Refined local Langlands correspondence

Let $G^*$ be a quasi-split connected reductive group over a local field, choose a Whittaker datum $\mathfrak w$ , and let $\varphi:L_F\to{}^LG^*$ be a relevant parameter. The refined local Langlands correspondence predicts a canonical internal parametrization of the compound packet of $\varphi$ by irreducible representations of the refined centralizer group:

\Pi_\varphi \;\longleftrightarrow\; \operatorname{Irr}\!\left(\pi_0(S_\varphi^+)\right).

Here the left side ranges over rigid inner twists of $G^*$ ; on the right, the central character of an irreducible representation records the inner form on which the corresponding packet member lives.

Quasi-split fiber

Restricting to $G^*(F)$ gives a parametrization by the appropriate center-quotiented component group. For tempered $\varphi$ , the conjecturally unique $\mathfrak w$ -generic member corresponds to the trivial representation.

Why rigidification is present

An ordinary inner twist does not retain enough cohomological information to normalize transfer factors and distinguish all packet members functorially. Pure inner twists suffice in some settings; rigid inner twists give a uniform framework for general connected reductive groups. The older notions of inner and pure inner forms are reviewed in Galois descent and forms.

Character identities

The parametrization is characterized not just as a bijection but through stable distributions and endoscopic character identities. The trivial component-group element gives a stable packet distribution, while other elements select endoscopic transfers. Dependence on the Whittaker datum and rigidifying cocycle is governed by explicit character twists.

Status

Refined correspondences are theorems for archimedean groups and many important nonarchimedean families, including broad classes of classical groups. The uniform statement above remains conjectural for a general reductive group.

References

Tasho Kaletha, “Representations of reductive groups over local fields,” §§2.2–2.3, 2022. arXiv.
Tasho Kaletha, “Rigid inner forms of real and $p$ -adic groups,” Annals of Mathematics 184 (2016), 559–632. DOI.