Taiwanese Journal of Mathematics

THE SEGAL-BARGMANN TRANSFORM FOR COMPACT QUOTIENTS OF SYMMETRIC SPACES OF THE COMPLEX TYPE

Brian C. Hall and Jeffrey J. Mitchell

Full-text: Open access

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 $\Gamma \backslash G/K$. 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.

Article information

Source
Taiwanese J. Math., Volume 16, Number 1 (2012), 13-45.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406526

Digital Object Identifier
doi:10.11650/twjm/1500406526

Mathematical Reviews number (MathSciNet)
MR2887850

Zentralblatt MATH identifier
1248.43005

Subjects
Primary: 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX]
Secondary: 81S30: Phase-space methods including Wigner distributions, etc. 35K05: Heat equation

Keywords
Segal-Bargmann transform heat operator symmetric space eigenfunctions

Citation

Hall, Brian C.; Mitchell, Jeffrey J. THE SEGAL-BARGMANN TRANSFORM FOR COMPACT QUOTIENTS OF SYMMETRIC SPACES OF THE COMPLEX TYPE. Taiwanese J. Math. 16 (2012), no. 1, 13--45. doi:10.11650/twjm/1500406526. https://projecteuclid.org/euclid.twjm/1500406526


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