Generalized orbifold Euler characteristic of symmetric products and equivariant Morava K–theory

Abstract

We introduce the notion of generalized orbifold Euler characteristic associated to an arbitrary group, and study its properties. We then calculate generating functions of higher order ($p$–primary) orbifold Euler characteristic of symmetric products of a $G$–manifold $M$. As a corollary, we obtain a formula for the number of conjugacy classes of $d$–tuples of mutually commuting elements (of order powers of $p$) in the wreath product $G\wr {\mathfrak{S}}_n$ in terms of corresponding numbers of $G$. As a topological application, we present generating functions of Euler characteristic of equivariant Morava K–theories of symmetric products of a $G$–manifold $M$.