Localized cohomological operations

Abstract: Cohomological operations in algebraic oriented cohomology theories of Levine-Morel (Steenrod operations in Chow groups; Adams operations in connective K-theory of Cai-Merkurjev; Landweber-Novikov operations and Vishik symmetric operations in algebraic cobordism) provide a useful tool to study algebraic cycles on projective homogeneous varieties G/P. In the talk, I will show how to extend these operations to a T-equivariant setup, where T is a split maximal torus of a semisimple linear algebraic group G over a field of characteristic zero. More generally, I will show how to extend it to structure algebras of moment graphs (rings of global sections of structure sheaves on moment graphs). I will explain a uniform algorithm that computes the usual (non-equivariant) operations for G/Ps using such extended (localized) operations and equivariant Schubert calculus techniques. This generalizes the approach suggested by Garibaldi-Petrov-Semenov for Steenrod operations. Examples include Adams operations, L.-N. operations and Vishik's Mod-p operations.

algebraic geometrygroup theorynumber theoryrepresentation theory

Audience: researchers in the topic

Algebraic groups and algebraic geometry: in honor of Zinovy Reichstein's 60th birthday