The local Langlands correspondence in families and Ihara's lemma for U(n)

Abstract

The goal of this paper is to reformulate the conjectural Ihara lemma for U(n) in terms of the local Langlands correspondence in families π˜Σ(⋅), as developed by Emerton and Helm. The reformulation roughly takes the following form. Suppose we are given an irreducible mod ℓ Galois representation r¯=r¯m, which is modular of full level (and small weight), and a finite set of places Σ, none of which divides ℓ. Then π˜Σ(rm) exists, and has a global realization as a localized module of ℓ-integral algebraic modular forms, where rm is the universal deformation of r¯ of type Σ. This is unconditional for n=2, where Ihara's lemma is an almost trivial consequence of the strong approximation theorem. We emphasize that throughout we will work with banal primes ℓ>n. That is, those for which #k(v)i≢1 (mod ℓ), where 1≤i≤n, for v∈Σ. A weakening of this assumption (quasi-banal) is ubiquitous in [CHT] (particularly in their Chapter 5 on the non-minimal case and Ihara), where the mod ℓ representation theory of GLn(Fv˜) plays a pivotal role. We will need actual banality, to ensure the functors we define are exact. Banality also removes many of the subtleties in the Emerton–Helm correspondence. For example, in yet unpublished work [Hel13], Helm shows that local Langlands in families do exist in the banal case, although we do not use his result. We should stress that our main result has been common knowledge among experts for some time, but its proof has never appeared in print.