Guillemin Sternberg Conjecture(proved) says that for symplectic manifold $(M,\omega)$ with $[Q,R]=0$ condiction, with compact group action $G$, such that $\mu:M\to \mathfrak g^*$ is regular at $0$, and $G$ action freely on $\mu^{-1}(0)$. Then $Ind^G(D_M^L)=Ind(D^{L_G}_{M//G})$.

Is there an example?

Here is an idea.

Let $M=\mathbb P^2$ $L=\mathcal O(1)$ and $G=S^1$ action on $M$ on the first coordinate, i.e. $\theta\cdot[z_1,z_2,z_3]=[\theta z_1,z_2,z_3]$. Then, $M//G=\mathbb P^1$.

Q: I donot know what is $L_G$. $\mathcal O(1)$ or trivial bundle?

By my understanding, I think $L_G=\mathcal O(1)$, then $Ind(D^{\mathcal O(1)}_{\mathbb P^1})=2$. Is this correct?