Segal-Bargmann transform on Hermitian symmetric spaces and Orthogonal Polynomials

Details Segal-Bargmann transform on Hermitian symmetric spaces and Orthogonal Polynomials Segal-Bargmann transform on Hermitian symmetric spaces and Orthogonal Polynomials

2002-06-26

arxiv.org/abs/math/0206275v1

Mathematics

Representation Theory

Scientific paper

Let $\mathcal{D}=G/K$ be a complex bounded symmetric domain of tube type in a complex Jordan algebra $V$ and let $\mathcal{D}_{\mathbb{R}}=H/L\subset \mathcal{D}$ be its real form in a formally real Euclidean Jordan algebra $J\subset V$. We consider representations of $H$ that are gotten by the generalized Segal-Bargmann transform from a unitary $G$-space of holomorphic functions on $\mathcal{D}$ to an $L^2$-space on $\mathcal{D_{\mathbf{R}}}$. We prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the Laplace transform. We find the extension to $\mathcal{D}$ of the spherical functions on $\mathcal{D}_{\mathbb{R}}$ and find the expansion in terms of the $L$-spherical polynomials on $\mathcal{D}$, which are Jack symmetric polynomials. We prove that the coefficients are orthogonal polynomials in an $L^2$-space, the measure being the Harish-Chandra Plancherel measure multiplied by the symbol of the Berezin transform. We prove the difference equation and recurrence relation for those polynomials by considering the action of the Lie algebra and the Cayley transform on the polynomials on $\mathcal D$.

Affiliated with

Also associated with

No associations

Rating

Segal-Bargmann transform on Hermitian symmetric spaces and Orthogonal Polynomials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Segal-Bargmann transform on Hermitian symmetric spaces and Orthogonal Polynomials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Segal-Bargmann transform on Hermitian symmetric spaces and Orthogonal Polynomials will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-195523