Unipotent representations of Lie incidence geometries

Abstract

Given a flag \(A\) of a spherical building \(\Delta \) of rank \(n\ge 3\), let \(\Gamma := \Delta _{J,A}\) be the residue of \(A\) in the \(J\)-shadow geometry \(\Delta _J\) of \(\Delta \), where \(J := \tau (A)\) is the type of \(A\). Let \(A^*\) be a flag opposite to \(A\) and \(G_{A^*}\) the stabilizer of \(A^*\) in the group \(G := \mathrm {Aut}(\Delta )\) of all type-preserving automorphisms of \(\Delta \). It is well known that \(G_{A^*}\) admits a maximal normal nilpotent subgroup \(U_{A^*}\), called the unipotent radical of \(G_{A^*}\) and acting regularly on the set of flags opposite to \(A^*\). As we will show in this paper, a representation \(\varepsilon _\Delta ^J\) of \(\Gamma \) in \(U_{A^*}\) can be given, every element of \(\Gamma \) being mapped onto a suitable subgroup of \(U_{A^*}\), containment between elements of \(\Gamma \) corresponding to containment between the corresponding subgroups. We call such a representation a unipotent representation. We develop some theory for unipotent representations and we examine a number of interesting cases, where a projective embedding of \(\Gamma \) can be obtained as a quotient of \(\varepsilon _\Delta ^J\) by factorizing over the derived subgroup of \(U_{A^*}\), while \(\varepsilon ^J_\Delta \) itself is dominant, namely it is not a proper image of any other representation of \(\Gamma \) in any group.