Unitary representations of unimodular Lie groups in Bergman spaces

TL;DR

This paper constructs strongly continuous unitary representations of unimodular Lie groups within Bergman spaces of specific pseudoconvex neighborhoods in their complexified manifolds, expanding the understanding of their complex-analytic representations.

Contribution

It introduces a method to realize unimodular Lie groups as unitary representations in Bergman spaces of pseudoconvex neighborhoods in their complexifications, a novel approach in representation theory.

Findings

Successfully constructs unitary representations in Bergman spaces

Establishes strong continuity of the representations

Provides a new geometric framework for group representations

Abstract

For an arbitrary unimodular Lie group $G$, we construct strongly continuous unitary representations in the Bergman space of a naturally constructed strongly pseudoconvex neighborhood of $G$ in the complexification of its underlying manifold.