Mirror symmetric Gamma conjecture for toric GIT quotients via Fourier transform

Let $\mathcal{X}=[({\mathbb{C}}^r\setminus Z)\mathrm{/}G]$ be a toric Fano orbifold. We compute the Fourier transform of the $G$-equivariant quantum cohomology central charge of any $G$-equivariant line bundle on ${\mathbb{C}}^r$ with respect to certain choice of parameters. This gives the quantum cohomology central charge of the corresponding line bundle on $\mathcal{X}$, while in the oscillatory integral expression it becomes the oscillatory integral in the mirror Landau-Ginzburg mirror of $\mathcal{X}$. Moving these parameters to real numbers simultaneously deforms the integration cycle to the mirror Lagrangian cycle of that line bundle. This computation produces a new proof the mirror symmetric Gamma conjecture for $\mathcal{X}$.