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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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