## LSU Doctoral Dissertations

#### Identifier

etd-07062009-114849

#### Degree

Doctor of Philosophy (PhD)

Mathematics

Dissertation

#### 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.

2009

#### Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

Gestur Ólafsson

COinS