Vol. 306, No. 2, 2020

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Remarks on the theta correspondence over finite fields

Dongwen Liu and Zhicheng Wang

Vol. 306 (2020), No. 2, 587–609
Abstract

S.-Y. Pan decomposed the uniform projection of the Weil representation of a finite symplectic-odd orthogonal dual pair, in terms of Deligne–Lusztig virtual characters, assuming that the order of the finite field is large enough. We use Pan’s decomposition to study the theta correspondence for dual pairs of this kind, following the approach of Adams and Moy and Aubert, Michel and Rouquier. Our results give the theta correspondence between unipotent representations and certain quadratic unipotent representations.

Keywords
theta correspondence, finite classical groups, Deligne–Lusztig characters, unipotent cuspidal representations
Mathematical Subject Classification 2010
Primary: 20C33
Secondary: 22E50
Milestones
Received: 16 August 2019
Revised: 13 December 2019
Accepted: 2 January 2020
Published: 13 July 2020
Authors
Dongwen Liu
School of Mathematical Sciences
Zhejiang University
Hangzhou
China
Zhicheng Wang
School of Mathematical Sciences
Zhejiang University
Hangzhou
China