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

TL;DR

This paper establishes the unitarity of the Segal-Bargmann transform on compact symmetric spaces, explores its behavior under space propagation, and constructs isometric isomorphisms for direct limits of these spaces.

Contribution

It proves the unitarity of the Segal-Bargmann transform on compact symmetric spaces and analyzes its compatibility with direct limits of such spaces.

Findings

Segal-Bargmann transform is a unitary isomorphism for compact symmetric spaces.

The transform behaves compatibly under propagation of symmetric spaces.

Constructs isometric isomorphisms between direct limits of Hilbert spaces.

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 $\{\cH_t(M_{n\C}),\delta_{n,m}\}$ 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 $\varinjlim L^2(M_n)^{K_n}\simeq \varinjlim \mathcal{H} (M_{n\mathbb{C}})^{K_n}$.