Abstract:In this talk I will present a miscellany of facts, some old and some new, concerning the maximal rank subgroups of a compact Lie group. These subgroups were classified long ago by Borel and de Siebenthal. They show up in the singularities of weight varieties, the index theory of homogeneous differential operators, and various induction maps among representation rings. Part of this is ongoing joint work with Megumi Harada and Gregory Landweber.