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

Abstract

Let $F$ be a finite extension of ${\mathbb{Q}}_p$. Let $W\left(k\right)$ denote the Witt vectors of an algebraically closed field $k$ of characteristic $\ell$ different from $p$, and let $\mathcal{𝒵}$ be the spherical Hecke algebra for ${GL}_n\left(F\right)$over $W\left(k\right)$. Given a Hecke character $\lambda :\mathcal{𝒵}\to R$, where $R$ is an arbitrary $W\left(k\right)$-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 principal series with Hecke character $\lambda$.

In the setting of mod-$\ell$ automorphic forms Clozel, Harris, and Taylor (2008) formulated 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.