The Index of a 1-Form on a Real Quotient Singularity

Let $G$ be a finite Abelian group acting (linearly) on space $\mathbb{R}^n$ and, therefore, on its complexification $\mathbb{C}^n$, and let $W$ be the real part of the quotient $\mathbb{C}^n/G$ (in the general case, $W \neq\mathbb{R}^n/G$). The index of an analytic 1-form on the space $W$ is expressed in terms of the signature of the residue bilinear form on the $G$-invariant part of the quotient of the space of germs of $n$-forms on $(\mathbb{R}^n,0)$ by the subspace of forms divisible by the 1-form under consideration.