Langlands-Shahidi gamma factors and local transfer in families

Let $G$ be a quasi-split $p$-adic group, defined over a non-archimedean local field of residue cardinality $q$, and $R$ be a $\mathbb{Z}[{\sqrt{q}}^{-1},{\mu }_{4p^{\mathrm{\infty }}}]$-algebra. We generalize the constructions of Shahidi local coefficients and Langlands-Shahidi gamma factors to the setting of co-Whittaker $R$-representations. For classical $p$-adic groups, we prove there is a unique integral morphism of endomorphism rings of Gelfand–Graev representations, interpolating the (standard) local generic transfer to a general linear group. This yields a local transfer morphism in families, which allows us to deduce properties of the local $\mathrm{\ell }$-adic generic transfer and congruences modulo primes $\mathrm{\ell }\mathrm{\ne }p$.