On a formula for the equivariant Euler characteristic of a $G$-sheaf

H. Fischbacher-Weitz and B. Köck computed the equivariant Euler characteristic of a $G-$sheaf on a $G$-curve $X$ over a field. Using a form of the Riemann-Roch theorem for quotient stacks proved by the second author we extend their computations to the cases where $dim(X)>1$.