Rigid Inner Forms
A rigid inner form (Kaletha) is an inner form of a connected reductive group together with a "rigidifying" cohomology class in a Galois gerbe. Rigid inner forms refine ordinary inner forms — and the intermediate pure inner twists — so that the local Langlands parametrisation across inner forms is canonical.
Motivation
For an inner form \(G^*\) of a quasi-split \(G\), the L-packet of an L-parameter \(\varphi\) is conjecturally indexed by characters of the S-group \(S_\varphi\). However:
- Pure inner twists: work when \(G\) has enough central cohomology to lift cocycles cleanly.
- General inner forms: lifting can fail, and the parametrisation is non-canonical.
Kaletha's solution (2016): replace \(G\) by a "Galois gerbe" \(\mathcal{G}\) whose cohomology \(H^1(F, \mathcal{G})\) surjects onto all inner forms (with kernel exactly the rigidification data).
Definition (sketch)
Let \(F\) be a local field with absolute Galois group \(\Gamma\). For a connected reductive \(G\) over \(F\), Kaletha defines:
- A Galois gerbe \(\mathcal{E}\) over \(\Gamma\) — a 2-cocycle in finite group cohomology controlling the "central kernel" data;
- Group cohomology \(H^1(F, \mathcal{E})\) computed with cocycle valued in a suitable inverse limit of fpqc gerbes;
- A rigid inner form is a class in \(H^1(F, \mathcal{E}_G)\), with a canonical surjection \[ H^1(F, \mathcal{E}_G) \;\twoheadrightarrow\; H^1(F, G_{\mathrm{ad}}). \]
The kernel of this surjection consists of classes that are trivial inner forms but carry non-trivial "rigid" data — used to refine the local Langlands parametrisation.
Properties
- Surjectivity: every inner form lifts to a rigid inner form (with multiple lifts possible).
- Refinement of pure: rigid inner forms refine pure inner twists. For \(G\) with simply-connected derived: pure = rigid; for non-simply-connected: rigid strictly contains pure.
- Galois functoriality: rigid inner forms are functorial in morphisms of reductive groups (with appropriate base-change conditions).
Examples
\(\mathrm{SL}_2\) over local field
\(G = \mathrm{SL}_2 / \mathbb{Q}_p\):
- \(Z(G) = \mu_2\); \(\pi_1(G) = 0\).
- Pure inner twists = inner forms: just two (split and quaternionic).
- Rigid inner forms: same two, but with explicit central data lifted via a Galois gerbe.
\(\mathrm{PGL}_n\)
\(G = \mathrm{PGL}_n / F\) (\(F\) a local field):
- \(\pi_1(G) = \mathbb{Z}/n\); so non-pure cocycles exist.
- Rigid inner forms strictly refine pure ones; the refinement parametrises central characters of L-packet members.
Twisted endoscopy
For twisted endoscopy (e.g.\ on \(\mathrm{GL}_n\) with outer automorphism): rigid inner forms are the natural setting to specify transfer factors that depend on central characters.
Local Langlands with rigid inner forms
Theorem/conjecture (Kaletha). For each rigid inner form \(\xi : G^* \to G\) (= rigid representative of an inner form), there is a canonical bijection: \[ \Pi_\varphi^{\mathrm{rigid}}(G^\xi) \;\cong\; \mathrm{Irr}(S_\varphi^+), \] where \(\Pi_\varphi^{\mathrm{rigid}}\) is the L-packet of rigid representations of \(G^{\xi}(F)\), and \(S_\varphi^+\) is the extended component group.
This is the most refined form of the local Langlands correspondence known, encompassing classical, pure, and rigid versions.
Computational tools
Cohomology of Galois gerbes is computable via:
- Tate-Nakayama for tori (after rigidification);
- Borovoi abelianisation for general reductive groups;
- Explicit cocycle representatives for low-dimensional / classical cases.
Role in Langlands
- Canonical L-packets: rigid inner forms give the cleanest framework for stating and proving local Langlands across inner forms.
- Stability of trace formula: rigid inner forms refine the matching between automorphic L-packets and Galois data.
- Endoscopic transfer: rigid inner forms simplify the Langlands-Shelstad-Kottwitz transfer factors.
References
- Kaletha, Rigid inner forms of real and p-adic groups (Annals 2016).
- Kaletha, Global rigid inner forms and multiplicities of discrete automorphic representations (Invent.\ Math.\ 2018).
- Dillery, Rigid inner forms of arbitrary reductive groups (preprint).