2005-09-30
Dan Edidin, University of Missouri-Columbia
Non-abelian localization in equivariant K-theory
The localization theorem for actions of diagonalizable groups is a
fundamental result in equivariant K-theory. The theorem can be stated
as follows: If T is a diagonalizable group acting on a space
X and h\in T has fixed locus X^h then the
direct image in equivariant K-theory i_*\colon G(X^h,T)\to
G(X,T) is an isomorphism after localizing at certain prime ideal
in the representation ring R(T). When X is smooth
the localization isomorphism has an explicit inverse which is
extremely useful for computation.
In this talk, I will explain how to obtain an explicit "localization"
(which actually involves completion) formula for actions of arbitrary
algebraic groups on smooth algebraic spaces defined over
C. Applications include a Riemann-Roch formula for geometric
quotients as well as a method to represent elements of equivariant
K-theory by cycles.
Much of the talk is based on a joint paper with William Graham, math.AG/0411213, to appear in Advances in Math.