Introduction to Equivariant K-theory - Dave Anderson

Special Year Seminar I
2:00pm|Simonyi 101
Topic: Introduction to Equivariant K-theory
Speaker: Dave Anderson
Affiliation: Institute for Advanced Study
Date: March 5, 2025

K-theory arose in the 1950s from Grothendieck’s formulation of the Riemann-Roch theorem – that is, from attempts to calculate spaces of sections of vector bundles on a variety X via intersection theory on X.  An equivariant version was introduced shortly afterwards, for varieties and vector bundles equipped with the action of a group G.  Like equivariant cohomology, equivariant K-theory has several advantages: On one hand, exploiting the symmetry manifested by the group action often leads to quite effective computational tools; on the other hand, the geometry of a variety X on which G acts can shed light on the representation theory of G.
These lectures will provide a concrete introduction to equivariant K-theory, focusing on the special features of the case where G=T is a torus, and in many ways complementing Bill Graham’s lectures on equivariant cohomology last month.  I will explain the localization theorem, including a K-theoretic version of equivariant multiplicity, and show how to use it to describe the equivariant K-theory of flag varieties and toric varieties.  As applications, we’ll derive the Weyl character formula (for irreducible representations of a semisimple Lie group) and Brion’s formula (for lattice points in a polytope).  I’ll also describe the equivariant Riemann-Roch theorem relating equivariant K-theory to equivariant cohomology and Chow groups.  Finally, I’ll state some positivity theorems for Schubert decompositions.