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 $\mathrm{\Omega }$ be an orbit. The following three invariants of $\mathrm{\Omega }\subset M$ can be characterized axiomatically: (1) the equivariant fundamental class $[\bar{\mathrm{\Omega }},M]\in H_G^{\ast }(M)$, (2) the equivariant Chern-Schwartz-MacPherson class $c(\mathrm{\Omega },M)\in H_G^{\ast }(M)$, and (3) the equivariant motivic Chern class $mC(\mathrm{\Omega },M)\in K_G(M)[y]$. 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 $\mathrm{\Omega }$ 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.