The universal unramified module for GL(n) and the Ihara conjecture

TL;DR

This paper constructs a universal unramified module for GL(n) over local fields, proves its properties, and applies these results to reduce a conjecture related to automorphic forms and the Ihara conjecture.

Contribution

It introduces the universal unramified module for GL(n), proves its flatness and embedding into its Whittaker space, and connects this to the Ihara conjecture in automorphic forms.

Findings

The universal unramified module embeds into its Whittaker space.

The module is flat over the base ring.

Application to reducing the Ihara conjecture to a genericity statement.

Abstract

Let $F$ be a finite extension of $\mathbb{Q}_p$. Let $W(k)$ denote the Witt vectors of an algebraically closed field $k$ of characteristic $\ell$ different from $p$ and $2$, and let $\mathcal{Z}$ be the spherical Hecke algebra for $GL_n(F)$ over $W(k)$. Given a Hecke character $\lambda:\mathcal{Z}\to R$, where $R$ is an arbitrary $W(k)$-algebra, we introduce the universal unramified module $\mathcal{M}_{\lambda,R}$. We show $\mathcal{M}_{\lambda,R}$ embeds in its Whittaker space and is flat over $R$, resolving a conjecture of Lazarus. It follows that $\mathcal{M}_{\lambda,k}$ has the same semisimplification as any unramified principle series with Hecke character $\lambda$. In the setting of mod-$\ell$ automorphic forms, Clozel, Harris, and Taylor formulate a conjectural analogue of Ihara's lemma. It predicts that every irreducible submodule of a certain cyclic module $V$ of mod-$\ell$ automorphic forms is generic. Our result on the Whittaker model of $\mathcal{M}_{\lambda,k}$ reduces the Ihara conjecture to the statement that $V$ is generic.