By Bent Ørsted and Henrik Schlichtkrull (Eds.)
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Extra resources for Algebraic and Analytic Methods in Representation Theory
Is homogeneous for Gelfand-Kirillov dimension, for examsimple. Then a result of O. Gabber [LS, Appendix] asserts is equidimensional. For further results on associated vari- eties, see [BE]. 1]. Assume g semisimple. Given J E PrimU(g), show that V(grj= J) is contained in the cone of nilpotent elements of g*. Lecture 3 Goldie rank and characteristic polynomials Let us describe more precisely how rk U(g)/J(A) depends on A E b*. First some notation is needed. , positive, simple) roots relative to our triangular decomposition of ~.
At least for characteristic zero, the B a l a - C a r t e r - D y n k i n theory gives a specific procedure for determining for each conjugacy class d an element w E W such that d N U~ is dense in Uw. The idea is as follows. Consider e Efl nilpotent. By the Jacobson-Morosov theorem, there exists an s[(2) triple (e, h, f) containing e. Let g- (~gi iEX be the decomposition of g into its t? eigenspaces with gi corresponding to h-eigenvalue i. Now [e, go] = g2 from s[(2) theory, and so dim g2 < dim go implies g~ -76 0.
The answer is that this dependence is essentially a polynomial one; that is, there is a finite set of polynomials the values of which determine these Goldie ranks. Moreover, for each X(O) it is appropriate to select a subset of these polynomials. Remarkably, this subset forms a basis for an irreducible representation of the Weyl group W for the pair (g, D), and this representation can be identified with the Springer representation Chapter 2 Orbital Varieties 55 attached to (_9 defined by quite different (geometrical) considerations.