Unipotent representations of Lie incidence geometries

Abstract

If a geometry $\Gamma$ is isomorphic to the residue of a point $A$ of a shadow geometry of a spherical building $\Delta$, a representation ${\epsilon }_{\Delta }^A$ of $\Gamma$ can be given in the unipotent radical $U_{A^{\ast }}$ of the stabilizer in $\mathrm{Aut}(\Delta )$ of a flag $A^{\ast }$ of $\Delta$ opposite to $A$, every element of $\Gamma$ being mapped onto a suitable subgroup of $U_{A^{\ast }}$. 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 $\Gamma$ can be obtained as a quotient of a suitable unipotent representation ${\epsilon }_{\Delta }^A$ by factorizing over the derived subgroup of $U_{A^{\ast }}$, while ${\epsilon }_{\Delta }^A$ itself is not a proper quotient of any other representation of $\Gamma$.

35 pages