The equivariant cohomology ring of regular varieties

Let $B$ denote the upper triangular subgroup of $SL_2(C)$, $T$ its diagonal torus and $U$ its unipotent radical. A complex projective variety $Y$ endowed with an algebraic action of $B$ such that the fixed point set $Y^U$ is a single point, is called regular. Associated to any regular $B$-variety $Y$, there is a remarkable affine curve $Z_Y$ with a $T$-action which was studied by the second author. In this note, we show that the coordinate ring of $Z_Y$ is isomorphic with the equivariant cohomology ring $H_T^*(Y)$ with complex coefficients, when $Y$ is smooth or, more generally, is a $B$-stable subvariety of a regular smooth $B$-variety $X$ such that the restriction map from $H^*(X)$ to $H^*(Y)$ is surjective. This isomorphism is obtained as a refinement of the localization theorem in equivariant cohomology; it applies e.g. to Schubert varieties in flag varieties, and to the Peterson variety studied by Kostant. Another application of our isomorphism is a natural algebraic formula for the equivariant push forward.