Induced character in equivariant K-theory, wreath products and pullback of groups

Let G be a finite group and let X be a compact G-space. In this note we study the (Z+ × Z/2Z)-graded algebra FqG (X) = ⊕n ≤ 0 qn · KG∫Gn(Xn) ⊗ C, defined in terms of equivariant K-theory with respect to wreath products as a symmetric algebra, we review some properties of FqG (X) proved by Segal and Wang. We prove a Kunneth type formula for this graded algebras, more specifically, let H be another finite group and let Y be a compact H-space, we give a decomposition of FqG × H (X × Y) in terms of FqG (X) and FqH (Y). For this, we need to study the representation theory of pullbacks of groups. We discuss also some applications of the above result to equivariant connective K-homology.