Type of Document Dissertation Author Wiboonton, Keng Author's Email Address kwiboo1@math.lsu.edu URN etd-07062009-114849 Title The Segal-Bargmann Transform on Inductive Limits of Compact Symmetric Spaces Degree Doctor of Philosophy (Ph.D.) Department Mathematics Advisory Committee

Advisor Name Title Gestur Ólafsson Committee Chair Hongyu He Committee Member Mark Davidson Committee Member Oliver Dasbach Committee Member William Adkins Committee Member Jacquelyn Sue Moffitt Dean's Representative Keywords

- heat equation
- compact symmetric spaces
- inductive limits
- Hilbert spaces of holomorphic functions
Date of Defense 2009-06-23 Availability unrestricted AbstractWe 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.

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