Flag Manifolds and Infinite Dimensional Kähler Geometry

Abstract

Let G be a connected, compact Lie group, A (generalized) flag manifold for G is the quotient of G by the centralizer of a torus. Hermitian symmetric spaces (e.g. complex projective spaces and Grassmannians), which are of the form G/C(T) for a circle T ⊂ G, are flag manifolds. The generic examples, though, are flag manifolds of the form G/T for T a maximal torus. The name derives from the manifold U(n)/diagonals of flags C0 = V0 ⊂ V1 ⊂ V2 ⊂ … ⊂ Vn = ℂ n in complex n-space. Flag manifolds enjoy many favorable geometric properties. They can be realized as coadjoint orbits of G, and thus carry an invariant symplectic form. There is also a complex description of flag manifolds as quotients of the complex group Gℂ. The symplectic and complex structures merge nicely: flag manifolds are homogeneous Kahler. Using these two properties—homogeneity and the Kahler condition—we easily compute curvature formulas. For special metrics flag manifolds are Kahler-Einstein. In any metric the Ricci curvature is positive, from which we deduce a vanishing theorem in cohomology. The full flag manifold G/T plays an important role in the representation theory of G a la Borel-Weil-Bott, and in that story the vanishing theorem plays a crucial part.