The Segal-Bargmann transform for compact quotients of symmetric spaces of the complex type

TL;DR

This paper extends the Segal-Bargmann transform to compact quotients of complex symmetric spaces, providing isometry and inversion formulas using novel methods due to the absence of a Gutzmer formula.

Contribution

It develops a new approach to analyze the Segal-Bargmann transform on compact quotients of complex symmetric spaces, paralleling known results for the dual compact case.

Findings

Derived explicit isometry formulas for the transform.

Established inversion formulas for the heat kernel-based transform.

Introduced techniques using double coset integrals and holomorphic change of variables.

Abstract

Let G/K be a Riemannian symmetric space of the complex type, meaning that G is complex semisimple and K is a compact real form. Now let {\Gamma} be a discrete subgroup of G that acts freely and cocompactly on G/K. We consider the Segal--Bargmann transform, defined in terms of the heat equation, on the compact quotient {\Gamma}\G/K. We obtain isometry and inversion formulas precisely parallel to the results we obtained previously for globally symmetric spaces of the complex type. Our results are as parallel as possible to the results one has in the dual compact case. Since there is no known Gutzmer formula in this setting, our proofs make use of double coset integrals and a holomorphic change of variable.