Alberto Minguez (Universität Wien): Local transfer for quasi-split classical groups and congruences mod ℓ

Let $\pi$ and $\pi’$ be two $\ell$-adic irreducible cuspidal representations of a quasi-split classical group $G$, and let $\sigma$ and $\sigma’$ be their respective Langlands’s transfers to $GL(N)$. Assume that $\pi$ and $\pi’$ are integral, and denote by $r_\ell(\pi)$ and $r_\ell(\pi')$ their reductions modulo $\ell$. We will show that if $r_\ell(\pi) \leq r_\ell(\pi’)$, then $r_\ell(\sigma$) and $r_\ell(\sigma’)$ have a unique generic irreducible component in common. This is joint work with Vincent Sécherre.