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Let S(oc) be the symmetric algebra over the complexification 9c of the real semisimple Lie algebra g. For u e S(C), a(u) is the corresponding differential operator on g. D(g) denotes the algebra generated by a(S( )) and multiplication by polynomials on gC. For any open set U C 9, Diff(U) is the algebra of differential operators with C??-coefficients on U. Let b be a Cartan subalgebra of g, b' the set of its regular points and 77 = 11caLp a, P some positive system of roots. Let W (b')G, G the connected adjoint group of B. Harish-Chandra showed that, for each D EDiff(W), there is a unique differential operator 8 (D) on b' such that (Df)Ib' = 8 (D)(flb) for all G-invariant fE C?(W), and that if D D()(b), then 8 (D) 7iT o D 077 for some DET(B). In particular a(u) = a(ulb), u 6 S(Bc) and invariant. We prove these results by different, yet simpler methods. We reduce evaluation of 8'(a(u)) (u e S(g), invariant) via Weyl's unitarian trick, to the case of compact G. This case is proved using an evaluation of a family of G-invariant eigenfunctions on: 7(HH T(H)JfG exp B(Hx, H')dx = c S s(s)exp B(sH, H'), H, H e B, c > 0. For G-invariant D e i)(g), we prove 07-1 o 8'(D) 0 77E6D(b) using properties of derivations E [(u), E] of 1)(g) induced by a(u) (u e S(a )) and of the algebra of polynomials on b invariant under the Weyl group. 1. Preliminaries. Our aim here is to give alternative proofs to some results of Harish-Chandra on radial components of invariant differential operators on a real semisimple Lie algebra [1], [2]. Let 9 be a semisimple Lie algebra over R, gc its complexification, t a Cartan subalgebra of 9, and tc complexification of t. Denote by 9' the set of regular elements of 9 and t' = ( n 9'. We have H e C' if and only if 77(H) / 0, Received by the editors May 2, 1972. AMS (MOS) subject classifications (1970). Primary 17B20; Secondary 17B99.

Invariant differential operators on a real semisimple Lie algebra and their radial components