The Borel-Moore homology of a topological space $X$ can be defined as the dual of the compactly supported cohomology of $X$. In this talk we will be concerned with the equivariant Borel-Moore homology groups of complex algebraic varieties which admit an algebraic action of a connected complex linear algebraic group $G$ such that $X$ is covered by a finite number of $G$-orbits. In this case, we will describe a filtration on the equivariant Borel-Moore homology as a graded $H^G(pt,\Q)$-module with the associated graded module the direct sum of the equivariant Borel-Moore homology groups of the orbits.

We will present applications of this development to certain equivariant embeddings of homogeneous spaces. This is a joint work with Aram Bingham (Tulane University) and Yildiray Ozan (Middle East Technical University, Turkey).