Local transfer for quasi-split classical groups and congruences mod $\ell$ (with an appendix by Guy Henniart)

Abstract

Let $G$ be the group of rational points of a quasi-split $p$-adic special orthogonal, symplectic or unitary group for some odd prime number $p$. Following Arthur and Mok, there are an integer $N⩾1$, a $p$-adic field $E$ and a local functorial transfer from isomorphism classes of irreducible smooth complex representations of $G$ to those of ${\mathrm{GL}}_N(E)$. By fixing a prime number $\ell$ different from $p$ and an isomorphism between the field of complex numbers and an algebraic closure of the field of $\ell$-adic numbers, we obtain a transfer map between representations with $\ell$-adic coefficients. Now consider a cuspidal irreducible $\ell$-adic representation $\pi$ of $G$: we can define its reduction mod $\ell$, which is a semisimple smooth representation of $G$ of finite length, with coefficients in a field of characteristic $\ell$. Let ${\pi }^{\prime }$ be a cuspidal irreducible $\ell$-adic representation of $G$ whose reduction mod $\ell$ is isomorphic to that of $\pi$. We prove that the transfers of $\pi$ and ${\pi }^{\prime }$ have reductions mod $\ell$ which may not be isomorphic, but which have isomorphic supercuspidal supports. When $G$ is not the split special orthogonal group ${\mathrm{SO}}_2$, we further prove that the reductions mod $\ell$ of the transfers of $\pi$ and ${\pi }^{\prime }$ share a unique common generic component.