The Segal-Bargmann Transform on Compact Symmetric Spaces and their Direct Limits

We study the Segal-Bargmann transform, or the heat transform, $H_t$ for a compact symmetric space $M=U/K$. We prove that $H_t$ is a unitary isomorphism H_t : L^2(M) \to \cH_t (M_\C) using representation theory and the restriction principle. We then show that the Segal-Bargmann transform behaves nicely under propagation of symmetric spaces. If $\{M_n=U_n/K_n,\iota_{n,m}\}_n$ is a direct family of compact symmetric spaces such that $M_m$ propagates $M_n$, $m\ge n$, then this gives rise to direct families of Hilbert spaces $\{L^2(M_n),\gamma_{n,m}\}$ and \{\cH_t(M_{n\C}),\delta_{n,m}\} such that $H_{t,m}\circ \gamma_{n,m}=\delta_{n,m}\circ H_{t,n}$. We also consider similar commutative diagrams for the $K_n$-invariant case. These lead to isometric isomorphisms between the Hilbert spaces $\varinjlim L^2(M_n)\simeq \varinjlim \mathcal{H} (M_{n\mathbb{C}})$ as well as $\varinjlim L^2(M_n)^{K_n}\simeq \varinjlim \mathcal{H} (M_{n\mathbb{C}})^{K_n}$

A local Paley-Wiener theorem for compact symmetric spaces

The Fourier coefficients of a smooth $K$-invariant function on a compact symmetric space $M=U/K$ are given by integration of the function against the spherical functions. For functions with support in a neighborhood of the origin, we describe the size of the support by means of the exponential type of a holomorphic extension of the Fourier coefficient

The c-function for non-compactly causal symmetric spaces and its relations to harmonic analysis and representation theory

We give an overview on the c-function of a non-compactly causal symmetric space G/H and explain its interplay with harmonic analysis and representation theory.Comment: 24 pages. Minor errors corrected, new format. To appear in the Karpelevic memorial volume (AMS series Translations 2

Differential Recursion Relations for Laguerre Functions on Hermitian Matrices

In our previous papers \cite{doz1,doz2} we studied Laguerre functions and polynomials on symmetric cones $\Omega=H/L$. The Laguerre functions $\ell^{\nu}_{\mathbf{n}}$, $\mathbf{n}\in\mathbf{\Lambda}$, form an orthogonal basis in $L^{2}(\Omega,d\mu_{\nu})^{L}$ and are related via the Laplace transform to an orthogonal set in the representation space of a highest weight representations $(\pi_{\nu}, \mathcal{H}_{\nu})$ of the automorphism group $G$ corresponding to a tube domain $T(\Omega)$. In this article we consider the case where $\Omega$ is the space of positive definite Hermitian matrices and $G=\mathrm{SU}(n,n)$. We describe the Lie algebraic realization of $\pi_{\nu}$ acting in $L^{2}(\Omega,d\mu_{\nu})$ and use that to determine explicit differential equations and recurrence relations for the Laguerre functions

Representation theory, Radon transform and the heat equation on a Riemannian symmetric space

Let X=G/K be a Riemannian symmetric space of the noncompact type. We give a short exposition of the representation theory related to X, and discuss its holomorphic extension to the complex crown, a G-invariant subdomain in the complexified symmetric space X_\C=G_\C/K_\C. Applications to the heat transform and the Radon transform for X are given