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 SIGMA, 2008, Volume 4, 017, 19 pages (Mi sigma270)

Branching Laws for Some Unitary Representations of $\mathrm{SL}(4,\mathbb R)$

Bent Ørsteda, Birgit Spehb

a Department of Mathematics, University of Aarhus, Aarhus, Denmark
b Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, NY 14853-4201, USA

Abstract: In this paper we consider the restriction of a unitary irreducible representation of type $A_{\mathfrak q}(\lambda)$ of $GL(4,\mathbb R)$ 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 $GL(2,\mathbb C)$, 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.

Keywords: semisimple Lie groups; unitary representation; branching laws

DOI: https://doi.org/10.3842/SIGMA.2008.017

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ArXiv: 0802.0974
MSC: 22E47; 11F70
Received: September 10, 2007; in final form January 27, 2008; Published online February 7, 2008
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Citation: Bent Ørsted, Birgit Speh, “Branching Laws for Some Unitary Representations of $\mathrm{SL}(4,\mathbb R)$”, SIGMA, 4 (2008), 017, 19 pp.

Citation in format AMSBIB
\Bibitem{OrsSpe08} \by Bent {\O}rsted, Birgit Speh \paper Branching Laws for Some Unitary Representations of $\mathrm{SL}(4,\mathbb R)$ \jour SIGMA \yr 2008 \vol 4 \papernumber 017 \totalpages 19 \mathnet{http://mi.mathnet.ru/sigma270} \crossref{https://doi.org/10.3842/SIGMA.2008.017} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2393310} \zmath{https://zbmath.org/?q=an:1135.22014} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000267267800017} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84857316955} 

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This publication is cited in the following articles:
1. Pevzner M., “Covariant quantization: spectral analysis versus deformation theory”, Japanese Journal of Mathematics, 3:2 (2008), 247–290
2. T. Kobayashi, “Hidden Symmetries and Spectrum of the Laplacian on an Indefinite Riemannian Manifold”, Contemporary Mathematics, 484 (2009), 73–87
3. Sekiguchi H., “Branching rules of Dolbeault cohomology groups over indefinite Grassmannian manifolds”, Proc Japan Acad Ser A Math Sci, 87:3 (2011), 31–34
4. Speh B., “Restriction of Some Representations of U(P,Q) to a Symmetric Subgroup”, Representation Theory and Mathematical Physics, Contemporary Mathematics, 557, eds. Adams J., Lian B., Sahi S., Amer Mathematical Soc, 2011, 371–388
5. Kobayashi T., “Branching Problems of Zuckerman Derived Functor Modules”, Representation Theory and Mathematical Physics, Contemporary Mathematics, 557, eds. Adams J., Lian B., Sahi S., Amer Mathematical Soc, 2011, 23–40
6. Harris, B; He, HY; Olafsson, G, “The continuous spectrum in discrete series branching laws”, International Journal of Mathematics, 24:7 (2013), 1350049
7. Gourevitch D., Sahi S., Sayag E., “Invariant Functionals on Speh Representations”, Transform. Groups, 20:4 (2015), 1023–1042
8. Oshima Y., “on the Restriction of Zuckerman'S Derived Functor Modules a(Q)(Lambda) To Reductive Subgroups”, Am. J. Math., 137:4 (2015), 1099–1138
9. Vargas J.A., “Associated symmetric pair and multiplicities of admissible restriction of discrete series”, Int. J. Math., 27:12 (2016), 1650100
10. Speh B., Zhang G., “Restriction to symmetric subgroups of unitary representations of rank one semisimple Lie groups”, Math. Z., 283:1-2 (2016), 629–647
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