LULU is syntomic

Abstract

Let G be a Chevalley group over a field $k$. Fix a maximal torus $T$ in G, along with opposite Borel subgroups B and $B^{-}$ satisfying $T=B\cap B^{-}$, and denote by $U:=R_u\left(B\right)$ and $L:=R_u\left(B^{-}\right)$ their respective unipotent radicals. We prove that the multiplication map $\mu :L\times U\times L\times U⟶G$ is syntomic and faithfully flat over any base field $k$.