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In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties. Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space V over a field F, which is a flag variety for the special linear group over F. Other flag varieties arise by considering partial flags, or by restriction from the special linear group to subgroups such as the symplectic group. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags. In the most general sense, a generalized flag variety is defined to mean a projective homogeneous variety, that is, a smooth projective variety X over a field F with a transitive action of a reductive group G (and smooth stabilizer subgroup; that is no restriction for F of characteristic zero). If X has an F rational point, then it is isomorphic to G/P for some parabolic subgroup P of G. A projective homogeneous variety may also be realised as the orbit of a highest weight vector in a projectivized representation of G. The complex projective homogeneous varieties are the compact flat model spaces for Cartan geometries of parabolic type. They are homogeneous Riemannian manifolds under any maximal compact subgroup of G, and they are precisely the coadjoint orbits of compact Lie groups. Flag manifolds can be symmetric spaces. Over the complex numbers, the corresponding flag manifolds are the Hermitian symmetric spaces. Over the real numbers, an R space is a synonym for a real flag manifold and the corresponding symmetric spaces are called symmetric R spaces.

According to basic results of linear algebra, any two complete flags in an n dimensional vector space V over a field F are no different from each other from a geometric point of view. That is to say, the general linear group acts transitively on the set of all complete flags. Fix an ordered basis for V, identifying it with Fn, whose general linear group is the group GL(n,F) of n × n invertible matrices. The standard flag associated with this basis is the one where the ith subspace is spanned by the first i vectors of the basis. Relative to this basis, the stabilizer of the standard flag is the group of nonsingular lower triangular matrices, which we denote by Bn. The complete flag variety can therefore be written as a homogeneous space GL(n,F) / Bn, which shows in particular that it has dimension n(n−1)/2 over F.

Note that the multiples of the identity act trivially on all flags, and so one can restrict attention to the special linear group SL(n,F) of matrices with determinant one, which is a semisimple algebraic group; the set of lower triangular matrices of determinant one is a Borel subgroup. If the field F is the real or complex numbers we can introduce an inner product on V such that the chosen basis is orthonormal. Any complete flag then splits into a direct sum of one dimensional subspaces by taking orthogonal complements. It follows that the complete flag manifold over the complex numbers is the homogeneous space

where U(n) is the unitary group and Tn is the n torus of diagonal unitary matrices. There is a similar description over the real numbers with U(n) replaced by the orthogonal group O(n), and Tn by the diagonal orthogonal matrices (which have diagonal entries ±1).

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. Hence, more generally, if G is a semisimple algebraic or Lie group, then the (generalized) flag variety for G is 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.

If G is a semisimple algebraic group (or Lie group) and V is a (finite dimensional) highest weight representation of G, then the highest weight space is a point in the projective space P(V) and its orbit under the action of G is a projective algebraic variety. This variety is a (generalized) flag variety, and furthermore, every (generalized) flag variety for G arises in this way. Armand Borel showed that this characterizes the flag varieties of a general semisimple algebraic group G: they are precisely the complete homogeneous spaces of G, or equivalently (in this context), the projective homogeneous G varieties.