On Normalizers of Parabolic Subgroups of Quaternionic Reflection Groups

Abstract

By work of Howlett and Muraleedaran--Taylor, a parabolic subgroup of a real or complex reflection group always admits a complement in its normalizer. In this note, we investigate this phenomenon for quaternionic reflection groups. Here, in contrast to the real and complex setting, we find that complements of parabolic subgroups do not exist in general. Indeed, there are infinitely many examples of quaternionic reflection groups in arbitrary rank greater than 2 with a parabolic subgroup that does not admit a complement in its normalizer. We give a full classification of parabolic subgroups of irreducible quaternionic reflection groups and describe their complements, if the latter exist.

28 pages; v2 added a remark about the connection with the Namikawa-Weyl group of the conic symplectic singularity V/G pointed out to us by Gwyn Bellamy; v3 added reference [BRS26]; to appear in J. Algebra