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 \{L²(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 L²(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 L²(M_n)^K_n into L²(M_m)^K_m for $m \geq n$. Hence, the direct limit of the Hilbert spaces \{L²(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.

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Applied Mathematics

Grantor

Mathematics

Year dc:date.available

2009

Author dc:creator

• Wiboonton, Keng

• heat equation
• compact symmetric spaces
• inductive limits
• Hilbert spaces of holomorphic functions

Statement dc:rights

• unrestricted
• Release the entire work immediately for access worldwide.

Identifier

etd-07062009-114849
https://repository.lsu.edu/gradschool_dissertations/562

OAI identifier oai:identifier

oai:repository.lsu.edu:gradschool_dissertations-1561