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 Zq^-1-modules of a quasi-split p-adic group 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 over Zq^-1, and the endomorphisms of a Gelfand-Graev representation for 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 G. Along the way, we show that the local Langlands correspondence for classical p-adic groups (1) preserves integrality of -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.