A compact manifold has a tangent bundle, and a natural
question is to find a replacement for the Chern classes of the tangent
bundle, in the case when the space is singular. The
Chern-Schwartz-MacPherson (CSM) classes are homology classes which
“behave like” the Chern classes of the tangent bundle, and are
determined by a functoriality property. The existence of these classes
was conjectured by Grothendieck and Deligne, and proved by MacPherson in
1970’s. The calculation of the CSM classes for Schubert cells and
Schubert varieties in flag manifolds was obtained only recently, and it
exhibited some unexpected features. For instance, these classes are
determined by a Demazure-Lusztig operator, and they are essentially
equivalent to certain Lagrangian cycles in the cotangent bundle of the
flag manifold, showing up in the proof of Kazhdan-Lusztig conjectures.
They are also equivalent to the stable envelopes of Maulik and Okounkov.
In this talk I will survey some of these developments. No prior
knowledge about the CSM classes will be assumed. This is joint work with
P. Aluffi, and ongoing joint work with P. Aluffi, J. Schurmann and C. Su.