The K-theory of abelian versus nonabelian symplectic quotients

TL;DR

This paper develops a K-theoretic method to compute the K-theory of symplectic quotients by a compact Lie group using the K-theory of quotients by its maximal torus, extending and refining previous cohomological results.

Contribution

It introduces a new K-theoretic approach to relate the K-theory of symplectic quotients by a Lie group to that by its maximal torus, avoiding torsion issues and dividing by group order.

Findings

Expresses K-theory of G-quotients in terms of T-quotients and Weyl group invariants

Uses Hodgkin's spectral sequence and Euler class e in the computation

Removes the need to divide by the Weyl group order in K-theoretic formulas

Abstract

We compare the K-theories of symplectic quotients with respect to a compact connected Lie group and with respect to its maximal torus, and in particular we give a method for computing the former in terms of the latter. More specifically, let G be a compact connected Lie group with no torsion in its fundamental group, let T be a maximal torus of G, and let M be a compact Hamiltonian G-space. Let M//G and M//T denote the symplectic quotients of M by G and by T, respectively. Using Hodgkin's Kunneth spectral sequence for equivariant K-theory, we express the K-theory of M//G in terms of the elements in the K-theory of M//T which are invariant under the action of the Weyl group, in addition to the Euler class e of a natural Spin^c vector bundle over M//T. This Euler class e is induced by the denominator in the Weyl character formula, viewed as a virtual representation of T; this is relevant for our proof. Our results are K-theoretic analogues of similar (unpublished) results by Martin for rational cohomology. However, our results and approach differ from his in three significant ways. First, Martin's method involves integral formulae, but the corresponding index formulae in K-theory are too coarse a tool, as they cannot detect torsion. Instead, we carefully analyze related K-theoretic pushforward maps. Second, Martin's method involves dividing by the order of the Weyl group, which is not possible in (integral) K-theory. We render this unnecessary by examining Weyl anti-invariant elements, proving a K-theoretic version of a lemma due to Brion. Finally, Martin's results are expressed in terms of the annihilator ideal of e^2, the square of the Euler class mentioned above. We are able to "remove the square", working instead with the annihilator ideal of e.