higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A flag in a vector space or a projective space is a nested system of linear/projective subspaces, one of each dimension from $0$ to $n-1$.
Given a field $k$, the space of all flags in an $n$-dimensional $k$-vector space has the structure of a projective variety over $k$, this is the flag variety.
If $k$ is the field of real or complex numbers, then it has a structure of a smooth manifold. It can be considered as the homogeneous space $SL(n,k)/B$ where $B$ is the subgroup of lower (or upper if you like) triangular matrices. By the Gram–Schmidt orthogonalization procedure, one shows the isomorphism as real manifolds $SL(n,\mathbb{C})/B\cong SU(n)/T$ where $T$ is the subgroup of the diagonal $n\times n$-matrices.
More generally, the generalized flag variety is the complex projective variety obtained as the coset space $G/T\cong G^{\mathbb{C}}/B$ where $G$ is a compact Lie group, $T$ its maximal torus, $G^{\mathbb{C}}$ the complexification of $G$, which is a complex semisimple group, and $B\subset G^{\mathbb{C}}$ is the Borel subgroup. It has a structure of a compact Kähler manifold. It is a special case of the larger family of coset spaces of semisimple groups modulo parabolics which includes, for example, Grassmannians. There are quantum, noncommutative and infinite-dimensional generalizations. Flag varieties have rich combinatorial and geometric structure and play an important role in representation theory and integrable systems.
character sheaf?, horocycle correspondence, Harish Chandra transform
D. Monk, The geometry of flag manifolds, Proceedings of the London Mathematical Society 1959 s3-9(2):253–286; doi:10.1112/plms/s3-9.2.253
Wikipedia, Generalized flag variety
M. Brion, Lectures on the geometry of flag varieties, pdf
Armand Borel, Linear algebraic groups, Springer
Flag varieties of loop groups are discussed in