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 spaceR n and thus on its complexificationC n . Let W be the real part of the quotientC n / G (in general W ≠ 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 Ω ω G ofΩ ω = ΩR n , 0 n / ω ∧ ΩR n , 0 n − 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 d f on the preimageπ − 1( W )of W under the natural quotient map.