Remarks on the theta correspondence over finite fields

Dongwen Liu and Zhicheng Wang

Vol. 306 (2020), No. 2, 587–609

DOI: 10.2140/pjm.2020.306.587

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