Topic Rational Forms of Reductive Groups
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Rational Forms of Reductive Groups

A rational form of a reductive group \(G\) over a field \(k\) is a \(k\)-form: a reductive group \(G'\) over \(k\) with \(G'_{\bar k} \cong G_{\bar k}\). Rational forms are classified by Galois cohomology of the automorphism group of \(G_{\bar k}\), and they organise the entire Langlands programme over fields.

Definition

Let \(G\) be a connected reductive group over a separably closed field \(\bar k\). A rational form of \(G\) over a subfield \(k \subseteq \bar k\) is a connected reductive \(k\)-group \(G'\) such that \(G' \otimes_k \bar k \cong G\).

\(G\) itself may not be defined over \(k\); the question is whether there is a \(k\)-form at all, and how many.

Classification

Fix a split connected reductive \(G_0\) over \(k\) (e.g., the Chevalley form). All other forms of \(G_0 \otimes \bar k\) are classified by: \[ \{\text{forms of } G_0 \text{ over } k\}/\cong \;\stackrel{\sim}{\longleftrightarrow}\; H^1(k, \mathrm{Aut}(G_0)). \]

Inner vs.\ outer decomposition

The exact sequence \[ 1 \;\to\; G_{0, \mathrm{ad}} \;\to\; \mathrm{Aut}(G_0) \;\to\; \mathrm{Out}(G_0) \;\to\; 1 \] gives a long exact sequence in cohomology: \[ 1 \to G_{0, \mathrm{ad}}(k) \to \mathrm{Aut}(G_0)(k) \to \mathrm{Out}(G_0)(k) \to H^1(k, G_{0, \mathrm{ad}}) \to H^1(k, \mathrm{Aut}(G_0)) \to H^1(k, \mathrm{Out}(G_0)). \]

So:

  1. The outer form structure of \(G\) is classified by a map \(\Gamma_k \to \mathrm{Out}(G_0)\) (modulo conjugacy).
  2. The inner forms within a fixed outer-form class are classified by \(H^1(k, G_{0, \mathrm{ad}}^c)\), where \(G_{0, \mathrm{ad}}^c\) is the adjoint group of the quasi-split form \(c\).

Quasi-split forms

A form \(G\) is quasi-split if it has a Borel subgroup defined over \(k\) (equivalently: \(H^1(k, B)\) is trivial for some Borel \(B \subseteq G\)).

Every outer-form class contains a unique quasi-split form (up to isomorphism); inner forms differ by a class in \(H^1(k, G_{\mathrm{ad}})\).

\(G_0\) (split) is the quasi-split form with trivial outer twist; \(\mathrm{U}(n)\) is the quasi-split form in the outer twist class of the non-trivial diagram automorphism.

Examples

\(\mathrm{GL}_n\)

\(\mathrm{Aut}(\mathrm{GL}_n) = \mathrm{PGL}_n \rtimes \mathbb{Z}/2\) (inner = \(\mathrm{PGL}_n\), outer = transpose-inverse).

Form \(\Gamma_k\)-action
Split \(\mathrm{GL}_n\) trivial
Inner forms = \(\mathrm{GL}_r(D)\) for \(D / k\) central simple of degree \(n\) non-trivial cocycle in \(\mathrm{PGL}_n\)
Outer form \(\mathrm{U}(n, K/k)\) non-trivial in \(\mathbb{Z}/2\); inner twist by Hermitian form

\(\mathrm{SL}_n\)

Inner forms: \(\mathrm{SL}_1(D)\) for division algebras \(D\) of degree \(n\); the norm-one subgroup of \(D^\times\).

Outer forms over \(\mathbb{R}\): special unitary groups \(\mathrm{SU}(p, q)\) and \(\mathrm{SU}^*(2n)\) (the quaternionic form).

\(E_6\)

Two classes of outer forms: \(E_6^{\mathrm{split}}\) (trivial Galois) and \({}^2 E_6\) (non-trivial Galois). Inner forms then refine each.

Tits index

A Tits index records the combinatorial data of a rational form:

  • The Dynkin diagram of the absolute root system;
  • The Galois orbits on the Dynkin diagram;
  • The distinguished vertices (= simple roots defined over \(k\)).

The Tits index gives a complete classification of reductive groups over a fixed field \(k\) (Tits, 1966).

Hasse principle

Theorem (Kneser-Harder-Chernousov). For a simply connected semisimple group \(G\) over a number field \(F\): \[ \Sha^1(F, G) \;:=\; \ker\Big(H^1(F, G) \to \prod_v H^1(F_v, G)\Big) \;=\; 1. \]

Equivalently, every \(G\)-torsor that is trivial locally everywhere is trivial globally.

For non-simply-connected groups: \(\Sha^1(F, G)\) can be non-trivial, and is computable via Borovoi's abelianised Galois cohomology.

Role in Langlands

  • Forms of \(G\) at all places: \(G\) over \(F\) determines forms \(G_v\) over each local completion \(F_v\); these forms organise the local L-packet structure.
  • Adelic local-global: \(G(\mathbb{A}_F) = \prod_v G(F_v)\) makes sense once each local form is fixed; rational structure ensures the product is well-defined.
  • Strong approximation: simply connected semisimple groups satisfy strong approximation (\(G(F) \cdot G(F_v)\) is dense in \(G(\mathbb{A}_F)\) for any place \(v\) of positive rank); this is the basis of many automorphic argument.

Mathlib

Not formalised; depends on Galois cohomology infrastructure.

References

  • Tits, Classification of algebraic semisimple groups (Proc.\ AMS Symp.\ 1966).
  • Platonov–Rapinchuk, Algebraic Groups and Number Theory (1994).
  • Borel–Serre, Théorèmes de finitude en cohomologie galoisienne (Comment.\ Math.\ Helv.\ 1964).
  • Conrad, Reductive group schemes (Notes, 2014), §7.

Prerequisite Topics

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