By Bent Ørsted and Henrik Schlichtkrull (Eds.)

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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.

5 Let A E X(T). i) Z~()~) contains a unique irreducible Gr-submodule, which we denote Lr()~). ii) Any finite dimensional simple G~-module is isomorphic to some such L~()~). iii) If also # E X ( T ) , then L~()~) ~_ Lr(#) iff A - # (mod p X ( T ) ) . 6 Set Xp~(T) = {A E X(T) I 0 _< (A, av> < p" for all simple roots a}. The elements in this set are called the p~-restricted weights. Note that any A E X ( T ) can be written uniquely )~ = )~0 + pr)~l with )~0 E Xpr(T), )k 1 E X(T). 6) Note that prA1 is trivial as a Gr-module.

3 we have, for all ,~ C X ( T ) , Hi(A) (~) ® St~ __ Hi(p~(A + p) - p), and for r >> 0, A E X ( T ) + the right hand side vanishes whenever i > 0. 2]. [-1 Chapter 1 Modular Representations 39 Let I E X ( T ) +. 4 (Weyl's character formula) ch H°(1) = J(e~+P)/J(eP), where J is the Z-endomorphism on J(e~) - E Z[X(T)] given by p E X(T). 1 we have ch H ° (A) = ~--~i( - 1)ich Hi(1). 10]. [3 11 Q u a n t u m groups In this section we introduce quantum groups and briefly sketch how many of the preceding results in the representation theory for algebraic groups have quantized counterparts.