Cuspidal Irreducible Representations of Connected Reductive p-adic Groups over Arbitrary Field
Let F be a non-archimedean field of residual characteristic p, G a connected reductive F-group and C a field of characteristic c. Cuspidal irreducible C-representations of the reductive p-adic group G (F) are totally mysterious if c = p, their existence is even known only when the characteristic of F is 0. But when c ≠ p, we conjecture that all cuspidal irreducible C-representations of G (F) are compactly induced from compact mod center open subgroups, because we can prove it in many cases.
All known examples of cuspidal irreducible complex representations of G (F) are of this form: cuspidal irreducible complex representations of G (F) of level 0 (Moy-Prasad, Morris), all cuspidal irreducible complex representations of G (F) if the semi-simple rank of G is 1 (Weissman), or (generalising Bushnell-Kutzko) if G = S L(n), or G is an inner form of G L(n), or G is a classical group or a quaternionic form of a classical group and p odd, or (generalising J.K. Yu), if G splits on a moderately ramified extension of F and p is prime to the order of the absolute Weyl group. The field of complex numbers has been replaced by an algebraically closed coefficient field of characteristic c ≠ p (many authors). In a work in progress with Henniart, we are able to drop the hypothesis that C is algebraically closed.