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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Tensor products of Minimal Holomorphic Representations
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by Genkai Zhang PDF
Represent. Theory 5 (2001), 164-190 Request permission


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:
  • 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)
  • © Copyright 2001 American Mathematical Society
  • Journal: Represent. Theory 5 (2001), 164-190
  • MSC (2000): Primary 22E46, 47A70, 32M15, 33C52
  • DOI:
  • MathSciNet review: 1835004