The $\ell$-modular Zelevinski involution

TL;DR

This paper extends the Zelevinski involution to modular representations of inner forms of GL(n,F), showing it preserves a unique irreducible component with the same cuspidal support, generalizing known complex case results.

Contribution

It introduces a new involution on irreducible modular representations of inner forms of GL(n,F), generalizing the Zelevinski involution to positive characteristic fields.

Findings

The involution preserves a unique irreducible component with the same cuspidal support.

The method applies to both local non-Archimedean fields and finite fields.

It generalizes the classical Zelevinski involution to modular representations.

Abstract

Let F be a non-Archimedean locally compact field with residual characteristic p, let G be an inner form of GL(n,F) for a positive integer n and let R be an algebraically closed field of characteristic different from p. When R has characteristic $\ell>0$, the image of an irreducible smooth R-representation $\pi$ of G by the Aubert involution need not be irreducible. We prove that this image (in the Grothendieck group of G) contains a unique irreducible term $\pi$* with the same cuspidal support as $\pi$. This defines an involution on the set of isomorphism classes of irreducible R-representations of G, that coincides with the Zelevinski involution when R is the field of complex numbers. The method we use also works for F a finite field of characteristic p, in which case we get a similar result.