On the Harishchandra homomorphism for infinite-dimensional Lie algebras

Abstract

A natural generalization of the finite-dimensional semisimple Lie algebras to the infinite-dimensional case is the Kac-Moody Lie algebra G(A) or more generally, the contragredient Lie algebra associated to an arbitrary square matrix A [9a, 111. The category a introduced in [la, b] to study the representations of finite-dimensional semisimple Lie algebras was extended [9b] to the case of Kac-Moody Lie algebras. In [lo], Kac and Kazhdan describe the irreducible subquotients of Verma modules. Using this a decomposition of the category B was obtained in [3]. In both [3] and [lo] the center 3 of the universal enveloping algebra plays no role, since there is no analog to the theory of central characters in the infinite-dimensional setup. Let G = G(A) be the Kac-Moody Lie algebra associated to a symmetrizable generalized Cartan matrix A, and let H = H(A) be the abelian diagonalizable subalgebra of G(A). In this article we introduce alalgebra % (&i(U(G), a)) of operators on the category P and a surjectio2 /I of % onto U(H). The Weyl group W acts on U(H) and the restriction of /3 to the cznter 3 of (li maps into the invariants U(EQW. In Section 3, we prove that /3 Ij is injective and that 3 injects into 3, i: 3 4 3. If G(A) is infinite dimensional, then the existence of the Casimir operator [5, 9b] implies that i(j) f 3, and hence it follows that the Harishchandra homomorphism is never surjective. In Section 4, we deal with the case of affme Kac-Moody Lie algebras. Using a theorem of Hochschild and Mostow [6], we deduce that U(H)W is the polynomial algebra on two generators. This disproves the conjecture in [3] that the image of the Harishchandra homomorphism separ$es the Weyl group orbits.* Further, we show that the restriction of p to 3 is an isomorphism onto U(H)W.