Generalized Flag Variety - Generalization To Semisimple Groups

Generalization To Semisimple Groups

The upper triangular matrices of determinant one are a Borel subgroup of SL(n,F), and hence the stabilizers of partial flags are parabolic subgroups. Furthermore, a partial flag is determined by the parabolic subgroup which stabilizes it, and partial flags belong to the same flag variety precisely when the corresponding parabolic subgroups are conjugate.

Hence, more generally, if G is a semisimple algebraic or Lie group, then a (generalized) flag variety for G is a conjugacy class of parabolic subgroups of G. It is therefore isomorphic, as a homogeneous space, to G/P where P is a parabolic subgroup of G. The correspondence between parabolic subgroups and generalized flag varieties allows each to be understood in terms of the other.

The extension of the terminology "flag variety" is reasonable, because points of G/P can still be described using flags. When G is a classical group, such as a symplectic group or orthogonal group, this is particularly transparent. If (V, ω) is a symplectic vector space then a partial flag in V is isotropic if the symplectic form vanishes on proper subspaces of V in the flag. The stabilizer of an isotropic flag is a parabolic subgroup of the symplectic group Sp(V,ω). For orthogonal groups there is a similar picture, with a couple of complications. First, if F is not algebraically closed, then isotropic subspaces may not exist: for a general theory, one needs to use the split orthogonal groups. Second, for vector spaces of even dimension 2m, isotropic subspaces of dimension m come in two flavours ("self-dual" and "anti-self-dual") and one needs to distinguish these to obtain a homogeneous space.