Algebraic and Analytic Methods in Representation Theory by Bent Ørsted and Henrik Schlichtkrull (Eds.)

By Bent Ørsted and Henrik Schlichtkrull (Eds.)

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Finally, for an appropriate choice of w C W, one has Lie U~ - m. T h e r e is a second n a t u r a l way to obtain irreducible components of 8(C). Fix u E C, and set B~ - {B E B I u C B}. An a r g u m e n t of N. Spaltenstein IS] based on B r u h a t decomposition (which we use later in a related context) shows how to move from component to component of B~. From this, one concludes t h a t Bu is equidimensional. Now let X be a S t a b c u orbit of an irreducible component of Bu × B~, hence in particular a finite union of pairs of irreducible components of B~.

3 Hi(E(r) ® ( p r _ 1 ) p ) ~ Hi(E)(r) ® Str. 11(iii)), we have H ~ - H~(G/GrB, 2 r ( - ) ) . 8. Hence, the tensor identity for H i ( G / G r B , - ) gives Hi(E(r) @ ( p r _ 1 ) p ) ~ Hi(G/GrB, E (r)) ® Str, and we are done if we show that H i ( G / G r B , E (r)) ~ Hi(E) (r). 1) One checks this easily for i - 0. , we need to check that H i ( G / G r B , I (r)) - 0 for i > 0 whenever I is an injective B-module. In fact, it is enough to consider I - k[B], and since k[B] (r) -~ Indaa;B(k) we have H i ( O / O r B , k[B] (r)) ~ Hi(G/Or, k).

4 we immediately reduce to the case M = D(A) with A C X(T)+\C. Moreover, since D ( A ) i s indecomposable, any f C E n d a ( D ( A ) ) may be written f = a . Id + f ' for some a E k and some nilpotent f ' E Enda(D(A)). Since T r ( f ' ) = 0, we have reduced the theorem to the following statement: If A e X(T)+\C, then Pl dim D(A). , PI(# + P, c~v} for some c~ e R+), then pldimH°(#) (we assume

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