Intertwining operators into Dolbeault cohomology representations

Abstract

Let G L be the quotient of a semisimple Lie group G by the centralizer L of a torus. The space of Dolbeault cohomology sections of a holomorphic line bundle over G L is a natural place to realize interesting irreducible unitary representations of G and was first studied for this purpose by Bott and Schmid. Zuckerman and Vogan later introduced derived functor modules to provide an algebraic analog of these representations. The authors give a nonzero integral intertwining operator from derived functor modules, realized in the Langlands classification, to the Dolbeault cohomology representations, under the assumption that L and G have the same real rank.