Localization of Cohomological Induction

Abstract

We give a geometric realization of cohomologically induced $(\mathfrak{g},K)$-modules. Let $(\mathfrak{h},L)$ be a subpair of $(\mathfrak{g},K)$. The cohomological induction is an algebraic construction of $(\mathfrak{g},K)$-modules from a $(\mathfrak{h},L)$-module $V$. For a real semisimple Lie group, the duality theorem by Hecht, Mili{\v{c}}i{\'c}, Schmid, and Wolf relates $(\mathfrak{g},K)$-modules cohomologically induced from a Borel subalgebra with $\mathcal{D}$-modules on the flag variety of $\mathfrak{g}$. In this article we extend the theorem for more general pairs $(\mathfrak{g},K)$ and $(\mathfrak{h},L)$. We consider the tensor product of a $\mathcal{D}$-module and a certain module associated with $V$, and prove that its sheaf cohomology groups are isomorphic to cohomologically induced modules.