By Bent Ørsted and Henrik Schlichtkrull (Eds.)

ISBN-10: 0126254400

ISBN-13: 9780126254402

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

**Example text**

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. Hence Z~(A) _~ Z~()~°) and L~(A) _~ L~(A°). 5(iii), which can be rephrased The finite dimensional simple Gr-modules are parametrized by the set of p~-restricted weights.

Hence, ~ o is an isomorphism. Standard degree shift arguments show that so are all ~ , i >_ 0. 5 Let A, # E X ( T ) +. , ExtlG(H°(p)*, H°(A)) = 0 for all A,# E X ( T ) +. Proof: Suppose first that A ~ -w0(#). 4. , we have found a retraction H ° (A) --+ E of the given embedding. If A < - w 0 ( # ) , we dualize the sequence and repeat the previous argument. [] Combining Serre duality and the Borel-Weil-Bott theorem, we obtain the following classical result. 6 Suppose char(k) - 0. i) H°(A) is an irreducible representation of G for all A C X ( T ) +.

6, it is a standard base change argument to derive H~(~)- 0 for ~ni > N. 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.

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

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