Duke Mathematical Journal

Cherednik algebras and differential operators on quasi-invariants

Yuri Berest, Pavel Etingof, and Victor Ginzburg

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Abstract

We develop representation theory of the rational Cherednik algebra ${\rm H}\sb c$ associated to a finite Coxeter group $W$ in a vector space $\mathfrak {h}$, and a parameter "c." We use it to show that, for integral values of "c," the algebra ${\rm H}\sb c$ is simple and Morita equivalent to $\mathscr {D}(\mathfrak {h})\#W$, the cross product of $W$ with the algebra of polynomial differential operators on $\mathfrak {h}$.

O. Chalykh, M. Feigin, and A. Veselov [CV1], [FV] introduced an algebra, $Q\sb c$, of quasi-invariant polynomials on $\mathfrak {h}$, such that $\mathbb {C}[\mathfrak {h}]\sp W\subset Q\sb c\subset \mathbb {C}[\mathfrak {h}]$. We prove that the algebra $\mathscr {D}(Q\sb c)$ of differential operators on quasi-invariants is a simple algebra, Morita equivalent to $\mathscr {D}(\mathfrak {h})$. The subalgebra $\mathscr {D}(Q\sb c)\sp W\subset \mathscr {D}(Q\sb c)$ of $W$-invariant operators turns out to be isomorphic to the spherical subalgebra $\mathbf {eH}\sb c\mathbf {e}\subset {\rm H}\sb c$. We show that $\mathscr {D}(Q\sb c)$ is generated, as an algebra, by $Q\sb c$ and its "Fourier dual" $Q\sb c\sp \flat$, and that $\mathscr {D}(Q\sb c)$ is a rank-one projective $(Q\sb c\otimes Q\sb c\sp \flat)$-module (via multiplication-action on $\mathscr {D}(Q\sb c)$ on opposite sides).

Article information

Source
Duke Math. J., Volume 118, Number 2 (2003), 279-337.

Dates
First available in Project Euclid: 23 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1082744649

Digital Object Identifier
doi:10.1215/S0012-7094-03-11824-4

Mathematical Reviews number (MathSciNet)
MR1980996

Zentralblatt MATH identifier
1067.16047

Subjects
Primary: 16S38: Rings arising from non-commutative algebraic geometry [See also 14A22]
Secondary: 14A22: Noncommutative algebraic geometry [See also 16S38] 17Bxx: Lie algebras and Lie superalgebras {For Lie groups, see 22Exx}

Citation

Berest, Yuri; Etingof, Pavel; Ginzburg, Victor. Cherednik algebras and differential operators on quasi-invariants. Duke Math. J. 118 (2003), no. 2, 279--337. doi:10.1215/S0012-7094-03-11824-4. https://projecteuclid.org/euclid.dmj/1082744649


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