nt.number theory - Hyperspecial subgroup of a product of semisimple algebraic groups

To allow all characteristics, the semisimplicity requirement on the Lie algebras should be replaced with the requirement that the $G_i$ are semisimple as $F$-groups. (In positive characteristic the Lie algebras of connected semisimple groups can often fail to be semisimple.) In fact, the simply connected and semisimplicity hypotheses are stronger than necessary; connected reductive is sufficient. Taking into account the definition of "hyperspecial maximal compact subgroup" (of the group of $F$-rational points), the question in this modified form immediately reduces to a more "algebraic" assertion having nothing to do with local fields, as follows.

Consider a reductive group scheme $mathcal{G}$ (with connected fibers, following the convention of SGA3) over a normal noetherian scheme $S$, and let $G$ be its fiber over the scheme $eta$ of generic points of $S$. Assume $G$ decomposes as a direct product $G = G_1 times G_2$ with $G_i$ necessarily connected reductive over $eta$. Then does there exist a pair of reductive closed $S$-subgroup schemes (with connected fibers) $mathcal{G}_i$ in $G_i$ such that $mathcal{G}_ 1 times mathcal{G}_2 = mathcal{G}$ compatibly with the given identification on generic fibers? Below is a proof that the answer is ``yes''.

First observe that such $mathcal{G}_i$ are necessarily unique if they exist, being the Zariski closures of the $G_i$ in $mathcal{G}$ (since $S$ is reduced). In view of this uniqueness, by descent theory it follows that to prove the existence we may work 'etale-locally on the base. (This step tends to ruin connectedness hypotheses, since $S' times_S S'$ is generally cannot be arranged to be connected, even if $S$ is connected and $S' rightarrow S$ is a connected 'etale cover.) Hence, we now assume that $mathcal{G}$ is $S$-split in the sense that it admits a (fiberwise) maximal $S$-torus $mathcal{T}$ that is $S$-split. We may also work separately over each connected component of $S$, so we can now assume $S$ is connected (as we will make no further changes to $S$ in the argument).

The maximal $eta$-torus $T := mathcal{T}_ {eta}$ in $G$ uniquely decomposes as $T = T_1 times T_2$ for necessarily $eta$-split maximal $eta$-tori $T_i$ in $G_i$. By the classification of split pairs (i.e., fiberwise connected reductive group equipped with a split maximal torus) over any base scheme in terms of root data, to give a direct product decomposition of $(mathcal{G}, mathcal{T})$ is the same as to decompose its root datum into a direct product (i.e., direct product of the $X$ and $X^{vee}$ parts, and corresponding disjoint union for the $R$ and $R^{vee}$ parts). By connectedness, the normal $S$ is irreducible (i.e., $eta$ is a single point), so the root data of $(mathcal{G}, mathcal{T})$ and its generic fiber $(G,T)$ are canonically identified. QED