#### Vol. 9, No. 5, 2016

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Lifting representations of finite reductive groups: a character relation

### Jeffrey D. Adler, Michael Cassel, Joshua M. Lansky, Emma Morgan and Yifei Zhao

Vol. 9 (2016), No. 5, 805–812
##### Abstract

Given a connected reductive group $\stackrel{˜}{G}$ over a finite field $k$, and a semisimple $k$-automorphism $\epsilon$ of $\stackrel{˜}{G}$ of finite order, let $G$ denote the connected part of the group of $\epsilon$-fixed points. Two of the authors have previously shown that there exists a natural lifting from series of representations of $G\left(k\right)$ to series for $\stackrel{˜}{G}\left(k\right)$. In the case of Deligne–Lusztig representations, we show that this lifting satisfies a character relation analogous to that of Shintani.

##### Keywords
finite reductive groups, Deligne–Lusztig representations, liftings, character relations
Primary: 20C33
Secondary: 20G40
##### Milestones
Revised: 1 November 2015
Accepted: 3 November 2015
Published: 25 August 2016

Communicated by Ken Ono
##### Authors
 Jeffrey D. Adler Department of Mathematics and Statistics American University Washington, DC 20016 %-8050 United States Michael Cassel Columbia University New York, NY 10027 %-7297 United States Joshua M. Lansky Department of Mathematics and Statistics American University Washington, DC 20016 %-8050 United States Emma Morgan Office of Institutional Research and Evaluation Tufts University Medford, MA 02155 United States Yifei Zhao Department of Mathematics Harvard University Cambridge, MA 02138 United States