Induced character in equivariant K-theory and wreath products

Abstract

Let $G$ be a finite group, $X$ be a compact $G$-space. In this note we study the $({\mathbb{Z}}_{+}\times \mathbb{Z}/2\mathbb{Z})$-graded algebra $${\mathcal{F}}_G^q(X)=\underset{n\ge 0}{⨁}{q}^n\cdot K_{G\wr {\mathfrak{S}}_n}(X^n)\otimes \mathbb{C},$$ defined in terms of equivariant K-theory with respect to wreath products as a symmetric algebra. More specifically, let $H$ be another finite group and $Y$ be a compact $H$-space, we give a decomposition of ${\mathcal{F}}_{G\times H}^q(X\times Y)$ in terms of ${\mathcal{F}}_G^q(X)$ and ${\mathcal{F}}_H^q(Y)$. 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.

New version with minor changes