Forms of Algebraic Groups
Two algebraic groups \(G, G'\) over a field \(k\) are called forms of each other if they become isomorphic after base change to a separable closure: \(G_{\bar{k}} \cong G'_{\bar{k}}\). They are classified by Galois cohomology.
Definition
Fix an algebraic group \(G_0\) over a separable closure \(k^{\mathrm{sep}}\). A form of \(G_0\) over \(k\) is an algebraic group \(G\) over \(k\) together with an isomorphism \(G_{k^{\mathrm{sep}}} \cong G_0\). Two forms are equivalent if they are \(k\)-isomorphic.
The set of equivalence classes is
\[ \{\text{forms of } G_0 \text{ over } k\} \;\leftrightarrow\; H^1\bigl(\operatorname{Gal}(k^{\mathrm{sep}}/k), \operatorname{Aut}(G_0)\bigr). \]
This is the Galois cohomology classification of forms.
Inner and outer forms
For a connected reductive group \(G\) over a field, the automorphism group fits into the exact sequence
\[ 1 \to \operatorname{Inn}(G) \to \operatorname{Aut}(G) \to \operatorname{Out}(G) \to 1. \]
Correspondingly, forms decompose:
- Inner forms of \(G\): those with image in \(H^1(\operatorname{Gal}, \operatorname{Inn}(G)) = H^1(\operatorname{Gal}, G^{\mathrm{ad}})\). Two inner forms have the same "shape" — same root datum + Dynkin diagram — and differ only in cocycle data.
- Outer forms: those whose image in \(\operatorname{Out}(G)\) is non-trivial. These can have a different Dynkin diagram from the split form (e.g. \(\mathrm{SU}_n\) is an outer form of \(\mathrm{SL}_n\); the Dynkin diagram of \(\mathrm{SU}_n\) is the twisted \(A_n\) labelled with Galois action).
Quasi-split forms
A reductive group \(G\) over \(k\) is quasi-split if it has a Borel subgroup defined over \(k\). Equivalently, \(G\) is the form obtained from the split form via the canonical "pinned" cocycle on the based root datum.
Every reductive group has a unique quasi-split inner form (its quasi-split form); inner forms of a quasi-split group \(G^*\) are classified by
\[ H^1\bigl(\operatorname{Gal}, G^*_{\mathrm{ad}}\bigr). \]
Examples
| split form | non-split forms | classification |
|---|---|---|
| \(\operatorname{SL}_n\) | \(\operatorname{SL}_1(D)\) for central simple algebras \(D\) of dim \(n^2\) | Brauer group \(\operatorname{Br}(k)\) of degree \(n\) |
| \(\operatorname{SO}_n\) (split) | non-split orthogonal groups | quadratic forms over \(k\) of dim \(n\) |
| \(\mathrm{PGL}_n\) | none (\(\operatorname{Aut} = \operatorname{Inn}\) for \(n \ne 2\), so inner forms only); these are \(\operatorname{PGL}_1(D)\) | Brauer group |
| \(\operatorname{SL}_n\) over a number field | inner forms = unitary groups + division-algebra forms | local-global compatibility |
Local–global principle (Hasse principle)
For some groups \(G\), a global form is determined by its local forms at all places: \(H^1(K, G) \to \prod_v H^1(K_v, G)\) is injective.
- Holds for simply connected semisimple groups over a number field (Kneser–Harder–Chernousov).
- Fails for tori and adjoint groups.
Modern refinements
Recent work introduces refined notions of forms for representation theory:
- Pure inner forms (Vogan) — refined to track more representation-theoretic data;
- Rigid inner forms (Kaletha) — uses Galois gerbes;
- Refined inner forms (Kaletha) — strictly finer than rigid.
These refinements are needed for the proper formulation of the local Langlands correspondence with character-theoretic precision.