Branching coefficients of holomorphic representations and Segal-Bargmann transform
| dc.creator | Zhang, Genkai | |
| dc.date | 2001-10-19 | |
| dc.date.accessioned | 2026-07-07T04:43:57Z | |
| dc.date.available | 2026-07-07T04:43:57Z | |
| dc.description | Let $\mathbb D=G/K$ be a complex bounded symmetric domain of tube type in a Jordan algebra $V_{\mathbb C}$, and let $D=H/L =\mathbb D\cap V$ be its real form in a Jordan algebra $V\subset V_{\mathbb C}$. The analytic continuation of the holomorphic discrete series on $\mathbb D$ forms a family of interesting representations of $G$. We consider the restriction on $D$ of the scalar holomorphic representations of $G$, as a representation of $H$. The unitary part of the restriction map gives then a generalization of the Segal-Bargmann transform. The group $L$ is a spherical subgroup of $K$ and we find a canonical basis of $L$-invariant polynomials in components of the Schmid decomposition and we express them in terms of the Jack symmetric polynomials. We prove that the Segal-Bargmann transform of those $L$-invariant polynomials are, under the spherical transform on $D$, multi-variable Wilson type polynomials and we give a simple alternative proof of their orthogonality relation. We find the expansion of the spherical functions on $D$, when extended to a neighborhood in $\mathbb D$, in terms of the $L$-spherical holomorphic polynomials on $\mathbb D$, the coefficients being the Wilson polynomials. | |
| dc.identifier | https://arxiv.org/abs/math/0110212 | |
| dc.identifier | http://arxiv.org/abs/math/0110212 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/62446 | |
| dc.subject | Representation Theory | |
| dc.title | Branching coefficients of holomorphic representations and Segal-Bargmann transform | |
| dc.type | text |