  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
$a_G\colon G\rightarrow\mathbb{Z}$ 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.
