Equivariant Cohomology and Equivariant Characteristic Numbers of a Homogeneous Space

Let G be a compact connected Lie group with maximal torus T , and H a closed subgroup containing T . We compute the equivariant cohomology ring and the equivariant characteristic numbers of the homogeneous space G/H under the natural action of the maximal torus T . The computation is based on the localization theorems of Borel and of Atiyah-Bott-Berline-Vergne. Let G be a compact connected Lie group with maximal torus T , and H a closed subgroup of G containing T . The quotient space G/H includes examples such as Grassmannians and flag manifolds. The torus T acts on G/H by left multiplication. One can work out the equivariant cohomology ring of this action from general theorems in the literature concerning equivariant cohomology ([8], Chap. III, §1, Prop. 1, p. 38). In this paper we have two goals. The first is to rederive, essentially from scratch, the rational equivariant cohomology ring of G/H , using only two classical theorems of Borel: the Borel localization theorem ([3], [1]) and Borel’s theorem on the rational cohomology ring of G/H ([4]). The main idea of our computation is to compare the equivariant cohomology of a space with that of the fixed point set of the action. In view of the many recent applications of equivariant cohomology, such an account is hopefully of some use. The advantage of this approach is that thereafter it leads quite easily, with the aid of the ABBV localization theorem (Atiyah-Bott [1], Berline-Vergne [2]), to formulas for equivariant characteristic numbers of the homogeneous space G/H , and this is our second goal. In addition to a general algorithm for computing equivariant characteristic numbers, we work out the formulas for the special cases of the projective space, the Grassmannian, and the complete flag manifold, all defined over the complex numbers. Let ET −→ BT be the universal T -bundle over the classifying space BT . If M is a manifold on which T acts, we denote by MT its homotopy quotient, sometimes called the Borel construction ([3], p. 52), MT := ET ×T M := (ET ×M)/ ∼, with the equivalence relation ∼ given by (et, x) ∼ (e, tx), for (e, x) ∈ ET ×M and t ∈ T. Date: January 31, 2001.