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

Moss, Gilbert

doi:10.2140/ant.2021.15.1181

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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

DOI: 10.2140/ant.2021.15.1181

Abstract

Let $F$ be a finite extension of $ℚ_{p}$ . Let $W (k)$ denote the Witt vectors of an algebraically closed field $k$ of characteristic $ℓ$ different from $p$ , and let $𝒵$ be the spherical Hecke algebra for ${GL}_{n} (F)$ over $W (k)$ . Given a Hecke character $λ : 𝒵 \to R$ , where $R$ is an arbitrary $W (k)$ -algebra, we introduce the universal unramified module $ℳ_{λ, R}$ . We show $ℳ_{λ, R}$ embeds in its Whittaker space and is flat over $R$ , resolving a conjecture of Lazarus. It follows that $ℳ_{λ, k}$ has the same semisimplification as any unramified principal series with Hecke character $λ$ .

In the setting of mod- $ℓ$ 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- $ℓ$ automorphic forms is generic. Our result on the Whittaker model of $ℳ_{λ, k}$ reduces the Ihara conjecture to the statement that $V$ is generic.

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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

Vol. 15, No. 5, 2021