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

For a space X acted by a finite group Γ, the product space X affords a natural action of the wreath product Γn = Γ n ⋊ Sn. The direct sum of equivariant K-groups ⊕ n≥0 KΓn(X )⊗C were shown earlier by the author to carry several interesting algebraic structures. In this paper we study the Kgroups K H̃Γn (X) of Γn-equivariant Clifford supermodules on X . We show that F Γ (X) = ⊕ n≥0 H̃Γn (X)⊗ 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 F Γ (X) , when Γ 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.