Equivariant Chern classes of singular algebraic varieties with group actions

We define equivariant Chern-Schwartz-MacPherson classes of a possibly singular algebraic G-variety over the base field ℂ, or more generally over a field of characteristic 0. In fact, we construct a natural transformation C_Z.ast;^G from the G-equivariant constructible function functor \cal{F}^G to the G-equivariant homology functor H _z.ast;^G or A_*^G (in the sense of Totaro-Edidin-Graham). This C_z.ast^G may be regarded as MacPherson's transformation for (certain) quotient stacks. The Verdier-Riemann-Roch formula takes a key role throughout.