Local Langlands in families: The banal case

TL;DR

This paper formulates a conjecture relating the center of smooth representations of p-adic groups to Langlands parameters, proves it for certain classical groups after inverting an integer, and explores properties of the local Langlands correspondence.

Contribution

It introduces a conjecture connecting representation centers and Langlands parameters, proves it for classical groups after inverting an integer, and establishes key properties of the local Langlands correspondence.

Findings

Proves the conjecture for classical p-adic groups after inverting an integer.

Shows the local Langlands correspondence preserves integrality of dic representations.

Demonstrates compatibility of the correspondence with automorphisms of  fixing q.

Abstract

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 $\sqrt{q}$.