Title page for ETD etd-07062009-114849


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
Abstract
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.

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