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The $n$-cohomology of representations with an infinitesimal character

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The n-cohomology of representations with an infinitesimal character COMPOSITIO MATHEMATICA WILLIAMCASSELMAN M. SCOTTOSBORNE The n-cohomology of representations with an infinitesimal character Compositio Mathematica, tome 31, no 2 (1975), p. 219-227. <> © Foundation Compositio Mathematica, 1975, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// implique l’accord avec les conditions générales d’utilisation ( Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 219 THE n-COHOMOLOGY OF REPRESENTATIONS WITH AN INFINITESIMAL CHARACTER William Casselman and M. Scott Osborne COMPOSITIO MATHEMATICA, Vol. 31, Fasc. 2, 1975, pag. 219-227 Noordhoff International Publishing Printed in the Netherlands 1 Let F be a field of characteristic 0, g a reductive Lie algebra over F, p a parabolic subalgebra with n as nilpotent radical and m a reductive complement in p to n. If V is an irreducible finite-dimensional g-module, then the Lie algebra cohomology H*(n, V) has a canonical m-module structure (see Section 2). Over an algebraic closure of F, this module decomposes completely into a direct sum of absolutely irreducible m-modules, and Kostant [5] has determined this decomposition com- pletely. In this paper we shall prove a partial generalization of Kostant’s result for the class of all g-modules with an infinitesimal character (Theorem 2.6 and its corollary). We include (in Section 4) an elementary derivation of the structure of H*(n, V) when V is finite-dimensional. One of the applications Kostant made of his result was to give a new proof of the Weyl character formula. In order

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