Induced modules and extensions of representations

Abstract

Let G be a connected affine algebraic group over an algebraically closed field k, and let B be a Borel subgroup of G. If V is a rational B-module we can ask for conditions under which V extends to G, that is, under which the action of B extends to a rational action of G on V. Since G/B is a complete variety, we have MtnlG_--__ M for any rational G-module M, cf. (1.3); so if any extension of V exists, it must be unique. The main result of this paper is that V is extendible to G if and only if it is extendible to every minimal parabolic subgroup P>B. Here "minimal" means that P properly contains B and that no other closed subgroup of P does so; equivalently, P has semisimple rank one. We also show that if P contains a Levi complement L, then V extends to P iff VILnn extends to L; so, for reductive G, the extendibility of V reduces to an SL 2 question. We obtain the main result as a corollary to a general theorem on induced modules. For any parabolic subgroup P >B, and any rational B-module V, let VIe denote the corresponding module induced from B to P. If P~ . . . . . P, is any sequence of such P's, we let VI e' ..... e. denote the P.-module vlv'lnlV2"-I r" obtained by successively inducing to P~_ 1, restricting to B, and inducing to P~. We prove the B-module structure of V[ e ...... e. depends only on the set-theoretic product P~ ... P., and that VI v' ..... e . _ VIG as P.-modules in case this product is G. At the same time we prove a "factorization" theorem for the affine coodinate ring R(G) of G. Let Px . . . . . P. be parabolic subgroups containing B such that G = P1... P.. Then we have R(G)~-R(P,)@n R(P,_ O@B"" @B R(PO, where in general for a right B-module M and a left B-module N, we denote by M@BN the set of fixed points of B in M @ N relative to the action b. (m@ n)= m b -1@ b n. The above results are in turn derived from simple considerations involving the Demazure desingularization of Schubert varieties [9], and are inspired by Kempf [12] and by Andersen's efforts [1] toward a reorganization of Kempfs work. A simplified version of Andersen's main reduction is given in the appendix.