Weierstrass sections for parabolic adjoint action in type A

Abstract

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