Elliptic characteristic classes of Schubert varieties and duality

ALGEBRAIC GEOMETRY & NUMBER THEORY SEMINAR

We modify the theory of Borisov and Libgober to define equivariant characteristic classes of Schubert varieties in the generalized flag varieties G/B. The resulting classes can be considered as functions depending on two sets of parameters: equivariant variables and Kaehler variables. There are two recursions which allow to compute inductively these classes: right recursion corresponding to geometric Demazure-Lusztig operation and left recursion induced by the R-matrix appearing in Yang-Baxter equation. When one passes from a group G to its Langlands' dual the recursions switch they roles. This allows to show that equivariant elliptic classes for Langlands dual groups coincide after a swap of equivariant variables with Kaehler variables. This duality is only on the numerical level. The geometric cause remains mysterious.