Quasi-split group

In mathematics, a quasi-split group over a field is a reductive group with a Borel subgroup defined over the field. Simply connected quasi-split groups over a field correspond to actions of the absolute Galois group on a Dynkin diagram.

Examples

All split groups (those with a split maximal torus) are quasi-split. These correspond to quasi-split groups where the action of the Galois group on the Dynkin diagram is trivial.

Lang (1956) showed that all simple algebraic groups over finite fields are quasi-split.

Over the real numbers, the quasi-split groups include the split groups and the complex groups, together with the orthogonal groups O_n,n+2, the unitary groups SU_n,n and SU_n,n+1, and the form of E₆ with signature 2.

References

• Lang, Serge (1956), "Algebraic groups over finite fields", American Journal of Mathematics, 78: 555–563, doi:10.2307/2372673, ISSN 0002-9327, JSTOR 2372673, MR 0086367