Equivariant K-theory, generalized symmetric products, and twisted Heisenberg algebra

Abstract

For a space X acted by a finite group \G, the product space $X^n$ affords a natural action of the wreath product \Gn. In this paper we study the K-groups K_{\tG_n}(X^n) of \Gn-equivariant Clifford supermodules on $X^n$. We show that \tFG =\bigoplus_{n\ge 0}K_{\tG_n}(X^n) \otimes \C is a Hopf algebra and it is isomorphic to the Fock space of a twisted Heisenberg algebra. Twisted vertex operators make a natural appearance. The algebraic structures on \tFG, when \G is trivial and X is a point, specialize to those on a ring of symmetric functions with the Schur Q-functions as a linear basis. As a by-product, we present a novel construction of K-theory operations using the spin representations of the hyperoctahedral groups.Comment: 33 pages, latex, references updated, to appear in Commun. Math. Phy