Abstract
Building on the classification of modules for algebraic groups with finitely many orbits on subspaces [9], we determine all faithful irreducible modules V for a connected simple algebraic group H, such that H≤SO(V) and H has finitely many orbits on singular 1-spaces of V. We do the same for H connected semisimple, and maximal among connected semisimple subgroups. 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 P1 of a classical group SOn, where P1 is the stabilizer of a singular 1-space. We find an interesting range of new examples, from a 5-dimensional module for A1 in characteristic p>5, to the spin module for B6 when p=2.
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