Tensor products of Minimal Holomorphic Representations

Zhang, Genkai

doi:10.1090/S1088-4165-01-00103-0

Tensor products of Minimal Holomorphic Representations
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by Genkai Zhang

Represent. Theory 5 (2001), 164-190

DOI: https://doi.org/10.1090/S1088-4165-01-00103-0

Published electronically: June 15, 2001

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Abstract:

Let $D=G/K$ be an irreducible bounded symmetric domain with genus $p$ and $H^{\nu }(D)$ the weighted Bergman spaces of holomorphic functions for $\nu >p-1$. The spaces $H^\nu (D)$ form unitary (projective) representations of the group $G$ and have analytic continuation in $\nu$; they give also unitary representations when $\nu$ in the Wallach set, which consists of a continuous part and a discrete part of $r$ points. The first non-trivial discrete point $\nu =\frac a2$ gives the minimal highest weight representation of $G$. We give the irreducible decomposition of tensor product $H^{\frac a2}\otimes \overline {H^{\frac a2}}$. As a consequence we discover some new spherical unitary representations of $G$ and find the expansion of the corresponding spherical functions in terms of the $K$-invariant (Jack symmetric) polynomials, the coefficients being continuous dual Hahn polynomials.

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