The universal unramified module for GL(n) and the Ihara conjecture
| Autor: | Moss, Gilbert |
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| Rok vydání: | 2019 |
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| Zdroj: | Alg. Number Th. 15 (2021) 1181-1212 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.2140/ant.2021.15.1181 |
| Popis: | 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. Comment: To appear in Algebra Number Theory. 29 pages |
| Databáze: | arXiv |
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