Quasi-split reductive group

Definition

Let $k$ be a field and let $G$ be a connected reductive $k$ -group. The group $G$ is quasi-split over $k$ if it contains a Borel subgroup defined over $k$ .

Every split reductive group is quasi-split, but the converse can fail: a quasi-split group need not have a split maximal torus. After a finite separable extension every connected reductive group becomes split and hence quasi-split.

Role among inner forms

An inner class over a local or global field has a quasi-split representative, unique up to isomorphism. The $L$ -group and the basic local Langlands parameter space are normally attached to this representative. Describing packets on the other inner forms requires additional data, such as a rigid inner twist.

Borel data and Galois action

For a quasi-split group one may choose a Borel pair $(B,T)$ over $k$ . The resulting Galois action on the based root datum defines the pinned action used in the L-group. Different rational Borel pairs are conjugate by $G(k)$ under the usual hypotheses, so the resulting outer action is canonical.

References

Armand Borel and Jacques Tits, “Groupes réductifs,” Publications Mathématiques de l'IHÉS 27 (1965), 55–150. Numdam.
Brian Conrad, Reductive Group Schemes, §§5.2–5.3. Author notes.