Topological Frobenius reciprocity and invariant Hermitian forms

TL;DR

This paper establishes a topological Frobenius reciprocity principle for certain representations on sheaf cohomology in flag manifolds, linking invariant Hermitian forms to geometric fibers.

Contribution

It provides a specific version of topological reciprocity in the regular, antidominant case and explores its implications for invariant Hermitian forms on sheaf cohomology.

Findings

Established a topological Frobenius reciprocity in the regular, antidominant case.

Connected invariant Hermitian forms on sheaf cohomology to those on geometric fibers.

Provided a framework for analyzing invariant forms in complex geometric settings.

Abstract

In his article "Unitary Representations and Complex Analysis", David Vogan gives a characterization of the continuous invariant Hermitian forms defined on the compactly supported sheaf cohomology groups of certain homogeneous analytic sheaves defined on open orbits in generalized flag manifolds. In the last section of the manuscript, Vogan raises a question about the possibility of a topological Frobenius reciprocity for these representations. In this article we give a specific version of the topological reciprocity in the regular, antidominant case and use it to study the existence of continuous invariant hermitian forms on the sheaf cohomology. In particular, we obtain a natural relationship between invariant forms on the sheaf cohomology and invariant forms on the geometric fiber.