Invariant Differential Operators on a Real Semisimple Lie Algebra and Their Radial Components

Let S(g ) be the symmetric algebra over the complexification 9 of the real semisimple Lie algebra g. For u £ S(g ), d(u) is the corresponding differential operator on g. 3)(g) denotes the algebra generated by d(S(g )) and multiplication by polynomials on g . For any open set U C t, Diff (LO is the algebra of differential operators with C°°-coefficients on U. Let t be a Cartan subalgebra of g, f) ' the set of its regular points and 77 = Baep a, P some positive system of roots. Let W (1)') , G the connected adjoint group of 8. Harish-Chandra showed that, for each D £ Diff (IT), there is a unique differential operator 8¡J(D) on b' such that (D/)|j' = &(fD)(fL) for all G-invariant feC^yW), and that if D 6J)(I|), then S¿(D) = n~X ° D ° n for some De$(i). In particular d(u) = d(izL), u e S(g ) and invariant. We prove these results by different, yet simpler methods. We reduce evaluation of 8j"(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: n(HMH')Lexp B(HX, H')dx = c 1 e(s)exp B(sH, H'), G SelV(gc,|)c) H, H' e g, e>0. For G-invariant D e 3)(g), We prove n~X ° S'(D) ° zre3)(i) using properties of derivations E [d(u), E] of J)(g) induced by d(u) (a f S(j )) and of the algebra of polynomials on d 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 [l], [2]. Let 9 be a semisimple Lie algebra over R, g^ its complexif¡cation, r] a Cartan subalgebra of g, and \)c complexification of \). Denote by g' the set of regular elements of g and £)' = i> n g'. We have H e i,' ¡f and only if zr(/7) ¿ 0, Received by the editors May 2, 1972. AMS(MOS) subject classifications (1970). Primary 17B20; Secondary 17B99.