Indexed Root Datum of an F-Reductive Group

An indexed root datum over \(F\) is a tuple \[ (X,D,X^\vee,D^\vee,D_0,\Gamma) \] where:

• \((X,D,X^\vee,D^\vee)\) is a based root datum;
• \(D_0\subseteq D\) is the distinguished subset, recording the anisotropic kernel;
• the absolute Galois group \(\Gamma_F=\operatorname{Gal}(F_s/F)\) acts continuously on \(X\), preserving \(D\) and \(D_0\), and dually on \(X^\vee\).

For a connected reductive \(F\)-group \(G\), choose a maximal \(F\)-split torus \(S\), a maximal \(F\)-torus \(T\supseteq S\), and a compatible basis \(D\) of the absolute roots. The resulting indexed root datum \[ {}_{F}\Psi(G)=(X,D,X^\vee,D^\vee,D_0,\Gamma_F) \] is independent of the choices up to isomorphism.

The group \(G\) is \(F\)-split exactly when \(D_0=\emptyset\) and the Galois action on the based root datum is trivial. It is quasi-split exactly when \(D_0=\emptyset\). It is anisotropic exactly when \(D_0=D\).