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

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

2002-04-09

arxiv.org/abs/math/0204104v1

Mathematics

Quantum Algebra

28 pages, latex

Scientific paper

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.

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