A Filtration on Equivariant Borel-Moore Homology

TL;DR

This paper introduces a filtration on the equivariant Borel-Moore homology of certain $G$-embeddings, linking it to the homologies of individual orbits, and applies this to compactifications and double flag varieties.

Contribution

It establishes a new filtration on equivariant Borel-Moore homology with a clear description of its associated graded module, extending to ordinary homology under specific conditions.

Findings

Filtration on equivariant Borel-Moore homology with associated graded as sum of orbit homologies

Descending filtration to ordinary homology when each orbit has a fixed point under a maximal torus

Applications to wonderful compactifications and double flag varieties

Abstract

Let $G/H$ be a homogeneous variety, and let $X$ be a $G$-equivariant embedding of $G/H$such that the number of $G$-orbits in $X$ is finite. We show that the equivariant Borel-Moore homology of $X$ has a filtration with associated graded module the direct sum of the equivariant Borel-Moore homologies of the $G$-orbits. If $T$ is a maximal torus of $G$ such that each $G$-orbit has a $T$-fixed point, then the equivariant filtration descends to give a filtration on the ordinary Borel-Moore homology of $X$. We apply our findings to certain wonderful compactifications as well as to double flag varieties.