# Finite Singular Orbit Modules for Algebraic Groups

## TL;DR

This paper classifies faithful irreducible modules for certain algebraic groups with finitely many orbits on singular 1-spaces, linking orbit structure to double coset finiteness, and presents new examples across various groups.

## Contribution

It provides a complete classification of modules with finitely many orbits on singular 1-spaces for specific algebraic groups, extending previous orbit classifications.

## Key findings

- Classified all faithful irreducible modules with finitely many orbits on singular 1-spaces.
- Connected orbit finiteness to double coset problems for subgroup pairs.
- Discovered new modules, including a 5-dimensional module for SL_2 and a spin module for B_6 in characteristic 2.

## Abstract

Building on the classification of modules for algebraic groups with finitely many orbits on subspaces, we determine all faithful irreducible modules for simple and maximal-semisimple connected algebraic groups that are orthogonal and have finitely many orbits on singular $1$-spaces. This question is naturally connected with the problem of finding for which pairs of subgroups $H,K$ of an algebraic group $G$ there are finitely many $(H,K)$-double cosets. This paper provides a solution to the question when $K$ is a maximal parabolic subgroup $P_1$ of a classical group $SO_n$. We find an interesting range of new examples ranging from a $5$-dimensional module for $SL_2$ to the spin module for $B_6$ in characteristic $2$.