Cherednik algebras and differential operators on quasi-invariants
Cherednik algebras and differential operators on quasi-invariants
复制标题
Cherednik 代数和准不变量的微分算子
DOI：
10.1215/s0012-7094-03-11824-4
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发表时间：
2001
影响因子：
2.5
通讯作者：
V. Ginzburg
中科院分区：
文献类型：
--
作者：
Y. Berest;P. Etingof;V. Ginzburg
我们发展了与向量空间\(h\)中的有限考克斯特群\(W\)相关联的有理切列尼克代数\(H\)的表示理论。它被用于表明，对于参数\(c\)的整数值，代数\(H\)是单的并且与\(D(h)\#W\)（\(W\)与\(h\)上的多项式微分算子代数的叉积）莫里塔等价。
我们进一步研究了由查利赫、费金和韦谢洛夫[CV]，[FV]引入的\(h\)上的拟不变多项式的代数\(Q\)，使得\(C[h]^W\subset Q\subset C[h]\)。我们证明了拟不变量上的微分算子代数\(D(Q)\)是一个单代数，与\(D(h)\)莫里塔等价。\(W\) -不变算子的子代数\(D(Q)^W\)结果同构于\(H\)中的球面子代数\(eHe\)。我们还表明，作为一个代数，\(D(Q)\)由\(Q\)及其“傅里叶对偶\(Q^*\)”生成，并且\(D(Q)\)是一个秩为\(1\)的投射\((Q - Q^*)\) -模（通过在\(D(Q)\)的两侧的乘法作用）。
We develop representation theory of the rational Cherednik algebra H associated to a finite Coxeter group W in a vector space h. It is applied to show that, for integral values of parameter `c', the algebra H is simple and Morita equivalent to D(h)#W, the cross product of W with the algebra of polynomial differential operators on h.
We further study an algebra Q of quasi-invariant polynomials on h introduced by Chalykh, Feigin, and Veselov [CV], [FV], such that C[h]^W \subset Q \subset C[h]. We prove that the algebra D(Q) of differential operators on quasi-invariants is a simple algebra, Morita equivalent to D(h). The subalgebra D(Q)^W of W-invariant operators turns out to be isomorphic to the spherical subalgebra eHe \subset H. We also show that D(Q) is generated, as an algebra, by Q and its `Fourier dual Q*, and that D(Q) is a rank one projective (Q-Q*)-module (via multiplication-action on D(Q) on opposite sides).