Characteristic classes of orbit stratifications, the axiomatic approach
Consider a complex algebraic group $G$ acting on a smooth variety $M$ with finitely many orbits, and let $Ω$ be an orbit. The following three invariants of $Ω\subset M$ can be characterized axiomatically: (1) the equivariant fundamental class $[\overlineΩ, M]\in H^*_G(M)$, (2) the equivariant Chern-Schwartz-MacPherson class $c(Ω, M)\in H^*_G(M)$, and (3) the equivariant motivic Chern class $mC(Ω, M) \in K_G(M)[y]$. The axioms for Chern-Schwartz-MacPherson and motivic Chern classes are motivated by the axioms for cohomological and K-theoretic stable envelopes of Okounkov and his coauthors. For $M$ a flag variety and $Ω$ a Schubert cell---an orbit of the Borel group acting---this implies that CSM and MC classes coincide with the weight functions studied by Rimanyi-Tarasov-Varchenko. In this paper we review the general theory and illustrate it with examples.