# 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 The field of complex numbers has been replaced by an algebraically closed coefficient field of characteristic *c* ≠ *p* (many authors).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.