On the variety of a highest weight module

Abstract

1.2. Let U(g) be the enveloping algebra of g. Detailed information on the primitive spectrum Prim U(g) of U(g) was summarized by Conjecture 7.4 of [ 181. Here a main question was to relate Prim U(g) with the Springer correspondence. Parts (i), (ii) of this conjecture were established in I, II and a partial solution to (iii) was given in [21] sufficient, e.g., to treat g of type A,. Meanwhile Borho and Brylinski [5] have established (iii) for induced ideals, while Barbasch and Vogan [ 1, 2] have established a version of (iii) in which %‘(gr I): I E Prim U(g) is replaced by a certain wavefront set (known to be contained in Y(gr 1)). This last work involves some case by case analysis and gives little indication as to why the Springer correspondence should arise in the study of Prim U(g).