论文标题
WeierStrass部分用于类型$ a $中的抛物线伴随动作
Weierstrass sections for parabolic adjoint action in type $A$
论文作者
论文摘要
“ Weierstrass部分”的概念来自Weierstrass的椭圆形曲线的规范形式。在著名的工作中[B. Kostant，关于多项式环的谎言组表示，Amer。 J. Math。 85（1963），327-404]构建了这样的部分，用于使用主s-triple在双重二元上进行半模子代数的作用。实际上，拥有一对“适应性对”是足够的，实际上[A。约瑟夫（Joseph）和D. shafrir，不变的多项式性，单污染和改编对，转化。第15组（2010年），没有。 4，851-882]对于代数的同伴作用，但不一定是还原性的代数。在目前的工作中，weierstrass部分是为抛物线亚代词的派生代数的伴随动作构建的，其nilradical type $ a $。起点是理查森的定理，它暗示了不变的子代数的多项式性。这里很少存在改编对。开发了一种新的结构，这主要是基于与“理查森组件”相关的年轻图表中的盒子组合的组合。给出了以其他类型扩展这种结构的指示。该建筑与Quivers有关系[T. Brustle，L.Hille，Lutz，C.M。 Ringel和G. Rohrle，$δ$滤光的模块，没有$ k [t]/tn $的Auslander代数的自传。代数。代表。理论2（1999），否。 3，295-312]和高表皮轨道品种[A. Joseph和A. Melnikov，$ SL（n）$中的高表面轨道品种的量化。几何和物理学中的轨道方法（Marseille，2000年），第165-196页，Progr。数学，213]。
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 $δ$-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].
