Let $(M^{2n},g)$ be a Riemannian manifold with spin structure. Let $G$ is a compact Lie group. Suppose that $G$ acts on $M$ smoothly and the metric is $G$-invariant. Also, assume that the $G$ action can be lifted to the spin structure(still called it $G$-action), then the $G$-action commutes with the Dirac operator. In particular, we have a $G$-action on the vector spaces $\ker D^+$ and $\ker D^-$. The equivariant index is defined by $$\operatorname{ind}_gD^+= \operatorname{tr}(g\vert_{\ker D^+}) - \operatorname{tr}(g\vert_{\ker D^-})$$ for any $g\in G$.
I think there should be another way to define an index which reflects the $G$-action. My thought is as follows. Let $\Gamma(S_{\pm})^G$ be the space of $G$-invariant section of $S_{\pm}$, where $S_{\pm}$ are spin bundles. If we restrict the Dirac operator on $\Gamma(S_+)^G$, then we have $D^+: \Gamma(S_+)^G \to \Gamma(S_-)^G $. It is natural to define another ``equivariant index" by $$\operatorname{ind}^GD^+= \ker D^+\vert_{\Gamma(S_+)^G} - \operatorname{coker} D^+\vert_{\Gamma(S_+)^G}.$$
My question is that is there any relationship between the usual equivariant index and $\operatorname{ind}^GD^+$ defined above?