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Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism
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| Venue: | Invent. Math |
| Citations: | 279 - 39 self |
BibTeX
@ARTICLE{Etingof_symplecticreflection,
author = {Pavel Etingof and Victor Ginzburg},
title = {Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism},
journal = {Invent. Math},
year = {},
pages = {348}
}
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Abstract
To any finite group Γ ⊂ Sp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ, of the algebra C[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of CP r, where r = number of conjugacy classes of symplectic reflections in Γ. The algebra Hκ, called a symplectic reflection algebra, is expected to be related to the coordinate ring of a universal Poisson deformation of the quotient singularity V/Γ. If Γ is the Weyl group of a root system in a vector space h and V = h ⊕ h ∗ , then the algebras Hκ are certain ‘rational ’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let Γ = Sn, the Weyl group of g = gl n. We construct a 1-parameter deformation of the Harish-Chandra homomorphism from D(g) g, the algebra of invariant polynomial differential operators on the Lie algebra g = gl n, to the algebra of Sn-invariant differential operators on the Cartan subalgebra C n with rational coefficients. The second order Laplacian on g goes, under our deformed homomorphism, to the Calogero-Moser
Keyphrases
harish-chandra homomorphism symplectic reflection algebra calogero-moser space weyl group smash product universal poisson deformation deformed homomorphism lie algebra gl quotient singularity vector space new multi-parameter deformation polynomial algebra root system invariant polynomial differential operator conjugacy class coordinate ring finite group sp symplectic reflection double affine hecke algebra rational coefficient sn-invariant differential operator cartan subalgebra symplectic vector space certain rational degeneration second order laplacian 1-parameter deformation
