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**Sample text**

2). D It is easy from this corollary to check that in characteristic zero the H°()in) are all irreducible. 6. We now want to compute the higher cohomology groups H / ()in), i > 0, n E Z. 2)) and Q is the quotient. 3) for various n C Z gives (cf. 3 Let n C Z. i) Hi(An) - 0 for i > 1. ii) If n >_ - 1 , then HI(An)- O. iii) If n <_ --1, then H°(An) - O. Moreover, H 1 ()~n) has a basis of elements wo, w l , . . 4 Let n >_ O. Then k, O, H°mG(HI(Am)'H°(An))- if m -- - n - 2, otherwise. ,n. L e t n e N.

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 ) +.

Andersen 24 The long exact sequence arising from the first of these gives Hi(G/P~,HI(A)) ~ Hi(G/P~,H°(Q)) for i _> 1. 4, we get Hi(G/P~, H°(Q)) ~ Hi+I(A) for all i >_ 0, and we are done. A) (cf. , the ~n are not necessarily isomorphisms). A ) for all i. A) for all i. If i < l(w), we see that Hi(A) - 0. A), where N = ]R+I and w0 E W is the maximal element in W (of length l(wo) - N). 3) (note that dim G / B - N). 3 remains true when char(k) - p > 0 for those A which satisfy (A + p, av) < p for all a E R.