Let $F$ be a $p$-adic field and $\mathbf{G}$ a connected reductive group over $F$, assumed to be quasi-split. Let $\mathbf{T}$ be a maximal split torus of $\mathbf{G}$ and $\mathbf{P}=\mathbf{M}\mathbf{N}$ an $F$-parabolic with its Levi decomposition ($\mathbf{M}\supset \mathbf{T}$). Let $\sigma$ be an irreducible supercuspidal representation of $M=\mathbf{M}(F)$.
Suppose $w\in W=W(\mathbf{T},\mathbf{G})$ is an element in the (relative) Weyl group, such that $wM=M$, i.e. if $n_w\in N:=N_{\mathbf{G}}(\mathbf{T})(F)$ is a representative of $w\in W$, we have $n_w M n_w^{-1}=M$. Then in this case $w\sigma$ is also a supercuspidal of $M$. It's the isomorphism class of the representation defined by $$(w\sigma)(m):=\sigma(n_w^{-1}mn_w).$$
My question is: suppose $\varphi_\sigma:WD_F\rightarrow {^L M}$ is the Langlands parameter of $\sigma$, then how to write down the parameter of $w\sigma$?
If $\mathbf{G}$ is split over $F$, then $\widehat{M}={^LM}\subset {^LG}=\widehat{G} $ and $W$ is just the absolute Weyl group, which is canonically isomorphic to the Weyl group $\widehat{W}$ of $\widehat{G}$, so $w$ corresponds canonically to some $\widehat{w}\in\widehat{W}$, and $\widehat{w}\widehat{M}=\widehat{M}$. I guess that the paramter of $w\sigma$ in this case is just the $\widehat{M}$-conjugacy class of $\widehat{w}\circ \varphi_{\sigma}$, i.e. $x\mapsto \widehat{w}(\varphi_{\sigma}(x))$. (Is this the correct answer?)
However, if $\mathbf{G}$ is not split, then the relative Weyl group $W$ is in general a subquotient of the absolute Weyl group. I guess that in this case I can take an arbitrary pullback of $w$ in the absolute Weyl group of $\mathbf{G}$, and this gives an element $\widehat{w}$ of the absolute Weyl group of $\widehat{G}$. Then the parameter of $w\sigma$ should be the $\widehat{M}$-conjugacy class of $x\mapsto \widehat{w}(\varphi_{\sigma}(x))$ (and the class doesn't depend on the choice of $\widehat{w}$). Am I right?
(Background: In local representation theory and harmonic analysis, while dealing with the parabolic induction, the condition whether $w\sigma\simeq \sigma$ or not for some $w$ is usually a crucial condition ("ramified") (especially when $\mathbf{P}$ is maximal and $w$ is the longest element in $W^M\backslash W^G/W^M$, cf. Silberger, Introduction to Harmonic Analysis on Reductive $p$-adic Groups, 5.4.2.2, 5.4.2.3, etc.). So it seems natural to consider how to translate this condition in the context of parameters, but I didn't find any related facts in the literature. )
Thanks a lot in advance for any help!
