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 work out the Atiyah-Bott-Berline-Vergne localization formula for the homogeneous space G/H under the natural action of the maximal torus T . The computation gives explicit formulas for the ordinary 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 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. In this note we apply the Atiyah-Bott-Berline-Vergne localization formula to calculate the equivariant characteristic numbers of G/H under this torus action (Section 13). As it well known, this yields the ordinary characteristic numbers of the homogeneous space G/H (Th. 14). However, our main interest is to derive a restriction formula (Th. 8) and an Euler class formula (Th. 12) in the equivariant cohomology of G/H , for it turns out that these formulas can be used to calculate the Gysin homomorphism in the ordinary cohomology of a flag bundle. The application to the Gysin homomorphism is an instance of a general principle relating the push-forward in ordinary cohomology to the localization formula in equivariant cohomology. As this more general principle is independent of homogeneous spaces, we treat it in a separate article [15]. In [13] a relative localization formula is used to calculate the Gysin map of an equivariant map. The restriction formula (Th. 8) and the Euler class formula (Th. 12) of the present article enter essentially in a calculation that allows us to recover a formula of Akyildiz and Carrell [2] for the Gysin map of a map between flag manifolds. Using techniques of symplectic quotients, Shaun Martin ([12], Prop. 7.2) has obtained a formula for the characteristic numbers of a Grassmannian similar to our Prop. 15. To simplify the exposition, we discuss first the homogeneous space G/T . The general case G/H follows by suitable modifications. Although we consider cohomology with real coefficients, everything goes through with rational coefficients. To minimize the pre-requisites and to establish the notations, in the first few sections we recall some results on associated bundles, the characteristic map, the actions of the Weyl group, and equivariant cohomology. I am grateful to Raoul Bott for suggesting the problem and for many helpful discussions. 1. Associated bundles and the characteristic map Suppose a torus T of dimension l acts freely on the right on a topological space X so that X −→ X/T is a principal T -bundle. A character of T is a multiplicative homomorphism of T into C. Let T̂ be the group of characters of T , with the multiplication of characters written additively: for α, β ∈ T̂ and t ∈ T , t := tt = α(t)β(t). Denote by Cγ the complex vector space C with an action of T given by the character γ : T −→ C . By the mixing construction, a character γ on T associates to the principal bundle X −→ X/T a complex line bundle L(X/T, γ) over X/T : L(X/T, γ) := X ×T Cγ := (X × Cγ)/T, Date: December 25, 2003.