A Deligne-Lusztig type correspondence for tame $p$-adic groups

arXiv2306.02093MaRDI QIDQ6439103

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Publication date: 3 June 2023

Abstract: We establish a generalization of the matrix simultaneous diagonalization theorem for disconnected reductive groups which relaxes both the semisimplicity condition and the commutativity condition. As an application, we prove the following basic results for mod $p$ Langlands parameters for quasi-split tame groups $G$ over a $p$-adic field $F$: (i.) All semisimple $L$-parameters $$ factor through the $L$-group of a maximal $F$-torus of $G$; (ii.) All semisimple mod $p$ $L$-parameters admit a de Rham lift of regular $p$-adic Hodge type; (iii.) A version of tame inertial local Langlands correspondnece: there exists a natural bijection between geometric conjugacy classes of inertial Deligne-Lusztig data and conjugacy classes of tame inertial types for $G$; (iv.) A group-theoretic description of irreducible components of the reduced Emerton-Gee stacks away from Steinberg parts: there exists a bijection between geometric conjugacy classes of based inertial Deligne-Lusztig data of niveau $1$ and isomorphism classes of parahoric Serre weights. We also generalize the definition of Herzig's explicit recipe for Serre weights (which is later extended by Gee-Herzig-Savitt and Le-Le Hung-Levin-Morra) for unramified groups to quasi-split tame groups.

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