According to the Harish-Chandra philosophy, cuspidal representations are the basic building blocks in the representation theory of finite reductive groups. Similarly for supercuspidal representations of p-adic groups. Self-dual representations play a special role in the study of parabolic induction. Thus, it is of interest to know whether self-dual (super)cuspidal representations exist. With a few exceptions involving some small fields, I will show precisely when a finite reductive group has irreducible cuspidal representations that are self-dual, of Deligne-Lusztig type, or both. Then I will look at implications for the existence of irreducible, self-dual supercuspidal representations of p-adic groups. This is joint work with Manish Mishra.
