On the Serre conjecture for Artin characters in the geometric case

Abstract

Let $G$ be a finite group and $A$ be a regular local ring on which $G$ acts. Under certain assumptions on $A$ and the action, Serre defined a function $a_G:G\to \mathbb{Z}$ which can be viewed as a higher dimensional analogue of Artin character, and conjectured that it is associated to a ${\mathbb{Q}}_{\ell }$-rational representation of $G$ for any prime $\ell$ invertible in $A$. We prove this conjecture in the equal characteristic case.