标题： 球面子代数上的多项式的分解 显示英文标题

标题： Decomposition of the polynomials over the spherical subalgebra

摘要： 给定复数向量空间 $\fh$的线性群的有限子群 $W \subset \GL(\fh)$，描述对称代数 $B= \sym(\fh^*)$作为 $G$的表示结构，以及作为在 $W$下的不变微分算子环中的模，以及在 $\fh$上的微分算子环 $\D(\fh)$中的结构，是一个被广泛研究的问题。 由于有理Cherednik代数$H_c(W,\fh)$和球面代数$eH_ce$分别是环$\D(\fh)$和环$\D(\fh)^W$的通用形变，其中$W$是不变微分算子，我们希望在\cite{ Nonk1, Nonk2, Nonk3}中不变微分算子上的模分解与有理Cherednik代数的球面子代数上的模分解之间建立类比。 环$\D_c= eH_ce$继承了$B$的自然分次，我们让$\D_c^0 \subset \D_c$和$\D_c^{-} \subset \D_c$分别为次数为 0 和严格负次数的元素子集。 我们的主要结果是，对于所有有限反射群，存在一个最低权描述，用于$B$的$\D_c$-模范畴，其中环$\Rc_c= \D_c^0 / \Dc_c^0 \cap \D_c \D_c^{-}$在其中扮演了卡丹代数的重要角色。 显示英文摘要

摘要： Given a finite subgroup $W \subset \GL(\fh)$ of the linear group of a finite-dimensional complex vector field $\fh$, it is a well-studied problem to describe the structure of the symmetric algebra $B= \sym(\fh^*)$ as a representation of $G$, and also as a module over the ring of invariant differential operators under $W$ in the ring $\D(\fh)$ of differential operators on $\fh$. Since the rational Cherednik algebra $H_c(W,\fh)$ and the spherical algebra $eH_ce$ are respectively universal deformations of the ring $\D(\fh)$ and the ring $\D(\fh)^W$of $W$-invariant differential operators, we would like to build an analogy between the decomposition of modules over the invariant differential operators in \cite{ Nonk1, Nonk2, Nonk3} and the decomposition of modules over the sperical subalgebra of the rational Cherednik algebra. The ring $\D_c= eH_ce$ inherits the natural grading of $B$, and we let $\D_c^0 \subset \D_c$ and $\D_c^{-} \subset \D_c$ be subset of elements of degree 0 end strictly negative degree, respectively. Our main result is that there is for all finite reflection groups a lowest weight description of the category of $\D_c$-modules of $B$ where the ring $\Rc_c= \D_c^0 / \Dc_c^0 \cap \D_c \D_c^{-}$ plays the very important role of Cartan algebra.