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A finiteness theorem for affine Lie algebras. (English) Zbl 0616.17009

Let \(G(A)\) be the Kac-Moody Lie algebra associated to a symmetrizable generalized Cartan matrix A, H be a Cartan subalgebra, \(\Phi\) be the root system with basis \(\Pi\), \(\Gamma =\sum_{\alpha \in \Phi}{\mathbb{Z}}\alpha\) be the root lattice, (.,.) be the bilinear form and \(\rho \in H^*\) be such that \(2(\rho,\alpha) = (\alpha,\alpha)\) for all \(\alpha\in \Pi\). For \(\lambda \in H^*\) define the following important set: \[ X_{\lambda} = \{\mu\in \lambda +\Gamma; \quad (\mu+\rho,\mu+\rho) = (\lambda+\rho,\lambda+\rho)\}. \] By using the fact that the Casimir operator \(\Omega\) acts as a scalar \((\lambda+\rho, \lambda+\rho)\) on a highest weight module M(\(\lambda)\), Kac proved that the highest weights of all irreducible subquotients of M(\(\lambda)\) belong to \(X_{\lambda}\). The action of \(\Omega\) is no longer a scalar on a module M in the category \({\mathcal O}\) of Bernstein-Gelfand-Gelfand, instead one has the decomposition \(M= \oplus_{c\in C}M_ c\) where \(M_ c=\{m\in M\); \((\Omega -cI)^ r_ m=0\) for some \(r\}\). In order to decompose \(M_ c\), an equivalence \(\sim\) was introduced on \(H^*\) in [V. V. Deodhar, O. Gabber and V. Kac, Adv. Math. 45, 92-116 (1982; Zbl 0491.17008)] and one gets \(M_ c=\oplus_{\Theta \subset X_{\lambda}}M_{\Theta}\) if \(c=(\lambda +\rho, \lambda +\rho)\), where \(M_{\Theta}\) is the submodule associated to an equivalence class \(\Theta\).
The main result of this article is to prove that in the case of an affine Kac-Moody Lie algebra, \(X_{\lambda}\) contains only finitely many equivalence classes (up to the action of the Weyl group W). The authors announce that the hyperbolic case has been settled by Deodhar and Moody.
Let us look at some technical parts of the paper. A complicated condition called (*) was introduced in Deodhar-Gabber-Kac’s paper and they proved that \(\{\lambda,\mu\}\) satisfies (*) iff the irreducible module \(L(\mu)\) occurs as a subquotient of \(M(\lambda)\); then \(\lambda \sim \mu\) iff there exists a sequence \(\lambda =\lambda_ 0,\lambda_ 1,...,\lambda_ m=\mu\) in \(H^*\) such that for every \(i=0,...,m-1\), one of the ordered pairs \(\{\lambda_ i,\lambda_{i+1}\}\) and \(\{\lambda_{i+1},\lambda_ i\}\) satisfies (*). Let \({\mathcal S}_{\lambda}\) be the set of equivalence classes contained in \(X_{\lambda}\) and \(\tilde W(\lambda) = \{w\in W\); \(w\lambda-\lambda\in \Gamma\}\), denote by \(W(\lambda)\) the subgroup of \(\tilde W(\lambda)\) generated by the reflections it contains. The main theorem asserts that if \(G(A)\) is an affine Kac-Moody Lie algebra then
(i) the number of \(\tilde W(\lambda)\)-orbits in \({\mathcal S}_{\lambda}\) is finite for all \(\lambda \in H^*;\)
(ii) if \((\lambda+\rho,\xi)=0\) then each orbit is a singleton.
(iii) if \((\lambda+\rho,\xi)\neq 0\) then each orbit is isomorphic to \(\tilde W(\lambda)/W(\lambda)\) (here \(\xi\) is the null root and the action of \(w\in \tilde W(\lambda)\) is \(w.\nu =w(\nu +\rho)-\rho)\).
Reviewer: L.Santharoubane

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B20 Simple, semisimple, reductive (super)algebras
17B35 Universal enveloping (super)algebras

Citations:

Zbl 0491.17008
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References:

[1] Bourbaki, N., Groupes et algèbres de Lie (1968), Hermann: Hermann Paris, Chaps. 4, 5, et 6 · Zbl 0186.33001
[2] Chari, V.; Ilangoven, S., On the Harish-Chandra homomorphism for infinite dimensional Lie algebras, J. Algebra, 90, 476-490 (1984) · Zbl 0545.17003
[3] Deodhar, V.; Gabber, O.; Kac, V., Structure of some categories of representations of infinite dimensional Lie algebras, Advan. in Math., 45, 92-116 (1982) · Zbl 0491.17008
[4] Garland, H.; Lepowsky, J., Lie algebra homology and the Macdonald-Kac formulas, Invent. Math., 34, 37-76 (1976) · Zbl 0358.17015
[5] Kac, V., Simple irreducible graded Lie algebras of finite growth, Math. USSR Izv., 1271-1311 (1968) · Zbl 0222.17007
[6] Kac, V., Infinite dimensional Lie algebras and Dedekind’s η-function, Functional Anal. Appl., 8, 68-70 (1974) · Zbl 0299.17005
[7] Kac, V.; Kazhdan, D., Structure of representations with highest weight of infinite dimensional Lie algebras, Advan. in Math., 34, 97-108 (1979) · Zbl 0427.17011
[8] Macdonald, I., Affine root systems and Dedekind’s η-function, Invent. Math., 15, 91-143 (1972) · Zbl 0244.17005
[9] Moody, R., A new class of Lie algebras, J. Algebra, 10, 211-230 (1968) · Zbl 0191.03005
[10] Moody, R., Euclidean Lie algebras, Canad. J. Math., 21, 1432-1454 (1969) · Zbl 0194.34402
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