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A Y(X algebraic group algebraically closed field B-socle Borel subgroup completes the proof conjugate Corollary denote derived functors descending filtration direct sum dominant weight easy to check epimorphism Ex_tp filt filtration for I=0 filtration with quotients filtration with successive follows G of type hence isomorphic hypotheses induction injective Kempf's Vanishing Theorem Lemma Let G long exact sequence module Moreover morphism non-zero obtain parabolic subgroup Proposition 3.2.6 R G(J rational G-module rechoose the filtration reductive group root system semisimple groups sequence of cohomology sequence of derived short exact sequence simple socle split extension subgroup of G submodule subset successive quotients suffices to prove suffices to show Suppose tensor identity true unique non-split extension Vanishing Theorem Weyl's Character Formula Y_(X Y(X_ Y(XJ Yn(X