Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 017, 19 pages      arXiv:0802.0974      http://dx.doi.org/10.3842/SIGMA.2008.017
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Branching Laws for Some Unitary Representations of SL(4,R)

Bent Ørsted a and Birgit Speh b
a) Department of Mathematics, University of Aarhus, Aarhus, Denmark
b) Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, NY 14853-4201, USA

Received September 10, 2007, in final form January 27, 2008; Published online February 07, 2008

Abstract
In this paper we consider the restriction of a unitary irreducible representation of type Aq(λ) of GL(4,R) to reductive subgroups H which are the fixpoint sets of an involution. We obtain a formula for the restriction to the symplectic group and to GL(2,C), and as an application we construct in the last section some representations in the cuspidal spectrum of the symplectic and the complex general linear group. In addition to working directly with the cohmologically induced module to obtain the branching law, we also introduce the useful concept of pseudo dual pairs of subgroups in a reductive Lie group.

Key words: semisimple Lie groups; unitary representation; branching laws.

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