A finiteness theorem for affine Lie algebras
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 [\textit{V. V. Deodhar}, \textit{O. Gabber} and \textit{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 \(w.\nu =w(\nu +\rho)-\rho)\).
scientific article; zbMATH DE number 3998053