On the coadjoint orbits of maximal unipotent subgroups of reductive groups

TL;DR

This paper studies the coadjoint orbits of maximal unipotent subgroups in reductive groups, providing a parametrization method, an explicit algorithm, and polynomial formulas for counting orbits over finite fields.

Contribution

It introduces a new parametrization of coadjoint orbits, proves key properties like separability and connectedness, and develops an algorithm for explicit calculations, extending previous results.

Findings

Parametrization of coadjoint orbits via minimal representatives.

An explicit algorithm for calculating orbit parametrization.

Polynomial formulas for counting orbits over finite fields.

Abstract

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).