Lifting banal representations of classical groups

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.