Rigid inner twist - Knowlpedia

Definition

Let $F$ be a local field, let $G^*$ be a quasi-split connected reductive group, and choose a finite central subgroup $Z\subset G^*$ . A rigid inner twist is an inner twist

\psi:G^*_{\overline F}\xrightarrow{\sim}G_{\overline F}

together with a cocycle $z$ in Kaletha's rigid cohomology set $Z^1(u\to W,Z\to G^*)$ , whose image in $Z^1(\Gamma_F,G^*_{\mathrm{ad}})$ , where $\Gamma_F$ is the absolute Galois group, is the cocycle $\sigma\mapsto\psi^{-1}\sigma(\psi)$ defining the underlying inner twist.

The extra cocycle is the rigidification. Forgetting it retains the inner form but loses information needed for canonical packet pairings.

Why the rigidification is used

For quasi-split groups a Whittaker datum normalizes transfer factors and the internal parametrization of an L-packet. For arbitrary inner forms, a rigid inner twist supplies the additional cohomological datum needed to extend those normalizations and to pair representations with the appropriate component group.

Different rigidifications of the same underlying inner twist need not give identical labels. Statements of the refined local Langlands correspondence must therefore include the rigidifying data, not merely the isomorphism class of $G$ .

Framework warning

Pure inner twists, rigid inner twists, and extended pure inner twists are related but not interchangeable frameworks. The notation for the gerbe $u\to W$ and the allowed central subgroup $Z$ also varies with the source.

References

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