Generalized Equivariant Cohomology and Stratifications

Abstract

For $T$ a compact torus and $E_T^{\ast }$ a generalized $T$-equivariant cohomology theory, we provide a systematic framework for computing $E_T^{\ast }$ in the context of equivariantly stratified smooth complex projective varieties. This allows us to explicitly compute $E_T^{\ast }(X)$ as an $E_T^{\ast }(\text{pt})$-module when $X$ is a direct limit of smooth complex projective $T_{\mathbb{C}}$-varieties with finitely many $T$-fixed points and $E_T^{\ast }$ is one of $H_T^{\ast }(\cdot ;\mathbb{Z})$, $K_T^{\ast }$, and $M{U}_T^{\ast }$. We perform this computation on the affine Grassmannian of a complex semisimple group.

V2: a few minor alterations