Local Langlands in families: The banal case

We state a conjecture, local Langlands in families, connecting the centre of the category of smooth representations on $\mathbb{Z}[\sqrt{q}^{-1}]$-modules of a quasi-split $p$-adic group $\mathrm{G}$ (where $q$ is the cardinality of the residue field of the underlying local field), the ring of global functions on the stack of Langlands parameters for $\mathrm{G}$ over $\mathbb{Z}[\sqrt{q}^{-1}]$, and the endomorphisms of a Gelfand-Graev representation for $\mathrm{G}$. For a class of classical $p$-adic groups (symplectic, unitary, or split odd special orthogonal groups), we prove this conjecture after inverting an integer depending only on $\mathrm{G}$. Along the way, we show that the local Langlands correspondence for classical $p$-adic groups (1) preserves integrality of $\ell$-adic representations; (2) satisfies an "extended" (generic) packet conjecture; (3) is compatible with parabolic induction up to semisimplification (generalizing a result of Moussaoui), hence induces a semisimple local Langlands correspondence; and (4) the semisimple correspondence is compatible with automorphisms of $\mathbb{C}$ fixing $\sqrt{q}$.