Galois Action on Root Data

For a reductive group \(G\) defined over a non-algebraically-closed field \(k\), the absolute Galois group \(\Gamma = \mathrm{Gal}(\bar k / k)\) acts on the root datum of \(G_{\bar k}\). This Galois module is the core invariant of \(G\) up to inner forms.

Setup

Let \(G\) be a connected reductive group over \(k\) with separable closure \(\bar k\) and absolute Galois group \(\Gamma\). Choose a maximal torus \(T \subseteq G\) defined over \(k\) (such tori exist). Then \(T_{\bar k}\) is a split torus over \(\bar k\), and the root datum \((X^*(T_{\bar k}), \Phi, X_*(T_{\bar k}), \Phi^\vee)\) is defined.

Galois action

\(\Gamma\) acts on \(T_{\bar k}\) via its action on \(\bar k\), hence acts on:

the character lattice \(X^*(T_{\bar k})\);
the cocharacter lattice \(X_*(T_{\bar k})\);
the set of roots \(\Phi\) (preserving it as a subset of \(X^*\));
the set of coroots \(\Phi^\vee\).

The pairing \(\langle -, - \rangle\) is \(\Gamma\)-invariant.

For a fixed pinning of \(G_{\bar k}\), the Galois action need not preserve the pinning; instead it acts on the based root datum through pinned automorphisms (i.e.\ through \(\mathrm{Aut}^{\mathrm{pin}}(G_{\bar k}) = \mathrm{Out}(G_{\bar k})\)).

Classification of forms

Theorem. Quasi-split forms of \(G\) over \(k\) (i.e.\ forms with a \(k\)-rational Borel) are in bijection with continuous homomorphisms \[ \Gamma \;\longrightarrow\; \mathrm{Out}(G_{\bar k}), \] up to conjugacy.

General forms are quasi-split forms further twisted by a class in \(H^1(k, G_{\mathrm{ad}})\) (an inner twist).

Inner vs.\ outer

A form \(G' / k\) of \(G\) is:

Inner: \(\Gamma\) acts trivially on the based root datum of \(G'_{\bar k}\) — the form is twisted purely "internally" by \(G_{\mathrm{ad}}\)-cocycles.
Outer: \(\Gamma\) acts non-trivially on the based root datum; the action factors through \(\mathrm{Out}(G_{\bar k})\).

\(\mathrm{GL}_n\) and \(\mathrm{Sp}_{2n}\) have no outer forms (their Dynkin diagrams have no automorphisms or only trivial ones for classical Dynkin reasons). \(\mathrm{U}(n)\) is the prototypical outer form (Galois flip of \(A_{n-1}\)).

Examples

\(\mathrm{GL}_n\)

\(\mathrm{Out}(\mathrm{GL}_n) = \mathbb{Z}/2\) (transpose-inverse automorphism for \(n \ge 2\)). If \(\Gamma \to \mathbb{Z}/2\) is the quadratic extension associated to \(K/k\), the resulting form is \(\mathrm{Res}_{K/k} \mathrm{GL}_n^{(\sigma)} = \mathrm{U}(n, K/k)\): the unitary group of the Hermitian form attached to \(K/k\).

\(D_4\) and triality

\(\mathrm{Out}(D_4) = S_3\). Continuous homomorphisms \(\Gamma \to S_3\) factor through finite quotients; for a cubic extension \(K/k\) with Galois group \(\mathbb{Z}/3\), one gets the trialitarian form \({}^3 D_4\) of \(\mathrm{Spin}_8\).

\(E_6\)

\(\mathrm{Out}(E_6) = \mathbb{Z}/2\); a quadratic extension gives the outer form \({}^2 E_6\), denoted \(E_6^{(2)}\).

Tori

For a torus \(T\): the entire root datum is just the character lattice (no roots), and the Galois action on \(X^*(T)\) classifies \(T\) over \(k\) (anti-equivalence with finite-rank free continuous \(\Gamma\)-modules).

Action on the dual group

Under root-datum duality, the Galois action on the root datum of \(G\) becomes a Galois action on the dual root datum of \(\hat G\), hence on \(\hat G\) by pinned automorphisms. This is precisely the action defining the L-group: \[ {}^L G \;=\; \hat G(\mathbb{C}) \;\rtimes\; \Gamma. \]

Tits indices

The combinatorial data of the Galois action plus the inner-form cocycle is encoded by a Tits index: a Dynkin diagram with:

\(\Gamma\)-orbits marked;
distinguished vertices (representing the \(k\)-rational simple roots);
arrows for the Galois action.

Tits indices give a complete classification of reductive groups over a fixed field, including their quasi-split data.

Role in Langlands

L-group construction: requires the Galois action on the dual root datum.
Local Langlands for non-split groups: L-parameters must respect the Galois action — they map into the L-group, not just \(\hat G\).
Global Langlands: the global Galois action on root data is the natural compatibility for adelic groups.