Representations on Partially Holomorphic Cohomology Spaces, Revisited

TL;DR

This paper revisits and updates a 1974 work on Plancherel formulae and Dolbeault cohomology for real reductive Lie groups, incorporating recent advances and refined methods in the geometric realization of unitary representations.

Contribution

It provides an updated exposition of classical results with new developments, connecting old and modern approaches to representation theory of real reductive Lie groups.

Findings

Refinements in Plancherel formulae for real reductive groups

Enhanced understanding of partial Dolbeault cohomology realizations

Connections between classical and modern geometric approaches

Abstract

This is a semi--expository update and rewrite of my 1974 AMS AMS Memoir describing Plancherel formulae and partial Dolbeault cohomology realizations for standard tempered representations for general real reductive Lie groups. Even after so many years, much of that Memoir is up to date, but of course there have been a number of refinements, advances and new developments, most of which have applied to smaller classes of real reductive Lie groups. Here we rewrite that AMS Memoir in in view of these advances and indicate the ties with some of the more recent (or at least less classical) approaches to geometric realization of unitary representations.