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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Specializing v to q, we have made k into an A-algebra. l) - K ? ,n. Proof: [Lul, Lu2]. Let now U denote the enveloping algebra over Q for the semisimple Lie algebra corresponding to (aij). Denote the standard generators of 0 by /~i,/~i, i - 1 , . . , n , and set /ti - [/~i,/~i]. r), i - 1 , . . , n, where E~) - E~/r! and similarly for/~/(~);see [Hul, Section 26]. 4 There is a Hopf algebra isomorphism (]z--%Uz/({Ki- 1 1 i - 1 , . . , n}) given by /~r) H image of E~r), /~(~) ~ /ti ~ image of F (~), image of [K/l; 0 ] .

In other words, any G~T-module that is induced from a finite dimensional B~T-module has a Zr-filtration. Moreover, [ 2 r V - 2~(#)] - dim V,. H. H. 5 Let A e X(T). Then Q~(A) has a Z~-filtration, and for each # E X(T) we have [Q~(~). 2~(~)1- [2~(~). L~(~)] (the latter symbol denoting composition factor multiplicity). " Note that IndTG~T -- 2r o Ind B~T. 4. , B' = woBwo, where w0 is (a representative in G of) the longest element in W. Set 2 ' - I n d ~ T T. Then, clearly, Z~ has properties completely analogous to Z~.

3) It turns out that the representation theory for Gr is quite similar to the one for G. W i t h o u t worrying too much about details and proofs (the reader is referred to [Ja] for these), we now give some of the highlights. 4 33 For any B~-module E, we set Z~(E) - IndCB~E. 4) Zr is an exact functor. The reason is that G~/B~ is affine, namely G~/B~ ~_ U¢, so that all higher sheaf cohomology of bundles on G~/Br vavishes. 5) dim Z~ (E) = pNr dim E, because dim k[U¢] = pN~ (cf. 6(iii)). 5 Let A E X(T).

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