Induced character in equivariant K-theory and wreath products

TL;DR

This paper explores the structure of a graded algebra arising from equivariant K-theory of wreath product groups acting on spaces, providing a decomposition formula and applications to equivariant connective K-homology.

Contribution

It introduces a decomposition of the algebra associated with product spaces in terms of individual components, extending the understanding of equivariant K-theory for wreath products.

Findings

Decomposition of alF^q_{G imes H}(X imes Y) in terms of alF^q_G(X) and alF^q_H(Y)

Analysis of representation theory of pullbacks of groups

Applications to equivariant connective K-homology

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 $$\mathcal{F}^q_G(X) = \bigoplus_{n\geq0} q^n \cdot K_{G\wr\mathfrak{S}_n}(X^n)\otimes\mathbb{C},$$ 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 $\mathcal{F}^q_{G\times H}(X\times Y)$ in terms of $\mathcal{F}^q_G(X)$ and $\mathcal{F}^q_H(Y)$. 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.