Regular Bernstein blocks

Abstract

Abstract For a connected reductive group G defined over a non-archimedean local field F, we consider the Bernstein blocks in the category of smooth representations of G ⁢ ( F ) {G(F)} . Bernstein blocks whose cuspidal support involves a regular supercuspidal representation are called regular Bernstein blocks. Most Bernstein blocks are regular when the residual characteristic of F is not too small. Under mild hypotheses on the residual characteristic, we show that the Bernstein center of a regular Bernstein block of G ⁢ ( F ) {G(F)} is isomorphic to the Bernstein center of a regular depth-zero Bernstein block of G 0 ⁢ ( F ) {G^{0}(F)} , where G 0 {G^{0}} is a certain twisted Levi subgroup of G. In some cases, we show that the blocks themselves are equivalent, and as a consequence we prove the ABPS Conjecture in some new cases.