Thanks! For those who might take a look at Lemma 4.1.7 of Ginzburg's note, the vector space on the last line should be $T_{\alpha}(\mu^{-1}(\lambda)/G)=T_{\alpha}(\mu^{-1}(\lambda))/Lie(G)$. The vector space $T_{\alpha}(\mu^{-1}(\lambda))$ is something coisotropic mentioned in the answer and $\text{Lie}(G)$ (value of fundamental vector fields at $\alpha$) is its orthogonal subspace with respect to the symplectic form of $T^*X$. Therefore quotienting out by $\text{Lie}(G)$ gives nondegeneracy.