Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 4

Special Issue: In Memory of Prof. Bertram Kostant

Guest Editors: Shrawan Kumar, Lizhen Ji, and Kefeng Liu

Branching laws for discrete series of some affine symmetric spaces

Pages: 1291 – 1320

DOI:  https://dx.doi.org/10.4310/PAMQ.2021.v17.n4.a4

Authors

Bent Ørsted (Department of Mathematics, Aarhus University, Aarhus, Denmark)

Birgit Speh (Department of Mathematics, Cornell University, Ithaca, New York, U.S.A.)

Abstract

In this paper we study branching laws for certain unitary representations. This is done on the smooth vectors via a version of the period integrals, studied in number theory, and also closely connected to the symmetry-breaking operators, introduced by T. Kobayashi. We exhibit non-vanishing symmetry breaking operators for the restriction of a representation $\Pi$ in the discrete spectrum for real hyperboloids to representations of smaller orthogonal groups. In the last part we discuss some conjectures for the restriction of representations in Arthur packets containing the representation $\Pi$ and the corresponding Arthur–Vogan packets to smaller orthogonal groups; these are inspired by the Gross–Prasad conjectures.

2010 Mathematics Subject Classification

Primary 22E46. Secondary 22E30, 22E45, 22E50, 53C30.

Dedicated to B. Kostant.

The research of B. Speh was partially supported by NSF grant DMS-1500644.

Received 17 July 2019

Accepted 14 October 2019

Published 22 December 2021