#### Vol. 15, No. 5, 2021

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The universal unramified module for $\mathrm{GL}(n)$ and the Ihara conjecture

### Gilbert Moss

Vol. 15 (2021), No. 5, 1181–1212
##### Abstract

Let $F$ be a finite extension of ${ℚ}_{p}$. Let $W\left(k\right)$ denote the Witt vectors of an algebraically closed field $k$ of characteristic $\ell$ different from $p$, and let $\mathsc{𝒵}$ be the spherical Hecke algebra for ${GL}_{n}\left(F\right)$ over $W\left(k\right)$. Given a Hecke character $\lambda :\mathsc{𝒵}\to R$, where $R$ is an arbitrary $W\left(k\right)$-algebra, we introduce the universal unramified module ${\mathsc{ℳ}}_{\lambda ,R}$. We show ${\mathsc{ℳ}}_{\lambda ,R}$ embeds in its Whittaker space and is flat over $R$, resolving a conjecture of Lazarus. It follows that ${\mathsc{ℳ}}_{\lambda ,k}$ 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 ${\mathsc{ℳ}}_{\lambda ,k}$ reduces the Ihara conjecture to the statement that $V$ is generic.

##### Keywords
Ihara's lemma, universal unramified module, mod-$\ell$ automorphic forms
##### Mathematical Subject Classification
Primary: 11F33
Secondary: 22E50, 22E55
##### Milestones
Received: 7 February 2020
Revised: 21 September 2020
Accepted: 13 November 2020
Published: 30 June 2021
##### Authors
 Gilbert Moss Department of Mathematics University of Utah Salt Lake City, UT United States