A program for branching problems in the representation theory of real reductive groups

Chapter
Part of the Progress in Mathematics book series (PM, volume 312)

Abstract

We wish to understand how irreducible representations of a group G behave when restricted to a subgroup G′ (the branching problem). Our primary concern is with representations of reductive Lie groups, which involve both algebraic and analytic approaches. We divide branching problems into three stages: (A) abstract features of the restriction; (B) branching laws (irreducible decompositions of the restriction); and (C) construction of symmetry breaking operators on geometric models. We could expect a simple and detailed study of branching problems in Stages B and C in the settings that are a priori known to be “nice” in Stage A, and conversely, new results and methods in Stage C that might open another fruitful direction of branching problems including Stage A. The aim of this article is to give new perspectives on the subjects, to explain the methods based on some recent progress, and to raise some conjectures and open questions.

Key words

branching law symmetry breaking operator unitary representation Zuckerman–Vogan’s \(A_{\mathfrak{q}}(\lambda )\) module reductive group spherical variety multiplicity-free representation 

MSC (2010):

Primary 22E46 Secondary 53C35 

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Notes

Acknowledgements

The author thanks J.-L. Clerc, T. Kubo, T. Matsuki, B. Ørsted, T. Oshima, Y. Oshima, M. Pevzner, P. Somberg, V. Souček, B. Speh for their collaboration on the papers which are mentioned in this article. This article is based on the lecture that the author delivered at the conference Representations of reductive groups in honor of David Vogan on his 60th birthday at MIT, 19-23 May 2014. He would like to express his gratitude to the organizers, Roman Bezrukavnikov, Pavel Etingof, George Lusztig, Monica Nevins, and Peter Trapa, for their warm hospitality during the stimulating conference.

This work was partially supported by Grant-in-Aid for Scientific Research (A) (25247006), Japan Society for the Promotion of Science.

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Kavli IPMU (WPI) and Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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