quasi-split algebraic group

Question

While reading papers, there usually an assumption "quasi-split" for reductive algebraic groups. To use their results I need to know which groups are quasi-split. For the case I am interested in suppose that $F$ is a non-degenrate quadratic form in $n\ge 2$ variables defined over a number field $k$. Is the group $SO(F)$ quasi-split over $k$?

Answer 1

It's probably helpful to give more detail, since the question involves the overlap between concrete earlier study of classical groups (Witt, Dieudonne, etc.) and the more general Borel-Tits theory of reductive algebraic groups over an arbitrary field $k$. The most useful overview of the entire classification in the latter case is given by J. Tits, Classification of algebraic semisimple groups, pp. 33-62, Proc. Symp. Pure Math. 9, Amer. Math. Soc., 1967 (proceedings of the 1965 summer institute at Boulder). In particular, his detailed tables (with references) at the end summarize the possibilities for each simple Lie type, with special attention to the real, $p$-adic, finite, and number field cases.

In the general Borel-Tits theory, the crucial case is semisimple (or even simple) algebraic groups. For a connected group $G$ (reductive, semisimple, simple) defined over a field $k$, one has an absolute root system and Dynkin diagram relative to a fixed maximal torus $T$ and Borel subgroup $B$ containing it over an algebraic closure of $k$. But one also has parallel data relative to $k$: a maximal $k$-split torus $S \subset T$, etc. One says $G$ is quasi-split over $k$ if there exists a Borel subgroup defined over $k$. This is automatic when $k$ is finite (by Lang's theorem) or more generally over fields of cohomological dimension 1. But in general there are lots of possibilities for a given Dynkin diagram, so the Tits classification gets complicated.

Over number fields the classification is fairly explicit in the tables given by Tits. Being quasi-split amounts to requiring that every vertex of the Dynkin diagram be circled (sometimes as part of a larger orbit, sometimes as a singleton). But the orthogonal groups (types $B, D$) do get complicated and require translation of some of the classical terminology about quadratic forms.

While the specifics in the survey by Tits are largely error-free, his statement and brief sketch of the main classification theorem were not entirely precise. His Bonn student Martin Selbach later wrote up more details in his thesis Klassifikationstheorie halbeinfacher algebraischer Gruppen, Bonner Math. Schriften 83 (1976). But in the number theory case one can probably just rely on the Tits article to get a clear sense of which are the quasi-split groups over number fields.