Cotangent Schubert Calculus

Schubert Calculus studies cohomology rings in (generalized) flag varieties, equipped with a distinguished basis - the fundamental classes of Schubert varieties - with structure constants satisfying many desirable properties. Cotangent Schubert Calculus studies the same rings, but with respect to different bases, arising from the theory of characteristic classes of singular varieties. Examples are the basis given by the Chern-Schwartz-MacPherson classes for Schubert cells in cohomology, and by the motivic Chern classes in K-theory. Their relation to the Schubert basis is governed by the Hecke algebra, and their study leads to many (proved and conjectural) positivity statements, and connections to other areas, such as Kazhdan-Lusztig theory. I will survey some of these recent developments, and highlight some outstanding questions.

This is based on several papers, mostly with P. Aluffi, J. Schurmann, and C. Su.