Normal structure of isotropic reductive groups over rings (Q6568826)

This paper is a further contribution of the authors in the classical problem of describing normal subgroups of reductive algebraic groups over rings; see the introduction of the paper for insights into the history of the problem.\N\NLet \(G\) be a reductive group scheme over a unital ring \(R\). Denote by \(G(R, \mathfrak{q})\) the principal congruence subgroup of \(G(R)\) of level \(\mathfrak{q}\), i.e. the kernel of the natural homomorphism \(\rho_{\mathfrak{q}} : G(R) \rightarrow G(R/\mathfrak{q})\). If the isotropic rank of \(G\) is \(\geq 1\) then \(G\) contains a pair of opposite strictly proper parabolic \(R\)-subgroups \(P^{\pm}\). The elementary subgroup \(E_P(R)\) is defined as the subgroup of \(G(R)\) generated by \(U_{P^{+}}(R)\) and \(U_{P^{-}}(R)\), where \(U_{P^{\pm}}\) is the unipotent radical of \(P^{\pm}\).\N\NAssume that the isotropic rank of \(G\) is \(\geq 2\). Then \(E(R) = E_P(R)\) is independent of the choice of \(P^{\pm}\) and is normal in \(G(R)\). Denote by \(U_{P^{\pm}}(\mathfrak{q})\) the intersection of \(G(R, \mathfrak{q})\) and \(U_{P^{\pm}}(R)\), and by \(E_{P}(R, \mathfrak{q})\) the normal closure of the subgroup generated by \(U_{P^{+}}(\mathfrak{q})\) and \(U_{P^{-}}(\mathfrak{q})\) in \(E(R)\). The full congruence subgroup \(C(R, \mathfrak{q})\) is defined as the full pre-image of the center of \(G(R/\mathfrak{q})\) under \(\rho_{\mathfrak{q}}\). For a maximal ideal \(\mathfrak{m}\) of \(R\) denote by \(\overline{R/\mathfrak{m}}\) the algebraic closure of the field \(R/\mathfrak{m}\). The main result of the paper is as follows (Theorem 1.1).\N\NLet \(G\) be a reductive group scheme over a ring \(R\) such that its isotropic rank is \(\geq 2\). Suppose that for every maximal ideal \(\mathfrak{m}\) of \(R\) the root system of \(G_{\overline{R/\mathfrak{m}}}\) is irreducible, and its structure constants are invertible in \(R\). Let \(P\) be a proper parabolic subgroup of \(G\). Then\N\begin{itemize}\N\item[(i)] \(E_{P}(R, \mathfrak{q}) = [G(R, \mathfrak{q}), E(R)]\) for any ideal \(\mathfrak{q}\) of \(R\). In particular, \(E_{P}(R, \mathfrak{q}) = E(R, \mathfrak{q})\) is independent of the choice of a parabolic \(R\)-subgroup \(P\).\N\item[(ii)] For any subgroup \(H \leq G(R)\) normalized by \(E(R)\), there exists a unique ideal \(\mathfrak{q}\) in \(R\) such that \(E(R, \mathfrak{q}) \leq H \leq C(R, \mathfrak{q})\).\N\end{itemize}\N\NThis theorem provides so-called sandwich description of subgroups of the group \(G\) normalized by the elementary subgroup \(E(R)\).