Geometric quantization and derived functor modules for semisimple Lie groups

Abstract

Let G be a semisimple Lie group, g the complexified Lie algebra, and X the flag variety of g . The mechanism of geometric quantization suggests that the various G-orbits in X should give rise to representations of G. On the other hand, Zuckerman's derived functor construction attaches algebraic representations of g to G-orbits. In this paper we show that geometric quantization leads to Fréchet representations of finite length, which are the maximal globalizations of the derived functor modules. We give two alternate realizations of the representations, as cohomology spaces of ∂̄ b complexes with hyperfunction coefficients, and as local cohomology groups along G-orbits in X. We use the latter realization to implement the duality between the derived functor modules and the Beilinson-Bernstein modules, as cup product between local cohomology group followed by evaluation over the fundamental cycle.