A mackey imprimitivity theory for algebraic groups

Abstract

Let G be an affine algebraic group over an algebraically closed field k and let H be a closed subgroup of G. If V is a rational H-module (a comodule for the coordinate ring of H) there is a now well-known notion of an induced module VI G for G, defined as the space Morph/~(G, V) of all H-equivariant morphisms from G to a finite dimensional subspace of V, with obvious G-action. The question arises, given a rational G-module M, how can one recognize M as an induced module VIG? For a finite group G the answer is the Mackey imprimitivity theorem: the module M is induced if and only if it is a direct sum of subspaces permuted transitively by G (with H the stabilizer of one of these subspaces, called V). One uses this result, for example, in proving the famous Mackey decomposition theorem which describes the restriction of any induced module to a second subgroup L as a direct sum of suitable induced modules; given the imprimitivity theorem, the proof is just a matter of grouping the summands permuted by G into their orbits under L. In the case of algebraic groups the situation is quite different. For some subgroups H, all G-modules are induced. This occurs, for example, if G is connected and k [G/H] ,= k, e.g., if H is parabolic. Also, if G is a connected unipotent group, then a rational G-module M is induced from some proper subgroup if and only if its endomorphism ring contains a two dimensional submodule E, for the conjugation action of G, with E___ k. 1 and such that k. l is precisely the subspace annihilated by the action of the Lie algebra of G on E [27] (cf. also (5.5) below). The latter is an application of a general criterion in case G/H is affine: a rational G-module M is induced if and only if there is an action of the coordinate ring A of G/H compatible with the action of G on both A and M. Note that this generalizes the imprimitivity theorem in the case of finite groups, since then A has a k-basis of [G:H] orthogonal idempotents permuted transitively by G.