By Bent Ørsted and Henrik Schlichtkrull (Eds.)

ISBN-10: 0126254400

ISBN-13: 9780126254402

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**Additional info for Algebraic and Analytic Methods in Representation Theory**

**Sample text**

As we shall see in the next lecture, the Spaltenstein-Steinberg theory allows one to describe them more precisely as follows. Let B be a Borel subgroup corresponding to b +. Then, for each w C W, there is a unique orbital variety V(w) the closure of which coincides with the B saturation set of n + N w(n+); that is, V(w) = {b(n + n w(n+))b-1 : b C B}. Moreover, the map w H V(w) of W to the set 12 of orbital varieties is surjective. Finally, there is a unique nilpotent orbit O(w) the closure of which coincides with the G saturation set of n + n w(n +), and the map w ~ O(w) of W to the set N" of nilpotent orbits is surjective.

1) Here, fA: G(A) --~ Ma(A) denotes the A-component of f. Note that IndCH(M) is a representation of G via x e G(R), g e G(A), f E Mor(G, Ma)® A, (gf)R: x ~ fR(g-lx), for any k-algebra A and any A-algebra R (we identify Mor(G, M a ) ® A with Mor(GA,MQA)). 6(iv)(a) (properly generalized). Our purpose in this and the next two sections is to explore the representations of Frobenius kernels. These are certain subgroup functors of G whose only point over the field k is 1: We continue to assume that k is an algebraically closed field and that G is a connected reductive algebraic group over k, but now this is considered as an algebraic group in the new sense (whose k-points form a connected reductive algebraic group in the old sense).

3). V(w) \ V2(w). 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.