Comptage de repr\'esentations cuspidales congruentes

Abstract

Let $F$ be a non-Archimedean locally compact field of residue characteristic $p$, $G$ be an inner form of $GL_n(F)$, $n\ge1$, and $\ell$ be a prime number different from $p$. We give a numerical criterion for an integral $\ell$-adic irreducible cuspidal representation $\tilde\rho$ of $G$ to have a super\-cuspidal irreducible reduction mod $\ell$, by counting inertial classes of cuspidal representations that are congruent to the inertial class of $\tilde\rho$, generalizing results by Vign{\'e}ras and Dat. In the case the reduction mod $\ell$ of $\tilde\rho$ is not super\-cuspidal irreducible, we show that this counting argument allows us to compute its length and the size of the supercuspidal support of its irreducible components. We define an invariant $w(\tilde\rho)\ge1$ | the product of this length by this size | which is expected to behave nicely through the local Jacquet-Langlands correspondence. Given an $\ell$-modular irreducible cuspidal representation $\rho$ of $G$ and a positive integer $a$, we give a criterion for the existence of an integral $\ell$-adic irreducible cuspidal representation $\tilde\rho$ of $G$ such that its reduction mod $\ell$ contains $\rho$ and has length $a$. This allows us to obtain a formula for the cardinality of the set of reductions mod $\ell$ of inertial classes of $\ell$-adic irreducible cuspidal representations $\tilde\rho$ with given depth and invariant $w$. These results are expected to be useful to prove that the local Jacquet-Langlands correspondence preserves congruences mod $\ell$.