Self-dual cuspidal representations

Let G be a connected reductive group defined over a finite or a non-archimedean local field F. We show that G(F) admits cuspidal representations when F is finite and supercuspidal representations when F is non-archimedean local. We also determine precisely when G(F) admits self-dual representations. For the results on self-duality, we assume some hypothesis on G. These hypotheses disallow G to have certain small rank factors when the field (in case F is finite) or the residue field (in case F is non-archimedean local) is of cardinality ≤ 5. When F is non-archimedean local and G is ramified, these hypotheses impose some additional restrictions on G. This is a joint work with Jeff Adler.