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Cherednik algebras and differential operators on quasi-invariants. (English) Zbl 1067.16047
Let $$W$$ be a finite Coxeter group generated by reflections in a finite-dimensional complex vector space $$\mathfrak h$$. The rational Cherednik algebras associated to $$W$$ is a family of associative algebras $$\{H_c(W)\}$$ parametrized by the set of $$W$$-invariant complex multiplicities $$c\colon R\to\mathbb{C}$$ on the system of roots $$R\subset{\mathfrak h}^*$$ of $$W$$. Specifically, for a fixed $$c\colon\alpha\to c_\alpha$$, the algebra $$H_c=H_c(W)$$ is generated by the vectors of $$\mathfrak{h,h}^*$$, and the elements of $$W$$ subject to the following relations $wxw^{-1}=w(x),\quad wyw^{-1}=w(y),\quad\forall y\in{\mathfrak h},\;x\in{\mathfrak h}^*,\;w\in W;$ $x_1x_2=x_2x_1,\quad y_1y_2=y_2y_1,\quad\forall y_1,y_2\in{\mathfrak h},\;x_1,x_2\in{\mathfrak h}^*;$ $yx-xy=\langle y,x\rangle-\sum_{\alpha\in R_+}c_\alpha\langle y,\alpha\rangle\langle\alpha^\vee,x\rangle s_\alpha,\quad\forall y\in{\mathfrak h},\;x\in{\mathfrak h}^*.$ Here, as usual, one writes $$\alpha^\vee\in{\mathfrak h}$$ for the coroot, $$s_\alpha\in W$$ for the reflection corresponding to the root $$\alpha\in R$$, $$R_+\subset R$$ for a choice of ‘positive’ roots in $$R$$, and $$\langle\cdot,\cdot\rangle$$ for the canonical pairing between $$\mathfrak h$$ and $${\mathfrak h}^*$$. Furthermore, each $$H_c$$ contains a distinguished subalgebra $$B_c:=eH_ce$$, where $$e:=\tfrac 1{|W|}\sum w$$ is the symmetrizing idempotent in $$\mathbb{C} W\subset H_c$$. We call $$B_c=B_c(W)$$ the spherical algebra associated to $$(W,c)$$.
In this paper, the authors develop representation theory of the rational Cherednik algebras $$H_c$$. The authors show that, for integral values of $$c$$, the algebra $$H_c$$ is simple and Morita equivalent to $${\mathcal D}({\mathfrak h})\#W$$, the cross product of $$W$$ with the algebra of polynomial differential operators on $$\mathfrak h$$.
For the algebra, $$Q_c$$, of quasi-invariant polynomials on $$\mathfrak h$$ introduced by O. A. Chalykh, M. Feigin and A. P. Veselov [Commun. Math. Phys. 126, No. 3, 597-611 (1990; Zbl 0746.47025) and Int. Math. Res. Not. 2002, No. 10, 521-545 (2002; Zbl 1009.20044)] the authors prove that the algebra $${\mathcal D}(Q_c)$$ of differential operators on quasi-invariants is a simple algebra, Morita equivalent to $${\mathcal D}(\mathfrak h)$$. The subalgebra $${\mathcal D}(Q_c)^W\subset{\mathcal D}(Q_c)$$ of $$W$$-invariant operators turns out to be isomorphic to the spherical subalgebra $$B_c$$. The authors show that $${\mathcal D}(Q_c)$$ is generated, as an algebra, by $$Q_c$$ and its “Fourier dual” $$Q_c^b$$, and that $${\mathcal D}(Q_c)$$ is a rank-one projective $$(Q_c\otimes Q_c^b)$$-module.

MSC:
 16S32 Rings of differential operators (associative algebraic aspects) 14A22 Noncommutative algebraic geometry 17B20 Simple, semisimple, reductive (super)algebras 20C08 Hecke algebras and their representations 20F55 Reflection and Coxeter groups (group-theoretic aspects) 14L24 Geometric invariant theory
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