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# Forum Mathematicum

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Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 30, Issue 1

# On two questions concerning representations distinguished by the Galois involution

Maxim Gurevich
/ Jia-Jun Ma
/ Arnab Mitra
• Corresponding author
• Department of Mathematics, Technion – Israel Institute of Technology, Haifa 3200003, Israel
• Email
• Other articles by this author:
Published Online: 2017-05-05 | DOI: https://doi.org/10.1515/forum-2016-0212

## Abstract

Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic 0. In this paper, we investigate two approaches which attempt to describe the irreducible smooth representations of ${\mathrm{GL}}_{n}\left(E\right)$ that are distinguished by its subgroup ${\mathrm{GL}}_{n}\left(F\right)$. One relates this class to representations which come as base change lifts from a quasi-split unitary group over F, while another deals with a certain symmetry condition. By characterizing the union of images of the base change maps, we show that these two approaches are closely related. Using this observation, we are able to prove a statement relating base change and distinction for ladder representations. We then produce a wide family of examples in which the symmetry condition does not impose ${\mathrm{GL}}_{n}\left(F\right)$-distinction, and thus exhibit the limitations of these two approaches.

Keywords: Distinguished representations

MSC 2010: 22E50; 11F70

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Revised: 2017-02-20

Published Online: 2017-05-05

Published in Print: 2018-01-01

Funding Source: Israel Science Foundation

Award identifier / Grant number: 756/12

Award identifier / Grant number: CUHK 405213

Funding Source: Israel Science Foundation

Award identifier / Grant number: 1138/10

Maxim Gurevich, partially supported by the ISF grant 756/12, and ERC StG grant 637912. Arnab Mitra, partially supported by postdoctoral fellowships funded by the Skirball Foundation via the Center for Advanced Studies in Mathematics at Ben-Gurion University of the Negev and the Department of Mathematics, Technion.

Citation Information: Forum Mathematicum, Volume 30, Issue 1, Pages 141–157, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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