The rigid refined local Langlands correspondence for rigid inner twists

Fix a connected quasi-split reductive group $G$ over a nonarchimedean local field, a finite central subgroup $Z$, a tempered $L$-parameter $L_F\to{}^{L}G$, and a Whittaker datum $\mathfrak{w}$ for $G$. Let $\Pi_\varphi^Z$ be the isomorphism classes of tempered representations $((\mathcal{T},\bar h),\pi)$ of $Z$-rigid inner twists of $G$, let $S_\varphi^+$ be the specified preimage of the centralizer of $\varphi$, and let $Z(\widehat{G/Z})^+$ be the preimage of $Z(\widehat G)^\Gamma$ in $\widehat{G/Z}$.

Rigid refined local Langlands conjecture. There is a bijection, depending on $\mathfrak{w}$,

$$\Pi_\varphi^Z\xrightarrow{\iota_{\mathfrak w}}\operatorname{Irr}(\pi_0(S_\varphi^+)),$$

which fits into a commutative diagram

$$\begin{array}{ccc} \Pi_\varphi^Z&\xrightarrow{\iota_{\mathfrak w}}&\operatorname{Irr}(\pi_0(S_\varphi^+))\\ \downarrow&&\downarrow\\ H^1(\mathcal E,Z\to G)&\longrightarrow&\pi_0(Z(\widehat{G/Z})^+)^*. \end{array}$$

Here the bottom map is the gerbe-theoretic analogue of the Tate–Nakayama isomorphism, the left map extracts the underlying torsor, and the right map is induced by central characters.

This is the rigid refinement of the local Langlands correspondence for inner forms. The cited works provide the relevant constructions and analogues in important settings, while the stated compatibility is presented here as a conjectural formulation.