Weierstrass sections for parabolic adjoint action in type $A$

TL;DR

This paper constructs a Weierstrass section for the adjoint action of a parabolic subalgebra's derived algebra on its nilradical in type A, using combinatorial methods based on Young tableaux, extending classical invariant theory.

Contribution

It introduces a new combinatorial construction of Weierstrass sections for a specific Lie algebra action where adapted pairs are rare, extending the theory beyond classical cases.

Findings

Constructed a Weierstrass section in type A using Young tableaux.

Demonstrated polynomiality of the invariant algebra via Richardson's theorem.

Indicated potential extensions to other Lie types.

Abstract

The notion of "Weierstrass Section", comes from Weierstrass canonical form for elliptic curves. In celebrated work [B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327-404] 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 [A. Joseph and D. Shafrir, Polynomiality of invariants, unimodularity and adapted pairs, Transform. Groups 15 (2010), no. 4, 851-882] works rather well for the coadjoint action of an algebraic, but not necessarily reductive Lie algebra. In the present work a Weierstrass section is constructed for the adjoint action of the derived algebra of a parabolic subalgebra on its nilradical in type $A$. The starting point is Richardson's theorem which implies the polynomiality of the invariant sub-algebra. Here adapted pairs seldom exist. A new construction is developed and this is mainly combinatorial based on joining boxes in the Young tableau associated to the "Richardson component". Indications are given for extending this construction in other types. The construction has relations to quivers [T. Brustle, L. Hille, Lutz, C.M. Ringel and G. Rohrle, The $\delta$-filtered modules without selfextensions for the Auslander algebra of $k[T]/Tn$. Algebr. Represent. Theory 2 (1999), no. 3, 295-312] and to hypersurface orbital varieties [A. Joseph and A. Melnikov, Quantization of hypersurface orbital varieties in $sl(n)$. The orbit method in geometry and physics (Marseille, 2000), 165-196, Progr. Math., 213].