Unipotent representations of Lie incidence geometries

TL;DR

This paper develops a theory of unipotent representations for Lie incidence geometries, showing how certain embeddings can be realized as quotients of these representations, revealing new structural insights.

Contribution

It introduces the concept of unipotent representations for Lie incidence geometries and explores their properties and applications in projective embeddings.

Findings

Unipotent representations can produce projective embeddings as quotients.

These representations are not proper quotients of other representations.

A theoretical framework for unipotent representations is established.

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 $\varepsilon_\Delta^A$ of $\Gamma$ can be given in the unipotent radical $U_{A^*}$ of the stabilizer in $\mathrm{Aut}(\Delta)$ of a flag $A^*$ of $\Delta$ opposite to $A$, every element of $\Gamma$ 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 $\Gamma$ can be obtained as a quotient of a suitable unipotent representation $\varepsilon_\Delta^A$ by factorizing over the derived subgroup of $U_{A^*}$, while $\varepsilon^A_\Delta$ itself is not a proper quotient of any other representation of $\Gamma$.