## 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.## References

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## Bibliographic Information

**Genkai Zhang**- Affiliation: Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden
- Email: genkai@math.chalmers.se
- 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: https://doi.org/10.1090/S1088-4165-01-00103-0
- MathSciNet review: 1835004