# Algebraic and Analytic Methods in Representation Theory by Bent Orsted

By Bent Orsted

This publication is a compilation of numerous works from well-recognized figures within the box of illustration thought. The presentation of the subject is exclusive in supplying a number of diversified issues of view, which should still makethe publication very priceless to scholars and specialists alike. offers numerous assorted issues of view on key subject matters in illustration concept, from the world over recognized specialists within the box

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**Example text**

APW, Section 9]) that any finite dimensional Uq-module V (of type 1) is a sum of weight spaces (note that replacing vi by - v i in the foregoing formulas defines an equally good character )~ and type 1 simply means that we work with all signs positive). We shall consider a slightly more general category of Uqmodules. Namely, define fq to be the category consisting of all Uq-modules V that satisfy i) V = ( ~ e z V~. ,n. ,i v- v-O, m >> O, i - 1 , . . , n ,~EX } is a Uq-submodule of V. By definition, F ( V ) E fq.

P). 3) hold also for p < h, cf. [An2, An3] Chapter 1 Modular Representations 7 25 Serre d u a l i t y a n d c o m p l e t e r e d u c i b i l i t y In this section, we shall prove Theorem Let A C X ( T ) . 1 Then, for each i > O, there is a G- isomorphism Hi(A) ~_ H N - i ( - A - 2p)*. 7]. 1 directly from the results of the previous sections. 3 we obtain the complete reducibility of all finite dimensional G-modules in characteristic zero. 2 g E G , v E E f E V*. ~ k. 2). [:] Let E and F be two B-modules.

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~. 2) for all A E X(T). We now have Let A,# E X(T). Then k, i f # - A + 2 ( p r - 1 ) p , i) Homa~T(2'~(A),2~(tt)) ~ O, otherwise. 6 ii) Ext~T(2'~(A), 2~(#)) -- 0. 5 noticing that the (contravariant) dual of Z~(A) is isomorphic to Zr(A + 2 ( / - 1)p). This also implies (i) via Frobenius reciprocity and some easy weight considerations.