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



Etale fundamental of a parahoric group scheme
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 $q$.


