Primitive ideals and orbital integrals in complex exceptional groups

Abstract

In [2], two related problems were studied for complex classical groups: determination of the primitive spectrum of the enveloping algebra (as a set), and Fourier inversion of unipotent orbital integrals. The first of these was solved completely (extending work of Joseph and others in SL(n, C) and small groups). Although our techniques presumably solve the second as well, up to determination of normalizing constants, we carried out the calculations only for special unipotent classes (see [ 181). We had, of course, tried to apply the same methods to the exceptional groups, but they seem to be inadequate. Since that work was done, however, Brylinski and Kashiwara and Beilinson and Bernstein have established the Kazhdan-Lusztig conjecture, giving character formulas for irreducible highest weight modules [3, 71. Because of a conjecture of Joseph proved in [24], this determines in principle the primitive ideals with a fixed regular integral infinitesimal character: they are in one-to-one correspondence with what Kazhdan and Lusztig call left cells in the Weyl group. However, the algorithm given by Kazhdan and Lusztig to compute these cells is enormously complicated, requiring one to compute roughly 1 I%‘]* polynomials of degrees on the order of half the number of positive roots. This is very unsatisfactory. However, it