Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism

By Pavel Etingof, V. Ginzburg. 2000

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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 ℙr, where r=number of conjugacy classes of symplectic reflections in $Γ$. The algebra Hκ, called a symplectic reflection algebra, is related to the coordinate ring of a Poisson deformation of the quotient singularity V/Γ. This leads to a symplectic analogue of McKay correspondence, which is most complete in case of wreath-products. If Γ is the Weyl group of a root system in a vector space ? and V=?⊕?*, 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 ?=??n. We construct a 1-parameter deformation of the Harish-Chandra homomorphism from ?(?)?, the algebra of invariant polynomial differential operators on ??n, to the algebra of Sn-invariant differential operators with rational coefficients on the space ℂn of diagonal matrices. The second order Laplacian on ? goes, under the deformed homomorphism, to the Calogero-Moser differential operator on ℂn, with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: ?(?)? ↠ spherical subalgebra in Hκ, where Hκ is the symplectic reflection algebra associated to the group Γ=Sn. This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of ‘quantum’ Hamiltonian reduction.In the ‘classical’ limit κ→∞, our construction gives an isomorphism between the spherical subalgebra in H∞ and the coordinate ring of the Calogero-Moser space. We prove that all simple H∞-modules have dimension n!, and are parametrised by points of the Calogero-Moser space. The family of these modules forms a distinguished vector bundle on the Calogero-Moser space, whose fibers carry the regular representation of Sn. Moreover, we prove that the algebra $H∞ is isomorphic to the endomorphism algebra of that vector bundle.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/86329e53f64f059bc79c4528b78000e8b9e2a46a

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Citations in the corpus

On singular Calogero–Moser spaces (2007, methodology): “Proof. The theorem is proved by showing that each irreducible H0,c-module is isomorphic to the regular representation of G4. By [ EG , Proposition 3.8], this is equivalent to the statement of the theorem.”
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Finite-dimensional representations of rational Cherednik algebras (2002, background): “Indeed, if J(λ i ) is the maximal proper submodule of M (λ i ) then KZ(J(λ i )) = D λ i+1 (by exactness of KZ).”
Projective-injective modules, Serre functors and symmetric algebras (2005, background): “As a vector space, H(V; ) is isomorphic to S(V ) C[W] S(V ), where S(V ) denotes the algebra of polynomial functions in V (This is the PBWtheorem [ EG02 , Theorem 1.3]).”
Refined knot invariants and Hilbert schemes (2013, background): “Rational Cherednik algebras were introduced in [18] as degenerations of the DAHA.”
Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves (2004, methodology): “this, note first that, as in [ EG , Appendix D], gr Uc ⊠ gr Ud inherits a Poisson structure from multiplication in Uc ⊠ Ud. For a, b, a ′ , b ′ ∈ C[h ⊕ h ∗ ] W this is given by”
On category O for cyclotomic rational Cherednik algebras (2011, background): “In [27] Etingof and Ginzburg introduced symplectic reflection algebras.”
Symplectic reflection algebras (2007, background): “It attempts to explain, from a mostly algebraic point of view, the why rather than the how.”