Etale Fundamental group of moduli of torsors under Bruhat-Tits group scheme over a curve

Abstract: Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $\mathcal{G}$ be a Bruhat-Tits group scheme on $X$ which is generically semi-simple and trivial. We show that the \'etale fundamental group of the moduli stack $\mathcal{M}_X(\mathcal{G})$ of torsors under $\mathcal{G}$ is isomorphic to that of the moduli stack $\mathcal{M}_X(G)$ of principal $G$-bundles. For any smooth, noetherian and irreducible stack $\mathcal{X}$, we show that an inclusion of an open substack $\mathcal{X}^\circ$, whose complement has codimension at least two, will induce an isomorphism of \'etale fundamental group. Over $\mathbb{C}$, we show that the open substack of regularly stable torsors in $\mathcal{M}_X(\mathcal{G})$ has complement of codimension at least two when $g_X \geq 3$. As an application, we show that the moduli space $M_X(\mathcal{G})$ of $\mathcal{G}$-torsors is simply-connected.