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

By Bent Ørsted and Henrik Schlichtkrull (Eds.)

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Extra info for Algebraic and Analytic Methods in Representation Theory

Example text

It follows that the vanishing H~(A) - 0 for i > N also holds for q any nonzero element in any field F. Let us return to the case where F - C and q E C\{0}. I f q i s not a root of unity, the Borel-Weil-Bott theorem holds for Uq and hence Lq(A) - H. H. Andersen 44 H°(A) for all A e X ( T ) +. Moreover, in this situation ~q is semisimple (as it follows from that theorem combined with quantized Serre duality, cf. 6). So suppose q is a primitive /th root of unity. 7 Determine ch Lq()~), A C X +. Now Lusztig has conjectured that if 1 - p _> h (h denoting the Coxeter number of R), then ch Lq()~) - ch L(A) for all A e Xp(T).

Notice it is not guaranteed that V(saw) is actually a component of n + N O, as it may be of codimension _> 1. 3) may be quite sizable. 3) iterated sumciently often eventually gives all the orbital varieties lying in the unique dense orbit O contained in GV. This was originally proved by Spaltenstein [S] using an argument of Steinberg involving Bruhat decomposition. It implies that O n n + is equidimensional. A similar argument can be used to show that O n n + is equidimensional. Using the characteristic polynomials introduced in Lecture 3, we can follow this operation more closely.

3. So, as in the previous two sections, we shall assume char(k) - p > 0. H. H. 2 Let V be a representation of G. Then we define V (r) to be the representation of G on the same vector space but with G-action given by g v - F~(g)v, g E G, v E V. 1 from Let E be a B-module. 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).