Duistermaat-Heckman measures in a non-compact setting

TL;DR

This paper extends the Duistermaat-Heckman formula to non-compact settings, enabling explicit calculations of pushforward measures and deriving classical multiplicity formulas for holomorphic discrete series.

Contribution

It introduces a Duistermaat-Heckman type formula applicable in non-compact contexts and applies it to compute measures and multiplicities in representation theory.

Findings

Derived explicit pushforward of Liouville measure in non-compact setting

Evaluated classical multiplicity formulas for holomorphic discrete series

Extended Duistermaat-Heckman formula beyond compact cases

Abstract

We prove a \dh type formula in a suitable non-compact setting. We use this formula to evaluate explicitly the pushforward of the Liouville measure via the moment map of both an abelian and a non-abelian group action. As an application we obtain the classical analogues of well-known multiplicity formulas for the holomorphic discrete series representations.