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

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.