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 Rn and thus on its complexification Cn. Let W be the real part of the quotient Cn/G (in general W≠Rn/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 ΩωG of Ωω=ΩRn,0n/ω∧ΩRn,0n−1. 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 π−1(W) of W under the natural quotient map.