The Segal-Bargmann transform on inductive limits of compact symmetric spaces
We construct the Segal-Bargmann transform on the direct limit of the Hilbert spaces $\{L^2(M_n)^{K_n}\}_n$ where $\{M_n = U_n/K_n\}_n$ is a propagating sequence of symmetric spaces of compact type with the assumption that $U_n$ is simply connected for each $n$. This map is obtained by taking the direct limit of the Segal-Bargmann tranforms on $L^2(M_n)^{K_n}, \ n = 1,2,...$. For each $n$, let $\widehat{U_n}$ be the set of equivalence classes of irreducible unitary representations of $U_n$ and let $\widehat{U_n/K_n} \subseteq \widehat{U_n}$ be the set of $K_n$-spherical representations. The definition of the propagation gives a nice property allowing us to embed $\widehat{U_n/K_n}$ into $\widehat{U_m/K_m}$ for $m \geq n$ in a natural way. With these embeddings, we can produce the unitary embeddings from $L^2(M_n)^{K_n}$ into $L^2(M_m)^{K_m}$ for $m \geq n$. Hence, the direct limit of the Hilbert spaces $\{L^2(M_n)^{K_n}\}_n$ is obtained in the category of Hilbert spaces and unitary embeddings and we can construct the Segal-Bargmann transform on the resulting limit in a canonical way.
heat equation, compact symmetric spaces, inductive limits, Hilbert spaces of holomorphic functions
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