Quaternionic discrete series

Gordon, Derek

doi:10.1090/S1088-4165-99-00045-X

Quaternionic discrete series
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by Derek Gordon

Represent. Theory 3 (1999), 32-57

DOI: https://doi.org/10.1090/S1088-4165-99-00045-X

Published electronically: June 2, 1999

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Abstract:

This work investigates the discrete series of linear connected semi- simple noncompact groups $G$. These are irreducible unitary representations that occur as direct summands of $L^{2}(G)$. Harish-Chandra produced discrete series representations, now called holomorphic discrete series representations, for groups $G$ with the property that, if $K$ is a maximal compact subgroup, then $G/K$ has a complex structure such that $G$ acts holomorphically. Holomorphic discrete series are extraordinarily explicit, it being possible to determine all the elements in the space and the action by the Lie algebra of $G$. Later Harish-Chandra parametrized the discrete series in general. His argument did not give an actual realization of the representations, but later authors found realizations in spaces defined by homology or cohomology. These realizations have the property that it is not apparent what elements are in the space and what the action of the Lie algebra $G$ is. The point of this work is to find some intermediate ground between the holomorphic discrete series and the general discrete series, so that the intermediate cases may be used to get nontrivial insights into the internal structure of the discrete series in the general case. The author examines the Vogan-Zuckerman realization of discrete series by means of cohomological induction. An explicit complex for computing the homology on the level of a $K$ module was already known. Also, Duflo and Vergne had given information about how to compute the action of the Lie algebra of $G$. The holomorphic discrete series are exactly those cases where the representations can be realized in homology of degree 0. The intermediate cases that are studied are those where the representation can be realized in homology of degree 1. Many of the intermediate cases correspond to the situation where $G/K$ has a quaternionic structure. The author obtains general results for ${\text {A}_{\mathfrak {q}}(\lambda )}$ discrete series in the intermediate case.

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