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

In the setting of mod- 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- automorphic forms is generic. Our result on the Whittaker model of λ,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