Cherednik algebras and differential operators on quasi-invariants

By Y. Berest, Pavel Etingof, V. Ginzburg. 2001

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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).

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2c7546e7779ae18f6e20323bfa712288045a26c2

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Citations in the corpus

Symplectic reflection algebras (2007, background): “This interesting category, which contains all the finite dimensional representations of H1,c for instance, was introduced and studied in [45], [11], [83] and [68].”
On primitive ideals (2002, methodology): “End C M, and the assignment a 7→ ˆ a gives an algebra imbedding: Hc/Ann M ֒→ End C M. The adjoint action of the subalgebra C[h] W on Hc being locally-nilpotent, see [ BEG ],”
Parabolic induction and restriction functors for rational Cherednik algebras (2008, background): “4.4. The simplicity of the spherical subalgebra for −1 . In [BEG1], it is shown that the algebra Hc(W, h) is simple if and only if c is not a singular value, and in this case Hc(W, h) is Morita equivalent to its spherical subalgebra eWHc(W, h)eW.”
Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra ? (2007, background): “The definition of a spherical subalgebra of a DAHA goes back to Etingof & Ginzburg [3], where a similar object was defined in the context of Cherednik algebras, see also [1].”
Combinatorics, symmetric functions, and Hilbert schemes (2002, methodology): “Gordon proves Theorem 7.2.4 using representation theory of Cherednik algebras, more specifically their rational degenerations Hc which have been studied by Dunkl [15], Opdam and Rouqier [unpublished], Berest, Etingof and Ginzburg [ 5 , 6], and Guay [unpublished] among others.”