algebraic groups - Image of a hyperspecial subgroup hyperspecial?

If by "surjective" you mean surjective in the usual sense (for example on $overline{F}$-points) then maybe you have a problem, because $G_1(F)$ may not surject onto $G_2(F)$. So for example $SL(2)$ surjects onto $PGL(2)$ but if $R$ is the integers of $F$ then $SL(2,R)$ is hyperspecial max compact but its image in $PGL(2,F)$ isn't (it's not even maximal, as $PGL(2,R)$ strictly contains the image of $SL(2,R)$).

However if $G_1to G_2$ is, say, a $z$-extension, then (by definition) the kernel is central in $G_1$ and has no $H^1$, so the long exact sequence shows $G_1(F)to G_2(F)$ is surjective. Moreover, if I've got things right, then I think that $G_1$ unramified forces the kernel to be unramified, and if you take a smooth integral model of $G_1$ with $G_1(R)$ equal to the hyperspecial you thought of, then the quotient of this model of $G_1$ by the Zariski closure of the kernel will also be unramified, and the same cohomology argument shows that $G_1(R)$ surjects onto $G_2(R)$, so in this case you win.

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