Nonabelian localization in equivariant K-theory and Riemann-Roch for quotients

TL;DR

This paper establishes a localization formula in equivariant algebraic K-theory for complex algebraic groups with finite stabilizers acting on smooth spaces, extending previous formulas to non-diagonalizable groups and deriving a Riemann-Roch theorem for quotients.

Contribution

It generalizes localization formulas in equivariant K-theory to non-diagonalizable groups and provides a Riemann-Roch formula for quotients of algebraic spaces by proper group actions.

Findings

Localization formula for arbitrary complex algebraic groups with finite stabilizers

Extension of Thomason and Nielsen formulas to non-diagonalizable groups

Riemann-Roch theorem for quotients of smooth algebraic spaces by proper group actions

Abstract

We prove a localization formula in equivariant algebraic $K$-theory for an arbitrary complex algebraic group acting with finite stabilizer on a smooth algebraic space. This extends to non-diagonalizable groups the localization formulas H.A. Nielsen in equivariant $K$-theory of vector bundles and R.W. Thomason for higher $K$-theory of equivariant coherent sheaves. As an application we give a Riemann-Roch formula for quotients of smooth algebraic spaces by proper group actions. This formula extends previous work of B. Toen for stacks with quasi-projective moduli spaces and the authors for quotients by diagonalizable groups.