  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.
