Self-dual cuspidal representations

Abstract

Let G G be a connected reductive group over a finite field f \mathfrak {f} of order q q . When q ≤ 5 q\leq 5 , we make further assumptions on G G . Then we determine precisely when G ( f ) G(\mathfrak {f}) admits irreducible, cuspidal representations that are self-dual, of Deligne-Lusztig type, or both. Finally, we outline some consequences for the existence of self-dual supercuspidal representations of reductive p p -adic groups.