Self-dual cuspidal representations

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