Moduli of $\mathcal{G}$-bundles under nonconnected group schemes and nondensity of essentially finite bundles

Abstract: We prove the existence of a projective good moduli space of principal $\mathcal{G}$-bundles under nonconnected reductive group schemes $\mathcal{G}$ over a smooth projective curve $C$. We also prove that the moduli stack of $\mathcal{G}$-bundles decomposes into finitely many substacks $\text{Bun}{\mathcal{P}}$ each admitting a torsor $\text{Bun}{\mathcal{G}\mathcal{P}}\to \text{Bun}{\mathcal{P}}$ under a finite group, for some connected reductive group schemes $\mathcal{G}_\mathcal{P}$ over $C$. We use this for the second purpose of the article: for any constant connected reductive group $G$, the subset of essentially finite $G$-bundles in the moduli of degree 0 semistable $G$-bundles over $C$ is not dense, unless $G$ is a torus or the genus of $C$ is smaller than 2. We do this by giving an upper bound on the dimension of the closure of the subset of essentially finite $G$-bundles.