Finite Singular Orbit Modules for Algebraic Groups

2019/07/15 by Rizzoli, Aluna
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1907.06755

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 P₁ of a classical group SO_n. We find an interesting range of new examples ranging from a 5-dimensional module for SL₂ to the spin module for B₆ in characteristic 2.

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