Weierstrass sections for parabolic adjoint action in type A

The notion of “Weierstrass Section”, comes from Weierstrass canonical form for elliptic curves. In celebrated work Kostant (1963) [26] constructed such a section the action semisimple Lie algebra on its dual using principal s-triple. Actually it is enough to have an “adapted pair” and indeed construction in Joseph Shafrir (2010) [25] works rather well co-adjoint algebraic, but not necessarily reductive algebra. present adjoint parabolic subgroup P nilradical m type A. starting point Richardson's theorem which asserts that admits dense orbit (in all types). For A explicit generator this was given Brüstle et al. (1999) [4]. It involves joining boxes Young tableau associated P. On other hand implies polynomiality (semi)-invariant subalgebra, where generators themselves are through ideals definition hypersurface orbital varieties Melnikov (2003) [24]. (Again relationship holds types.) adapted pairs seldom exist. Thus entirely new developed mainly combinatorial. initial step Ringel ([24]). significantly modified take account invariant suggested Benlolo Sanderson (2001) [3]. Indications as how one might extend types.