On the Serre conjecture for Artin characters in the geometric case

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 $a_G\text{\hspace{0.17em}⁣}:G\to \mathbb{Z}$ which can be viewed as a higher dimensional analogue of Artin character, and conjectured that it is associated to a ${\mathbb{Q}}_{\mathrm{\ell }}$-rational representation of $G$ for any prime $\mathrm{\ell }$ invertible in $A$. We prove this conjecture in the equal characteristic case.