Kaletha Refined Inner Forms
Kaletha's refined inner forms (also called isocrystal inner forms or B(G)-inner forms) parametrise inner forms of a quasi- split group \(G\) by basic elements of the Kottwitz set \(B(G)\) (or its global analogue). The framework unifies rigid inner twists with the Kottwitz isocrystal classification.
Setup
Let \(F\) be a non-archimedean local field, \(G\) a connected reductive quasi-split \(F\)-group. Recall:
- Inner forms of \(G\): classified by \(H^1(F, G_{\mathrm{ad}})\).
- Basic strata \(B(G)_{\mathrm{basic}}\): in bijection with \(H^1(F, G_{\mathrm{ad}})\) (Kottwitz).
- Rigid inner twists: classified by \(H^1(F, \mathcal{E}_G)\) (Kaletha's gerbe).
Refined inner form
A refined inner form is an inner form \(G^*\) of \(G\) together with a chosen basic element \([b] \in B(G)_{\mathrm{basic}}\) realising \(G^* = G_b\) as the centraliser of the Newton point of \(b\).
Equivalently: a pair \((G^*, b)\) where \(G^*\) is an inner form and \(b\) is a basic cocycle in \(G(\breve F)\) inducing the inner twist.
Comparison with rigid inner twists
Theorem (Kaletha, Rigid inner forms vs isocrystals 2018). There is a canonical map \[ \text{rigid inner twists of } G \;\longrightarrow\; \text{refined inner forms of } G, \] realising the isocrystal classification of inner forms within the rigid framework.
The map is functorial and unifies the two classifications: refined inner forms are a "geometric" description (via \(B(G)\)) of the same data that rigid inner twists capture cohomologically (via the gerbe).
Examples
\(\mathrm{SL}_2 / \mathbb{Q}_p\)
\(B(\mathrm{SL}_2)_{\mathrm{basic}} = \{[1]\}\) (trivial) for the quasi-split form (which is split for \(\mathrm{SL}_2\)). So the only refined inner form is \(\mathrm{SL}_2\) itself.
Inner forms of \(\mathrm{SL}_2\) over \(\mathbb{Q}_p\): \(\mathrm{SL}_2\) and \(\mathrm{SL}_1(D)\) for the quaternion division algebra \(D\). But \(B(\mathrm{SL}_2)\) only captures \(\mathrm{SL}_2\) (because \(\mathrm{SL}_2\) is simply connected); the second form needs a \(z\)-extension.
\(\mathrm{PGL}_n / F\)
\(\pi_1(\mathrm{PGL}_n) = \mathbb{Z}/n\); \(B(\mathrm{PGL}_n)_{\mathrm{basic}} = \mathbb{Z}/n\). Each refined inner form of \(\mathrm{PGL}_n\) is \(\mathrm{PGL}_r(D)\) for a central simple \(F\)-algebra \(D\) of degree dividing \(n\).
The labelling is exactly by the Brauer invariant in \(\mathrm{Br}(F)[n] = \mathbb{Z}/n\).
Global refined inner forms
For a global field \(F\): Kaletha defines global refined inner forms via a global Galois gerbe. The constraint:
- Locally at each place \(v\), the global form restricts to a local refined inner form;
- Local data are compatible via a "Hasse-principle gerbe" controlling the global rigidification.
This gives a clean framework for global automorphic representations across inner forms.
Local-global compatibility
Theorem (Kaletha, global rigid inner forms). The local rigid inner forms at all places of \(F\) and the global rigid inner form fit into an exact sequence \[ 1 \to H^1_{\mathrm{global}}(F, \mathcal{E}_G) \to \prod_v H^1(F_v, \mathcal{E}_G) \to (\text{Hasse-principle gerbe defect}) \to 0. \]
This is the modern global Hasse principle for inner forms, refining the classical Borovoi/Sansuc statements.
Role in Langlands
- Multiplicities of discrete automorphic representations: Kaletha's global rigid framework gives canonical formulas for multiplicities in terms of L-parameters and S-groups.
- Endoscopic transfer factors: refined inner forms give canonical normalisations of transfer factors.
- Compatibility with Arthur's classification: Arthur's endoscopic classification (for classical groups) is naturally stated in terms of refined inner forms.
References
- Kaletha, Rigid inner forms vs isocrystals (J.\ EMS 2018).
- Kaletha, Global rigid inner forms and multiplicities of discrete automorphic representations (Invent.\ Math.\ 2018).
- Kaletha-Minguez-Shin-White, Endoscopic classification of representations: inner forms of unitary groups (preprint, 2014).