Characterisation of the poles of the \(\ell \)-modular Asai L-factor. (Caractérisation des pôles du facteur d’Asai \(\ell\)-modulaire.) (English. French summary) Zbl 1473.22014
Summary: Let \(F/F_0\) be a quadratic extension of non-archimedean local fields of odd residual characteristic, set \(G=\mathrm{GL}_n(F)\), \(G_0=\mathrm{GL}_n(F_0)\) and let \(\ell\) be a prime number different from the residual characteristic of \(F\). For a complex cuspidal representation \(\pi\) of \(G\), the Asai \(L\)-factor \(L_{\text{As}}(X,\pi)\) has a pole at \(X=1\), if and only if \(\pi\) is \(G_0\)-distinguished. In this paper, we solve the problem of characterising the occurrence of a pole at \(X=1\) of \(L_{\text{As}}(X,\pi)\) when \(\pi\) is an \(\ell\)-modular cuspidal representation of \(G\); we show that \(L_{\text{As}}(X,\pi)\) has a pole at \(X=1\), if and only if \(\pi\) is a relatively banal distinguished representation, namely \(\pi\) is \(\mathrm{G}_{\circ}\)-distinguished but not \(\vert\det_{F_0}\)-distinguished. This notion turns out to be an exact analogue for the symmetric space \(G/G_0\) of Mínguez and Sécherre’s notion of banal cuspidal \(\overline{\mathbb{F}_\ell}\)-representation of \(G_0\).
Along the way, we compute the Asai \(L\)-factor of all cuspidal \(\ell\)-modular representations of \(G\) in terms of type theory and prove new results concerning lifting and reduction modulo \(\ell\) of distinguished cuspidal representations. Finally, we determine when the natural \(G_0\)-period on the Whittaker model of a distinguished cuspidal representation of \(G\) is non-zero.
Along the way, we compute the Asai \(L\)-factor of all cuspidal \(\ell\)-modular representations of \(G\) in terms of type theory and prove new results concerning lifting and reduction modulo \(\ell\) of distinguished cuspidal representations. Finally, we determine when the natural \(G_0\)-period on the Whittaker model of a distinguished cuspidal representation of \(G\) is non-zero.
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
Keywords:
Asai L-factor; distinguished representations of \(p\)-adic groups; modular representations of \(p\)-adic groupsReferences:
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