Local Langlands in families: The banal case

Abstract We state a conjecture, local Langlands in families , connecting the centre of the category of smooth representations on Z ⁢ [ q − 1 ] \mathbb{Z}[\sqrt{q}^{-1}] -modules of a quasi-split 𝑝-adic group G \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 G \mathrm{G} over Z ⁢ [ q − 1 ] \mathbb{Z}[\sqrt{q}^{-1}] , and the endomorphisms of a Gelfand–Graev representation for G \mathrm{G} . For a class of classical 𝑝-adic groups (symplectic, unitary, or odd special orthogonal groups), we prove this conjecture after inverting an integer depending only on G \mathrm{G} . Along the way, we show that the local Langlands correspondence for classical 𝑝-adic groups (1) preserves integrality of ℓ-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 ℂ fixing q \sqrt{q} .