The Bethe algebra of a quantum integrable system is a commutative algebra acting on a vector space. We consider three versions: the rational, trigonometric, and elliptic cases. We show that these algebras can be identified with the equivariant cohomology, K-theory, and elliptic cohomology rings of some flag manifolds acting on themselves by multiplication. The key objects in the identification are the equivariant characteristic classes of singular varieties. As a byproduct, we find an interpolation characterization of Chern-Schwartz-MacPherson classes, as well as define the polynomials in CSM theory that should be analogous to Schur polynomials in fundamental class theory.