Lifting banal representations of classical groups

TL;DR

This paper proves that for banal primes, all smooth irreducible mod $ar{bF}_ell$ representations of classical groups over local fields can be lifted to characteristic zero, with applications to Howe duality.

Contribution

It establishes the liftability of mod $ar{bF}_ell$ representations for banal primes and extends results to more classical groups, also proving Howe duality in this context.

Findings

Every smooth irreducible mod $ar{bF}_ell$ representation admits a lift to characteristic zero.

Results apply to classical groups of symplectic, orthogonal, or unitary type.

Proves Howe duality in the strongly banal case for certain dual pairs.

Abstract

Let $\mathrm{G}$ be a symplectic or a split orthogonal group over a local non-archimedean field $\mathrm{F}$. A prime $\ell$ is called banal with respect to $\mathrm{G}$ if it does not divide the cardinality of the $k$-points of $\mathrm{G}$, where $k$ is the residue field of $\mathrm{F}$. In this paper we show that for every banal prime $\ell$, any smooth irreducible $\overline{\mathbb{F}}_\ell$-representation of $\mathrm{G}(\mathrm{F})$ admits a lift to $\overline{\mathbb{Q}}_\ell$. We also state similar results for more general classical groups of symplectic, orthogonal or unitary type. As an application we prove Howe-duality in the strongly banal case for symplectic-orthogonal or unitary dual pairs.