Isometry theorem for the Segal-Bargmann transform on noncompact symmetric spaces of the complex type

TL;DR

This paper proves isometry and surjectivity theorems for the Segal-Bargmann transform on noncompact symmetric spaces of complex type, extending results from the compact case through analytic continuation and singularity cancellation.

Contribution

It establishes the first isometry and surjectivity results for the Segal-Bargmann transform in the noncompact complex setting, paralleling the compact case.

Findings

Proves isometry and surjectivity theorems for the transform.

Shows integral extension to large R despite singularities.

Demonstrates cancellation of singularities enables the extension.

Abstract

We consider the Segal-Bargmann transform for a noncompact symmetric space of the complex type. We establish isometry and surjectivity theorems for the transform, in a form as parallel as possible to the results in the compact case. The isometry theorem involves integration over a tube of radius R in the complexification, followed by analytic continuation with respect to R. A cancellation of singularities allows the relevant integral to have a nonsingular extension to large R, even though the function being integrated has singularities.