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

We establish a "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 concerning mod $p$ Langlands parameters for quasi-split tame groups $G$ over a $p$-adic field $F$: (1) semisimple $L$-parameters $\operatorname{Gal}_F\to {^ L\!G}({\bar{\mathbb{F}}_p})$ factor through the $L$-group of a maximal $F$-torus of $G$; (2) All semisimple mod $p$ $L$-parameters admit a de Rham lift of regular $p$-adic Hodge type; (3) A version of tame inertial local Langlands correspondnece; and (4) A group-theoretic description of irreducible components of the reduced Emerton-Gee stacks away from Steinberg parts. We also propose generalizations of the explicit recipe for Serre weights (after Herzig) and the geometric Breuil-M\'ezard for tame groups.