Parametrizing Real Even Nilpotent Coadjoint Orbits Using Atlas
نویسنده
چکیده
Suppose GR is the real points of a complex connected reductive algebraic group G defined over R. (The class of such groups is exactly the class treated by the software package atlas.) Write g R for the dual of the Lie algebra of GR and N R for the nilpotent elements in gR. Then GR acts with finitely many orbits on N via the coadjoint action, and it is clearly very desirable to parametrize these orbits in a way that atlas can manipulate. For instance, it is definitely not sufficient to produce a list of tables of such orbits for, say, G simple. The purpose of these notes is to describe such a parametrization of even nilpotent orbits. This can be extracted from [ABV, Chapter 20], but we offer a more easily accessible treatment here and make the atlas-based algorithms explicit and effective. The parametrization is in terms of certain closed orbits of a symmetric group K on partial flag varieties P for G. Since the geometry of K orbits on partial flag varieties is “dual” to the study of translation families of Harish-Chandra modules with singular infinitesimal character, the description of K\P is of independent interest. We give complete details in Section 2 before turning to the parametrization of nilpotent orbits in Section 3.
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