Finite Singular Orbit Modules for Algebraic Groups (1907.06755v1)

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 $2$.