The K-theory of abelian versus nonabelian symplectic quotients
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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Our results are K-theoretic analogues of similar (unpublished) results by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.
