Lectures on quasi-invariants of Coxeter groups and the Cherednik algebra

Abstract

The paper an elementary introduction for non-specialists to the theory of quasi-invariants of Coxeter groups. The main object of study is the variety X_m of quasi-invariants for a finite Coxeter group, which arose in a work of O.Chalykh and A.Veselov about 10 years ago, as the spectral variety of the quantum Calogero-Moser system. Despite being singular, this variety has very nice properties (Cohen-Macaulay, Gorenstein, simplicity of the ring of differential operators, explicitly given Hilbert series). It is interesting that although the definition of X_m is completely elementary, to understand the geometry of X_m it is helpful to use representation theory of the rational degeneration of Cherednik's double affine Hecke algebra, and the theory of integrable systems.

28 pages, latex

References in corpus (1)

• Action of Coxeter groups on m-harmonic polynomials and KZ equations

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• Explicit Computations for the Classical and Quantum Integrability of the 3-Dimensional Rational Calogero-Moser System
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