Generalized Verma modules, the Cartan-Helgason theorem, and the Harish-Chandra homomorphism

Abstract

This paper is a natural sequel to [6(a)]. H ere we complete the solution of [l, Problem 35, p. 3361, in the intended algebraic spirit. (See [6(c)] for another kind of solution of the same problem.) Specifically, let g = k @ p be a Cartan decomposition of a real semisimple Lie algebra, a a Cartan subspace of p, g = k @ a @ n a corresponding Iwasawa decomposition, G the universal enveloping algebra of g, Gk the centralizer of k in G, A the universal enveloping algebra of a, and p: Gk + A the HarishChandra homomorphism, defined to be the projection to A with respect to the decomposition G = A @ (kG + Gn). The problem is to prove “purely algebraically” Harish-Chandra’s theorem [2, Sect. 41 that the image of p equals the algebra Aw of translated (by half the sum of positive restricted roots) restricted Weyl group invariants in A. The proof should be valid for semisimple symmetric Lie algebras with splitting Cartan subspaces, over arbitrary fields of characteristic zero (see [l, Sect. I.131 and [6(a)]). In [6(a)], such a proof was given for the fact that p(Gk) C A, . (The subalgebra n will be denoted u in the body of this paper, since the symbol n has another natural use; see Section 2.) Now the assertion that ,p(Gk) is all of A, is equivalent to the assertion of Chevalley’s polynomial restriction theorem [3(a), p. 430, Theorem 6.10]-that the restriction map from p to a takes the algebra of k-invariant polynomial functions on p onto the algebra of polynomial functions on a invariant under the restricted Weyl group W. In the paper, we give two suitably general proofs of Chevalley’s theorem; see Theorem 5.1. Actually, one of these proofs is nothing more than a modification of Harish-Chandra’s classical proof presented in [3(a), pp. 4334341; see the Appendix. In the body of this paper, we present a proof similar in spirit to the Kostant-Steinberg-Varadarajan proof of Chevalley’s