On the coadjoint orbits of maximal unipotent subgroups of reductive groups

Simon M. Goodwin, Peter Mosch, Gerhard Roehrle

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)


Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \u its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \u* of \u. This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which has been used for G of rank at most 8, except E8. When G is defined and split over the field of q elements, for q the power of a good prime for G, this algorithmic parametrization is used to calculate the number k(U(q), \u*(q)) of coadjoint orbits of U(q) on \u*(q). Since k(U(q), \u*(q)) coincides with the number k(U(q)) of conjugacy classes in U(q), these calculations can be viewed as an extension of the results obtained in our earlier paper. In each case considered here there is a polynomial h(t) with integer coefficients such that for every such q we have k(U(q)) = h(q).
Original languageEnglish
Pages (from-to)399-426
JournalTransformation Groups
Issue number2
Publication statusPublished - 8 Jul 2015

Bibliographical note

14 pages; v2 23 pages; to appear in Transformation Groups


  • math.GR
  • 20G40, 20E45


Dive into the research topics of 'On the coadjoint orbits of maximal unipotent subgroups of reductive groups'. Together they form a unique fingerprint.

Cite this