TITLE: Etale fundamental of a parahoric group scheme
QUESTION [7 upvotes]: Let $p:X\rightarrow Y$ be a double cover of curves, denote by $$SU_n:=(p_*SL_n(\mathcal O_X))^{\tilde{\sigma}}$$
i.e. the $\tilde{\sigma}-$invariant part, the action of $\tilde{\sigma}$ is given by $$\tilde{\sigma}(g)=\,^t(g\circ\sigma)^{-1}$$ where $\sigma$ is the involution induced by the double cover. $SU_n$ is well knowing to be a parahoric group shceme in the sens of Bruhat-Tits. 
My question: What is $\pi_1(({SU_n})_\eta)$ ? (the algebraic fundamental group). where $\eta$ is the generic point of $Y$.
Thanks

REPLY [2 votes]: There is an exact sequence
$$ \pi_1 ((SU_n)_{\overline{\eta}}) \to \pi_1((SU_n)_{\eta}) \to \operatorname{Gal}(\overline{\eta}|\eta)$$
Over $\overline{\eta}$, $p_* SL_n$ is just $SL_n \times SL_n$. The involution $\tilde{\sigma}$ acts by switching the two $SL_n$s and then doing an performing some automorphism, so the involution invariants $SU_n$ are just the diagonal $SL_n$ embedded by the graph of the automorphism.
In characteristic $0$, $\pi_1(SL_n)$ is trivial, so the fundamental group is just the Galois group of the base field. In characteristic $p$ things are more complicated as the fundamental group is infinite, as witnessed by the family of etale covers $g \to g^{-1} \operatorname{Frob}_q(g)$ which is Galois of group $SL_n(\mathbb F_q)$ for each $q$.