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Square integrable highest weight representations

Published online by Cambridge University Press:  18 May 2009

Karl-Hermann Neeb
Karl-Hermann Neeb
Affiliation:
Mathematisches Institut, Universität Erlangen-N¨urnberg, Blsmarckstrasse 1½, D-90154 Erlangen, Germany
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If G is the group of holomorphic automorphisms of a bounded symmetric domain, then G has a distinguished class of irreducible unitary representations called the holomorphic discrete series of G. These representations have been studied by Harish-Chandra in [7]. On the Lie algebra level, the Harish-Chandra modules corresponding to the holomorphic discrete series representations are highest weight modules. Even for G as above, it turns out that not all the unitary highest weight modules belong to the holomorphic discrete series but there exists a condition on the highest weight which characterizes the holomorphic discrete series among the unitary highest weight representations. They can be defined as those unitary highest weight representations with square integrable matrix coefficients.

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Type
Research Article
Information
Glasgow Mathematical Journal , Volume 39 , Issue 3 , September 1997 , pp. 296 - 321
Copyright
Copyright © Glasgow Mathematical Journal Trust 1997

References

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