Geometric analysis on small unitary representations of GL(N,R)

Details Geometric analysis on small unitary representations of GL(N,R) Geometric analysis on small unitary representations of GL(N,R)

2010-02-16

arxiv.org/abs/1002.3006v3

Journal of Functional Analysis 260, (2011), pp.1682-1720

Mathematics

Representation Theory

Corrections of the previous version

Scientific paper

10.1016/j.jfa.2010.12.008.

The most degenerate unitary principal series representations {\pi}_{i{\lambda},{\delta}} (with {\lambda} \in R, \delta \in Z/2Z) of G = GL(N,R) attain the minimum of the Gelfand-Kirillov dimension among all irreducible unitary representations of G. This article gives an explicit formula of the irreducible decomposition of the restriction {\pi}_{i{\lambda},{\delta}}|_H (branching law) with respect to all symmetric pairs (G,H). For N=2n with n \geq 2, the restriction {\pi}_{i{\lambda},{\delta}}|_H remains irreducible for H=Sp(n,R) if {\lambda}\neq0 and splits into two irreducible representations if {\lambda}=0. The branching law of the restriction {\pi}_{i{\lambda},{\delta}}|_H is purely discrete for H = GL(n,C), consists only of continuous spectrum for H = GL(p,R) \times GL(q,R) (p+q=N), and contains both discrete and continuous spectra for H=O(p,q) (p>q\geq1). Our emphasis is laid on geometric analysis, which arises from the restriction of `small representations' to various subgroups.

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