Split groups and quasi-split groups.

Question

By definition, a quasi-split group over a field is a reductive group with a Borel subgroup defined over the field. A split group is a quasi-split group which has split torus ($T = \mathbb{G}_m^n$, $\mathbb{G}_m$ is the multiplicative group). Are there some examples of quasi-split groups which are not split? Thank you very much.

Answer 1

Here are two standard examples of non-split but quasi-split groups:

• Non-split tori (a dumb example), e.g. U(1) over $\mathbf{R}$. If $T$ is any torus, then $T$ is a Borel subgroup of itself, and this is clearly defined over the ground field.

• A slightly less dumb example: the semisimple group $SU(2, 1)$ over $\mathbf{R}$, defined as the group of 3x3 complex matrices of determinant 1 satisfying $g J \bar{g}^t = J$ where $J$ is the matrix $\begin{pmatrix} &&1 \\&1 \\ 1 \end{pmatrix}$. The intersection of this group with the upper-triangular matrices in $\operatorname{GL}_3(\mathbf{C})$ is a Borel, but there is no split maximal torus.