An algebraic formula for the index of a 1-form on a real quotient singularity

Let a finite abelian group $G$ act (linearly) on the space $\mathbb{R}^n$ and thus on its complexification $\mathbb{C}^n$. Let $W$ be the real part of the quotient $\mathbb{C}^n/G$ (in general $W \neq \mathbb{R}^n/G$). We give an algebraic formula for the radial index of a 1-form on the real quotient $W$. It is shown that this index is equal to the signature of the restriction of the residue pairing to the $G$-invariant part $\Omega^G_\omega$ of $\Omega_\omega= \Omega^n_{\mathbb{R}^n,0}/\omega \wedge \Omega^{n-1}_{\mathbb{R}^n,0}$. For a $G$-invariant function $f$, one has the so-called quantum cohomology group defined in the quantum singularity theory (FJRW-theory). We show that, for a real function $f$, the signature of the residue pairing on the real part of the quantum cohomology group is equal to the orbifold index of the 1-form $df$ on the preimage $\pi^{-1}(W)$ of $W$ under the natural quotient map.