Regular Bernstein blocks

Publication date: 22 September 2019

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)

. Bernstein blocks whose cuspidal support involves a regular supercuspidal representation are called

e x t i t r e g u l a r

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)

is isomorphic to the Bernstein center of a regular depth-zero Bernstein block of

G^{0} (F)

, where

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.

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