2. Classification of Unramified Reductive Groups

Definition 2.1

If G G is quasi-split and splits over an unramified extension (that is, if G G satisfies Assumptions 1.5 and 1.7), then G G is said to be an unramified reductive group.

Let G G be an unramified reductive group. It is classified by data (called root data)

( X ∗ , X ∗ , Φ , Φ ∨ , σ ) . (X^*,X_*,\Phi ,\Phi ^{\vee },\sigma ).

The data is as follows:

  • X ∗ X^* is the character group of a Cartan subgroup of G G .

  • X ∗ X_* is the cocharacter group of the Cartan subgroup.

  • Φ ⊂ X ∗ \Phi \subset X^* is the set of roots.

  • Φ ∨ ⊂ X ∗ \Phi ^{\vee } \subset X_* is the set of coroots.

  • σ \sigma is an automorphism of finite order of X ∗ X^* sending a set of simple roots in Φ \Phi to itself.

    σ \sigma is obtained from the action on the character group induced from the Frobenius automorphism of Gal ⁡ ( F u n / F ) \operatorname {Gal}(F^{un}/F) on the maximally split Cartan subgroup in G G .

The first four elements ( X ∗ , X ∗ , Φ , Φ ∨ ) (X^*,X_*,\Phi ,\Phi ^{\vee }) classify split reductive groups G G over F F . For such groups σ = 1 \sigma =1 .