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By Leonard E Dickson

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5 Hi(A) ~ H i - I ( G / P ~ , H ~ ( A ) ) for all i >_ 1. Pro@ Let again 0 --~ A ~ I. be an injective B-resolution. 1 the complex 0 ~ H°(Io) ~ H°(I1) - . . is exact except in degree 1, where its cohomology is H~ (A). If we set Q - I0/,~ ~ Ker(I1 ~ means that we have exact sequences 0 and no(z ) /2), this H. H. 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.

Iv) Let A~ E C ~, where C ~ is the alcove obtained from reflecting C in the {~ E X ] (~ + p,a~)} - p } wall H short root (here, a0 E R+ is the highest we are assuming that R is indecomposable). A' + pet. 6 Determine ch D(A) for all ~ E X ( T ) +. 7 Let )~ E Xl. Then Qq(£) = Dq(2(1- 1)p + w0A). (Contrary to our usual convention we have stated this result in its quantized version. 10 says that Qq()~) has a good filtration. Since Stl is self-dual, we see that Qq(/~)* is also injective (cf. 8), and hence Qq()~)* has also a good filtration.

KL] D. Kazhdan and G. from. afifine Lie algebras, J. Amer. Math. Soc. 6 (1993), 905-947, 949-1011:7 (1994), 335-381,383-453. [Kel] G. Kempf, Linear systems on homogeneous spaces, Ann. of Math. (2), 103 (1976), 557-591. [Ke2] ~ , Representations of algebraic groups in prime characteristics, Ann. Sci. Ecole Norm. Sup. 14 (1981), 61-74. [Lull G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990). 257-296. [Lu2] ~ , Introduction to quantum groups, Progress in Mathematics, BirkhSuser, 1993.

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Algebraic invariants by Leonard E Dickson

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