  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.
