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Tensor products of minimal holomorphic representations


Author: Genkai Zhang
Journal: Represent. Theory 5 (2001), 164-190
MSC (2000): Primary 22E46, 47A70, 32M15, 33C52
DOI: https://doi.org/10.1090/S1088-4165-01-00103-0
Published electronically: June 15, 2001
MathSciNet review: 1835004
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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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Additional Information

Genkai Zhang
Affiliation: Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden
Email: genkai@math.chalmers.se

DOI: https://doi.org/10.1090/S1088-4165-01-00103-0
Keywords: Bounded symmetric domains, weighted Bergman spaces, unitary highest weight representations, invariant differential operators, tensor product, irreducible decomposition, Clebsch-Gordan coefficients
Received by editor(s): May 23, 2000
Received by editor(s) in revised form: April 10, 2001
Published electronically: June 15, 2001
Additional Notes: Research supported by the Swedish Natural Science Research Council (NFR)
Article copyright: © Copyright 2001 American Mathematical Society

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