Segal–bargmann Transform for Compact Quotients

Abstract. Let G be a connected complex semisimple group, assumed to have trivial center, and let K be a maximal compact subgroup of G. Then G/K, with a fixed G-invariant Riemannian metric, is a Riemannian symmetric space of the complex type. Now let Γ 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 Γ\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.