Goresky-MacPherson calculus for the affine flag varieties

TL;DR

This paper applies Goresky-MacPherson calculus to affine flag varieties, calculating their equivariant cohomology and revealing geometric structures like a quadric cone and paraboloid vertices.

Contribution

It introduces a novel application of fixed point arrangement techniques to affine flag varieties, providing explicit descriptions of their equivariant cohomology and geometric properties.

Findings

Equivariant cohomology part generated by degree 2 forms a quadric cone.

Vertices of moment map images lie on a paraboloid.

Explicit geometric description of the spectrum of the full equivariant cohomology ring.

Abstract

We use the fixed point arrangement technique developed by Goresky-MacPherson to calculate the part of the equivariant cohomology of the affine flag varieties generated by degree 2. This turns out to be a quadric cone. We also describe the spectrum of the full equivariant cohomology ring as an explicit geometric object. We use our results to show that the vertices of the moment map images of the affine flag varieties lie on a paraboloid.