The character of the total power operation

Abstract

We compute the total power operation for the Morava $E$–theory of any finite group up to torsion. Our formula is stated in terms of the ${GL}_n\left({\mathbb{Q}}_p\right)$–action on the Drinfel’d ring of full level structures on the formal group associated to $E$–theory. It can be specialized to give explicit descriptions of many classical operations. Moreover, we show that the character map of Hopkins, Kuhn and Ravenel from $E$–theory to ${GL}_n\left({\mathbb{Z}}_p\right)$–invariant generalized class functions is a natural transformation of global power functors on finite groups.