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 $M=U\mathrm{/}K$. We prove that $H_t$ is a unitary isomorphism $H_t:L^2(M)\to {\text{cH}}_t(M_{\text{C}})$ using representation theory and the restriction principle. We then show that the Segal-Bargmann transform behaves nicely under propagation of symmetric spaces. If $\{M_n=U_n\mathrm{/}{K}_n,{\iota }_{n,m}{\}}_n$ 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 $\{{\text{cH}}_t(M_{n\text{C}}),{\delta }_{n,m}\}$ 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$\mathbb{C}$})$aswellas$\varinjlim L^2(M_n)^{K_n}\simeq \varinjlim $\mathcal{H}$ (M_{n$\mathbb{C}$})^{K_n}$.