Invariants of the coadjoint action on the basic varieties of the unitriangular group

We find the generators of the fields of invariants of the coadjoint action of the unitriangular group on the basic varieties and basic cells. It is proved that the transcendental degree of the field of invariants on a basic cell coincides with the number of factors in the special factorization of the associated element of the Weyl group as a product of reflections.