Reductive group schemes, the Greenberg functor, and associated algebraic groups

Abstract

Let $A$ be an Artinian local ring with algebraically closed residue field $k$, and let $\mathbf{G}$ be an affine smooth group scheme over $A$. The Greenberg functor $\mathcal{F}$ associates to $\mathbf{G}$ a linear algebraic group $G:=(\mathcal{F}\mathbf{G})(k)$ over $k$, such that $G\cong\mathbf{G}(A)$. We prove that if $\mathbf{G}$ is a reductive group scheme over $A$, and $\mathbf{T}$ is a maximal torus of $\mathbf{G}$, then $T$ is a Cartan subgroup of $G$, and every Cartan subgroup of $G$ is obtained uniquely in this way. The proof is based on establishing a Nullstellensatz analogue for smooth affine schemes with reduced fibre over $A$, and that the Greenberg functor preserves certain normaliser group schemes over $A$. Moreover, we prove that if $\mathbf{G}$ is reductive and $\mathbf{P}$ is a parabolic subgroup of $\mathbf{G}$, then $P$ is a self-normalising subgroup of $G$, and if $\mathbf{B}$ and $\mathbf{B}'$ are two Borel subgroups of $\mathbf{G}$, then the corresponding subgroups $B$ and $B'$ are conjugate in $G$.