The K-theory of Abelian versus Nonabelian Symplectic Quotients

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 quotient of M by G and by T , respectively. Using Hodgkin’s Künneth 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 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 Our results are K-theoretic analogues of similar (unpublished) results by Martin for rational cohomology.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 formulæ, but the corresponding index formulæ 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, the square of the Euler class mentioned above. We are able to “remove the square,” working instead with the annihilator ideal of e.