What are the Borels/parabolics of the orthogonal or symplectic groups?

Does anyone know where I can find info about the Borel subgroups and parabolic subgroups of algebraic groups? I know what they are for the general linear group and for the special linear group you can just take the determinant 1 elements of a Borel for the general linear group. But I'm assuming that this doesn't always work because when I tried it for the orthogonal group I got that it was just diagonal matrices that square to the identity.

Ideally I'm looking for a reference which gives explicit computations of Borels, the root system, and parabolics.

Solution 1:

If you choose your bilinear forms nicely (so that the standard maximal flag includes a maximal totally isotropic subspace and a subspace minimal with respect to the quotient being totally isotropic), then you can indeed just do the intersection.

You can see that something like this has to work because a Borel subgroup of $H$ is contained in a Borel subgroup of $G$ whenever $H$ is a closed subgroup of $G$. The trick is which Borel subgroup!

At any rate this is handled nicely in Malle–Testerman (2011) on page 38.

• Malle, Gunter; Testerman, Donna. Linear algebraic groups and finite groups of Lie type. Cambridge Studies in Advanced Mathematics, 133. Cambridge University Press, Cambridge, 2011. xiv+309 pp. ISBN: 978-1-107-00854-0 MR2850737