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

Abstract

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 $\{\cH_t(M_{n\C}),δ_{n,m}\}$ 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 $\underrightarrow{\lim }{L}^2(M_n)\simeq \underrightarrow{\lim }\mathcal{H}(M_{n\mathbb{C}})$ as well as $\underrightarrow{\lim }{L}^2(M_n{)}^{K_n}\simeq \underrightarrow{\lim }\mathcal{H}(M_{n\mathbb{C}}{)}^{K_n}$.