I've been reading Kamnitzer's survey Symplectic resolutions, symplectic duality, and Coulomb branches. Here the Higgs branch is defined as a projective GIT quotient, but I couldn't figure out how the construction gives you a symplectic structure on the variety.

Let me recall the construction given in Section 2.4 of the survey article. We begin with a reductive group $G$ and its representation $N$. Then the cotangent bundle $T^*N=N^*\oplus N$ is equipped with a Hamiltonian action of $G$. Let $\Phi:T^*N\rightarrow \mathfrak{g}^*$ be the moment map. Choose a character $\chi$ of $G$. Then the *Higgs branch* (which we denote by $Y$) is the projective GIT quotient
$$Y=\text{Proj}\left( \bigoplus_{n\ge 0} \mathbb{C}[\Phi^{-1}(0)]^{G,n\chi}\right).$$
Let $X$ be the GIT quotient
$X=\text{Spec}\left(\mathbb{C}[\Phi^{-1}(0)]^{G}\right)$. Then the canonical map $Y\rightarrow X$ is sometimes a symplectic resolution. I was wondering how this projective GIT quotient of Higgs branch tells you its symplectic structure (assuming the smoothness of Higgs branch).