Rigid Inner Twists
A rigid inner twist (Kaletha) is the cocycle-level counterpart of a rigid inner form: a 1-cocycle valued in a "Galois gerbe" \(\mathcal{E}_G\) that rigidifies the central data of an inner twist. Rigid inner twists are the foundation of modern local Langlands across inner forms.
Setup
Let \(F\) be a local field with absolute Galois group \(\Gamma\), and \(G\) a connected reductive group over \(F\). Pure inner twists are \(Z^1(F, G)\), but the natural map \[ H^1(F, G) \;\to\; H^1(F, G_{\mathrm{ad}}) \] is not always surjective — some inner forms have no pure inner twist representative. Kaletha's solution:
Galois gerbe \(\mathcal{E}_G\)
Kaletha constructs a "Galois gerbe" \(\mathcal{E}_G\) — a sheaf of groupoids over the étale site of \(\mathrm{Spec}\, F\) — fitting into an exact sequence \[ 1 \;\to\; \mathcal{C} \;\to\; \mathcal{E}_G \;\to\; G \;\to\; 1, \] where \(\mathcal{C}\) is the "canonical band" coming from the pro-finite inverse system of finite multiplicative-type subgroups of \(Z(G^{\mathrm{sc}})\) (the centre of the simply connected cover).
\(\mathcal{C}\) is engineered so that \(H^2(F, \mathcal{C})\) absorbs the obstructions to lifting inner forms to pure inner twists.
Rigid inner twists
A rigid inner twist is a class in \[ H^1(F, \mathcal{E}_G) \;:=\; H^1_{\mathrm{rig}}(\Gamma, \mathcal{E}_G), \] the rigid cohomology of \(\mathcal{E}_G\). The natural projection \(\mathcal{E}_G \twoheadrightarrow G\) gives a map \[ H^1(F, \mathcal{E}_G) \;\to\; H^1(F, G), \] and Kaletha's main theorem says this map is surjective and canonically maps onto all inner twists modulo precisely controlled central data.
Properties
- Surjection onto inner forms: the composite \(H^1(F, \mathcal{E}_G) \to H^1(F, G_{\mathrm{ad}})\) is surjective; every inner form has a rigid representative.
- Refinement of pure: pure inner twists embed into rigid inner twists as \(H^1(F, G) \hookrightarrow H^1(F, \mathcal{E}_G)\) via the natural quotient.
- Kottwitz formula: \(H^1(F, \mathcal{E}_G)\) is computable via Kaletha's local Tate-Nakayama duality (an extension of classical Tate-Nakayama).
Examples
Quasi-split \(G\)
Trivial rigid inner twist: the identity cocycle. Rigid inner twists of \(G\) parametrise all inner forms (with central data) starting from the quasi-split form.
Non-quasi-split inner forms
For inner forms not arising from pure inner twists (e.g.\ certain non-split forms of \(\mathrm{Spin}\) groups): rigid inner twists provide the cocycle representatives that ordinary \(H^1(F, G)\) misses.
Connection with Kottwitz isocrystal formalism
Kaletha's rigid inner twist formalism is closely related to the Kottwitz isocrystal formalism for inner forms. There is a natural map \[ H^1(F, \mathcal{E}_G) \;\to\; B(G)_{\mathrm{basic}}, \] identifying rigid inner twists with basic \(\sigma\)-conjugacy classes (modulo central character data).
This gives a uniform framework comparing:
- Local Langlands (parameters into \({}^L G\));
- Inner forms (\(H^1(F, G_{\mathrm{ad}})\));
- Rigid inner twists (\(H^1(F, \mathcal{E}_G)\));
- Kottwitz set (\(B(G)\)).
Role in Langlands
- Canonical L-packet bijection: \(\Pi_\varphi^{\mathrm{rig}}(G^{\xi}(F)) \cong \mathrm{Irr}(S_\varphi^+)\) uses rigid inner twists \(\xi\) as the parametrising data.
- Endoscopic transfer factors: the precise normalisation across inner forms requires rigid inner twists.
- Global Langlands: Kaletha (2018) extends to global rigid inner forms, with a global Galois gerbe and a "Hasse-principle-like" exact sequence.
References
- Kaletha, Rigid inner forms of real and p-adic groups (Annals 2016).
- Kaletha, Rigid inner forms vs isocrystals (J.\ EMS 2018).
- Kaletha, Global rigid inner forms and multiplicities of discrete automorphic representations (Invent.\ Math.\ 2018).