4 2 , we can write Fi = βi − αi β − α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From n = i − 1, we have λ = 2(Fi + Fi−1) Fi+1 + Fi , k0 = −2(2Fi+1 − aFi) a(F 2 i + F 2 i+1) − 4FiFi+1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let t be a real variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We define the functions Λ and K0 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Λ(t) = 2(βt − αt + βt−1 − αt−1) βt+1 − αt+1 + βt − αt , K0(t) = −2(β − α)(2(βt+1 − αt+1) − a(βt − αt)) a((βt − αt)2 + (βt+1 − αt+1)2) − 4(βt − αt)(βt+1 − αt+1) − 2, We have λ = Λ(i) and k0 = K0(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using these function, we can calculate as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='dtΛ = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4 log β(a + 2)(β − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(βt+1 − αt+1 + βt − αt)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='dt(Λ2 − 2Λ) = 8 log β(a + 2)(β − α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(1 − a)βt − (1 − a)αt + 3βt−1 − 3αt−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(βt+1 − αt+1 + βt − αt)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='dtK0 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2(β − α)(a2 − 4) log β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(a2 − 4)2(β2t+1 + α2t+1 − 2)2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2β3t+2 + 2α3t+2 − aβ3t+1 − aα3t+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='+(3a + 4)βt+1 + (3a + 4)αt+1 − (2a + 6)βt − (2a + 6)αt� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='dt(Λ2 − 2Λ + 4K0) = 8 log β(β − α) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(a + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(1 − a)βt − (1 − a)αt + 3βt−1 − 3αt−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(βt+1 − αt+1 + βt − αt)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='(a2 − 4)(β2t+1 + α2t+1 − 2)2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='2β3t+2 + 2α3t+2 − aβ3t+1 − aα3t+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='+(3a + 4)βt+1 + (3a + 4)αt+1 − (2a + 6)βt − (2a + 6)αt� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='Clearing the denominator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we can calculate as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (βt+1 − αt+1 + βt − αt)3(a − 2)(β2t+1 + α2t+1 − 2)2 8(a + 2) log β(β − α) d dt(Λ2 − 2Λ + 4K0) = (β6t+3 − α6t+3) − (β6t+2 − α6t+2) + (a − 2)(1 − a)(β5t+2 − α5t+2) + 3(a − 2)(β5t+1 − α5t+1) + 2(β4t+3 − α4t+3) + (a − 4)(β4t+2 − α4t+2) + (11 − 2a)(a − 2)(β3t+2 − α3t+2) − (8a + 1)(a − 2)(β3t+1 − α3t+1) + (−11a + 5)(β2t+1 − α2t+1) + 17(β2t − α2t) + (8a − 14)(a − 2)(βt+1 − αt+1) + (12a − 6)(a − 2)(βt − αt) + 14(β − α) The coefficient on the left hand side is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Using the fact that βt − αt is monotonically increasing, we can calculate that the right hand side is also positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This shows that 8µ = (Λ2 − 2Λ + 4K0)(i) is monotonically increasing with respect to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3, we consider the case when i = 1, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We assume i = j = 1 and n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If we consider 8µ to be a function of a by (*), 8µ is monotonically increasing with respect to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Under this assumption, we have 8µ = −4a2 a3 − 3a − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Differentiating this as a function of the real variable a, from a ≥ 3, we know that 8µ is monotonically increasing with respect to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 23 Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When i = j, we consider s-modules of g that are neither a highest weight module nor a lowest weight module containing a root vector about the root of type B obtained by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The modules are complementary series representations, except for the following five types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For these exceptions, the modules are unitary principal series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (a, i, n) = (4, 1, 0), (3, 1, 0), (3, 2, 1), (3, 3, 2), (3, 4, 3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We use Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='6, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='3, and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, when a = 5, i = 1, n = 0, we have 8µ = − 25 27 > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, when a ≥ 5, for any i, n, the module for a, i, n is a complementary series representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Next, when a = 4, i = 1, n = 0, we have 8µ = − 32 25 < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Hence the module for this is a unitary principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' On the other hand, when a = 4, i = 2, n = 1, we have 8µ > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, when a = 4, the module for a, i, n is a complementary series representation except when i = 1, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Finally, when a = 3, 8µ < −1 when i = 1, 2, 3, 4 and n = i − 1, and in these four cases the module is a unitary principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When n = i − 2 or i = 5, we have 8µ > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we know that the module is a complementary series representation in other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the above, with five exceptions, neither a highest weight module nor a lowest weight module containing a root vector about the root of type B is a complementary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Next, we consider the case i = j − 1 or i = j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From symmetry, it is sufficient to consider the case i = j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In this case, λ can be written λ = 2Fn Fi+1 with n as an integer such that i ≥ n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' On the other hand, for k0, we have k0 = −2(2Fi+1 − aFi) a(F 2 i + F 2 i+1) − 4FiFi+1 − 2 as for i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' As with i = j, 8µ is determined by i, n and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The next lemma is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We assume i = j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If we consider 8µ to be a function of n by (*), 8µ is monotonically decreasing with respect to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In the following, we consider whether 8µ is monotonically increasing with respect to i when n = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' In this case, we have λ = 2Fi Fi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When i = j, n = i − 1, we already know λ = 2(Fi + Fi−1) Fi+1 + Fi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 24 We rewrite as λ1 = 2Fi Fi+1 , λ2 = 2(Fi + Fi−1) Fi+1 + Fi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Let t be a real variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We define Λ1, Λ2 and K0 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Λ1(t) = 2(βt − αt + βt−1 − αt−1) βt+1 − αt+1 + βt − αt , Λ2(t) = 2(βt − αt) βt+1 − αt+1 , K0(t) = −2(β − α)(2(βt+1 − αt+1) − a(βt − αt)) a((βt − αt)2 + (βt+1 − αt+1)2) − 4(βt − αt)(βt+1 − αt+1) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We have λ1 = Λ1(i), λ2 = Λ2(i), k0 = K0(i), and 8µ = (Λ2 1 − 2Λ1 + 4K0)(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When i = j, 8µ = (Λ2 2 − 2Λ2 + 4K0)(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We compare d dtΛ1 and d dtΛ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Since d dt(Λ2 1 − 2Λ1 + 4K0) = 2(Λ1 − 1) d dtΛ1 + 4 d dtk0 and 0 < Λ1 − 1 < 1, the smaller the value of d dtΛ1, the larger the value of d dt(Λ2 1 − 2Λ1 + 4K0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We know that d dt(Λ2 2−2Λ2+4K0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If we show d dtΛ1 > d dtΛ2, we also know (Λ2 2−2Λ2+4K0) is monotonically increasing with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, we know 8µ is also monotonically increasing with respect to i when i = j − 1, n = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' d dtΛ1 > d dtΛ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we show that d dtΛ1 − d dtΛ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' we have d dtΛ1 − d dtΛ2 = 4 log β(a + 2)(β − α) (βt+1 − αt+1 + βt − αt)2 − 4 log β(β − α) (βt+1 − αt+1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Calculating this, we have (βt+1 − αt+1 + βt − αt)2(βt+1 − αt+1)2 4 log β(β − α) � d dtΛ1 − d dtΛ2 � =(β2t+3 + β2t+3) + (β2t+2 + α2t+2) − (β2t+1 + α2t+1) − (β2t + α2t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The coefficient on the left hand is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We can easily calculate to know that βt + αt is monotonically increasing with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From this, we know the right hand side is also positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, d dtΛ1 > d dtΛ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We assume i = j − 1, n = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If we consider 8µ to be a function of i by (*), 8µ is monotonically increasing with respect to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We assume i = 0, j = 1, and n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' If we consider 8µ to be a function of a by (*), 8µ is monotonically increasing with respect to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Under this assumption, 8µ = −4 a − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' This is monotonically increasing with respect to a ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When i = j −1, We consider s-modules containing a root vector about the root of type B that are neither highest weight modules nor lowest weight modules obtained by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The modules are complementary series representations, except for the following 23 types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For these exceptions, the modules are unitary principal series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' (a, i, n) =(a′, 0, 0) (6 ≤ a′ ≤ 18), (5, 0, 0), (5, 1, 1), (4, 0, 0), (4, 1, 1), (3, 0, 0), (3, 1, 1), (3, 1, 0), (3, 2, 2), (3, 3, 3), (3, 4, 4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We use Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='6, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='8, and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' First, when a = 18, i = 0, n = 0, 8µ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Therefore, when a ≥ 18, the modules for a, i, n are complementary series representations except when (a, i, n) = (18, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Then, when 6 ≤ a ≤ 17, i = 0, n = 0, from 8µ = − 4 3 > −1, the module for this pair is a unitary principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' On the other hand, when 6 ≤ a ≤ 17, 8µ < −1 except when (a, i, n) = (a, 0, 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=', the module about a, i, n is a complementary series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When a = 5, if (a, i, n) = (5, 0, 0), (5, 1, 1), then the modules are unitary principal series representations, and the others are complementary series repre- sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When a = 4, if (a, i, n) = (4, 0, 0), (4, 1, 1), then the modules are unitary principal series representations, and the others are complementary series repre- sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' When a = 3, if (a, i, n) = (3, 0, 0), (3, 1, 1), (3, 1, 0), (3, 2, 2), (3, 3, 3), (3, 4, 4), then the modules are unitary principal series representations, and the others are complementary series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' From the above, 23 unitary principal series representations are obtained, and the rest are all complementary series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' We consider modules obtained by (1) of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The modules are neither highest weight modules nor lowest weight modules and contain root vectors about roots of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' The modules are complementary series representations, except those enumerated by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' For the exceptions, the modules are unitary principal series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' It can be shown from Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='5 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 26 Acknowledgements I would like to express my appreciation to my supervisor, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Hisayosi Matu- moto for his thoughtful guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' References [CM93] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Collingwood, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold, 1993 [Dyn57] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' Dynkin, Semisimple subalgebras of simple Lie algebras, American Mathematical Society Translations: Series 2, 6, 1957, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf'} +page_content=' 111–245 [GOW02] M.' metadata={'source': 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