On equivariant indices of 1-forms on varieties

Given a $G$-invariant holomorphic $1$-form with an isolated singular point on a germ of a complex-analytic $G$-variety with an isolated singular point ($G$ is a finite group), its equivariant homological index and (reduced) equivariant radial index are defined as elements of the ring of complex representations of the group. We show that these indices coincide on a germ of a smooth complex analytic $G$-variety. This makes it possible to consider the difference between them as a version of the equivariant Milnor number of a germ of a $G$-variety with an isolated singular point.