Unipotent representations of Lie incidence geometries

Abstract

If a geometry $Γ$ is isomorphic to the residue of a point $A$ of a shadow geometry of a spherical building $Δ$, a representation $\varepsilon_Δ^A$ of $Γ$ can be given in the unipotent radical $U_{A^*}$ of the stabilizer in $\mathrm{Aut}(Δ)$ of a flag $A^*$ of $Δ$ opposite to $A$, every element of $Γ$ being mapped onto a suitable subgroup of $U_{A^*}$. We call such a representation a unipotent representation. We develope some theory for unipotent representations and we examine a number of interesting cases, where a projective embedding of a Lie incidence geometry $Γ$ can be obtained as a quotient of a suitable unipotent representation $\varepsilon_Δ^A$ by factorizing over the derived subgroup of $U_{A^*}$, while $\varepsilon^A_Δ$ itself is not a proper quotient of any other representation of $Γ$.