Symplectic reflection algebras

Pavel Etingof and Victor Ginzburg

Introduction Given an action of a finite group Γ on a smooth affine algebraic variety X, one can study the algebraic structure of the orbifold X/Γ through its coordinate ring ℂ[X]^{Γ}, the ring of Γ-invariant regular functions on X. It is known, however, that if the Γ-action on X is not free the (possibly singular) space X/Γ does not necessarily reflect the features of the Γ-action quite adequately. For instance, one is lead to introduce various ”stringy” topological invariants, cf. [Ba], which do not coincide with the ordinary topological invariants of X/Γ, but are rather related to certain ”resolutions” of X/Γ. A standard approach to this problem, see e.g. [Co], is to replace the commutative coordinate ring ℂ[X]^{Γ} by the smash-product algebra ℂ[X]#Γ. The two algebras are Morita equivalent, hence lead to equivalent theories, provided the Γ-action on X is free, but are not equivalent in general. Although the algebra ℂ[X]#Γ is non-commutative, it is believed that the ”right” geometry of the Γ-action on X can be read-off from the ”non-commutative algebraic geometry” of ℂ[X]#Γ. This is the approach that we adopt in the present paper.

Specifically, we will study deformations of some interesting orbifolds X/Γ through non-commutative deformations of the corresponding algebra ℂ[X]#Γ. We will see that a systematic use of this ideology leads, via Morita equivalence, to various new results in conventional ”commutative” algebraic geometry which have been unaccessible by purely ”commutative” methods.

Another fascinating feature of our approach is that genuine deformations of X/Γ coming from ”commutative” algebraic varieties stand on an equal footing with purely ”quantum” deformations that are not associated with any algebraic varieties whatsoever. ”Commutative” and ”quantum” deformations only differ by the values of deformation parameters. Thus, ”quantum” deformations may be viewed as quantizations of ”commutative” ones, and Kontsevich’s quantization plays a role here. On the other hand, the algebras arising in such ”quantum” deformations are closely related to the algebras of invariant polynomial differential operators on semisimple Lie algebras and Quiver varieties. Thus, on the ”quantum” side, we enter the realm of representation theory, which was our original motivation. Symplectic reflection algebras. Let V be a finite dimensional vector space over ℂ, and T V its tensor algebra. Let Γ ⊂ GL(V) be a finite group, and (T V)#Γ the smashproduct of T V with ℂΓ, the group algebra of Γ. Given a skew-symmetric ℂ-bilinear pairing κ : V × V → ℂΓ, put

H_{κ} := (T V #Γ)/I⟨x ⊗ y − y ⊗ x − κ(x, y) ∈ T^{2}V ⊕ ℂΓ⟩_{x,y∈V} ,  (1)

where I⟨… ⟩ stands for the two-sided ideal in (T V)#Γ generated by the indicated set. Thus, H_{κ} is an associative algebra.

In the special case κ = 0, the situation simplifies drastically since all the generators x ∈ V commute with each other. The resulting algebra is therefore isomorphic to the smash-product algebra: H_{κ=0} = (SV) # Γ. Note that this algebra has a natural grading obtained by placing ℂΓ in grade degree zero, and V in grade degree 1.

In general, if κ ≠ 0, the defining relations, see (1), become inhomogeneous, hence assigning ℂΓ grade degree zero, and V grade degree 1, one only gets an increasing filtration F_{•}(H_{κ}) on the corresponding algebra H_{κ}, such that, for any i, j ≥ 0, we have: F_{i}(H_{κ}) · F_{j}(H_{κ}) ⊂ F_{i+j}(H_{κ}). Let gr(H_{κ}) = ⨁_{i} F_{i}(H_{κ})/F_{i−1}(H_{κ}) denote the associated graded algebra. It is immediate from (1) that, for any x, y ∈ V ⊂ gr(H_{κ}), in gr(H_{κ}) we have: xy − yx = 0. Therefore the tautological imbedding: V ↪ gr(H_{κ}) extends to a well-defined and surjective graded algebra homomorphism: H_{κ=0} = (SV)#Γ ⟶ gr(H_{κ}). We say that Poincaré-Birkhoff-Witt (PBW-)property holds for H_{κ} if this morphism is an isomorphism.

From now on we assume, in addition, that V is a symplectic vector space with symplectic 2-form ω, and that Γ ⊂ Sp(V). The triple (V, ω, Γ) is said to be indecomposable if there is no ω-orthogonal direct sum decomposition: V = V_{1} ⊕ V_{2}, where V_{i} are Γ-stable proper symplectic vector subspaces in V.

Definition 2 An element s ∈ Γ is called a symplectic reflection if rk(Id − s) = 2.

Let S denote the set of symplectic reflections in Γ. The group Γ acts on S by conjugation. For each s ∈ S, write ω_{s} for the (degenerate) skew-symmetric form on V which coincides with ω on Im(Id − s), and has Ker(Id − s) as the radical (recall that there is an ω-orthogonal direct sum decomposition V = Im(Id − s) ⊕ Ker(Id − s)).

Theorem 3 Assume that (V, ω, Γ) is an indecomposable triple. Then, PBW-property holds for H_{κ} if and only if there exists a constant t ∈ ℂ, and an Ad Γ-invariant function c : S → ℂ , s ↦ c_{s}, such that the pairing κ has the form:

κ(x, y) = t · ω(x, y) · 1 + ∑_{s∈S} c_{s} · ω_{s}(x, y) · s , ∀x, y ∈ V .

Remarks (i) Our proof of the Theorem shows that the algebra H_{κ} associated to a pairing κ of the type described by the formula above always satisfies the PBW-property, even if the triple (V, ω, Γ) is decomposable.

(ii) An analogue of Theorem 3 holds in a more general setup, where V is any (not necessarily symplectic) vector space, and Γ ⊂ GL(V) is a finite subgroup containing no elements s ∈ Γ such that rk(Id − s) = 1. The proof of Theorem 3 shows that, in this generality, PBW-property holds for H_{κ} if and only if κ(x, y) = ω(x, y) · 1 + ∑_{s∈S} ω_{s}(x, y) · s, ∀x, y ∈ V, where ω is a (possibly degenerate) skew-symmetric Γ-invariant form on V, symplectic reflections are defined as in Definition 2 (involving no reference to any symplectic structure), and {ω_{s}}_{s∈S} stands for an arbitrary Γ-equivariant collection of skew-symmetric forms such that: Ker(Id − s) ⊂ Radical(ω_{s}) , ∀s ∈ S.

Theorems 5–7 below also hold in this generality. ♢ Let C denote the vector space of Ad Γ-invariant functions on S. We have

dim C = number of Ad Γ-conjugacy classes in S .  (4)

From now on, we assume that the pairing κ in (1) has the form described in Theorem 3, for a certain t ∈ ℂ and c ∈ C. The corresponding algebra H_{t,c} := H_{κ} will be referred to as a symplectic reflection algebra.

Let e = \frac{1}{|Γ|} ∑_{g∈Γ} g ∈ ℂΓ, be the ‘symmetriser’ idempotent, viewed as an element of H_{t,c}. The algebra eH_{t,c}e ⊂ H_{t,c} will be called the spherical subalgebra in H_{t,c}. For t = 0 and c = 0, we have: eH_{0,0}e = e(SV #Γ)e ≃ (SV)^{Γ}, the algebra of Γ-invariants. Furthermore, for any (t, c) ∈ ℂ ⊕ C, Theorem 3 yields: gr(eH_{t,c}e) ≃ (SV)^{Γ}.

The space H_{t,c}e has an obvious left H_{t,c}-module structure, and right eH_{t,c}e-module structure. Given an algebra A, and a left, resp. right, A-module M, we set: M^{∨} = Hom_{A-left}(M, A), resp. ^{∨}M = Hom_{A-right}(M, A). Theorem 5 (i) eH_{t,c}e is a finitely generated Gorenstein algebra without zero divisors.

(ii) H_{t,c}e is a finitely generated, Cohen-Macaulay right eH_{t,c}e-module.

(iii) We have: ^{∨}(H_{t,c}e) = eH_{t,c}, and (eH_{t,c})^{∨} = H_{t,c}e. In particular, H_{t,c}e is reflexive. (vi) The left H_{t,c}-action induces an algebra isomorphism: H_{t,c} ⟶^{∼} End_{eH_{t,c}e}(H_{t,c}e).

For any t ≠ 0, let A_{t} = T V/I⟨x · y − y · x − t · ω(x, y)⟩_{x,y∈V} be the Weyl algebra of the symplectic vector space (V, t · ω); the algebras A_{t} are isomorphic to each other, for all t ≠ 0. We observe that, for c = 0 (and any t ≠ 0), we have: H_{t,0} = A_{t}#Γ. We show that, for any fixed t ≠ 0, the family {H_{t,c}}_{c∈C} gives a universal deformation of the algebra A_{t}#Γ. Furthermore, the spherical subalgebra eH_{t,0}e is isomorphic to At^{Γ}, the subalgebra of Γ-invariants in the Weyl algebra, and (for any fixed t ≠ 0) the family {eH_{t,c}e}_{c∈C} gives a universal deformation of the algebra At^{Γ}. Theorem 6 For any c ∈ C, the algebra eH_{t,c}e is commutative if and only if t = 0.

Example: Kleinian singularity. Let Γ ⊂ SL_{2}(ℂ) be a finite subgroup acting on V = ℂ^{2}. The action on V ∖ {0} is free, and any element of Γ ∖ {1} is a symplectic reflection. This case has been studied by Crawley-Boevey and Holland [CBH]. ♢ Quasi-classical case: κ = (0, c). Assume next that t = 0 and c ≠ 0. The algebra H_{0,c} , c ≠ 0, has some very interesting special features making it quite different from other members of the family {H_{κ}}_{κ∈ℂ⊕C} . First of all, the algebra H_{0,c} has a large center, Z_{0,c} = Z(H_{0,c}), as opposed to the algebras H_{κ} for generic κ. In more detail, the standard filtration on H_{0,c} induces a filtration on Z(H_{0,c}), and we have:

gr Z(H_{0,c}) = Z(H_{0,0}) = Z(SV #Γ) = (SV)^{Γ} .

Further, for any c ∈ C, we use the construction of Hayashi [Ha], to make both the spherical subalgebra eH_{0,c}e and the center Z_{0,c} Poisson algebras, which reduce to the standard Poisson structure on (SV)^{Γ}, if c = 0. Unlike the case c = 0, the center of H_{0,c} is harder to describe explicitly. To this end, we generalize the classical Satake isomorphism for an affine Hecke algebra (as proposed by Lusztig), and show that the map: z ↦ z·e establishes a Poisson algebra isomorphism: Z_{0,c} ⟶^{∼} eH_{0,c}e, which we call the Satake isomorphism.

Let R denote the coherent sheaf on Spec Z_{0,c} corresponding to H_{0,c}e, viewed as a (finitely generated) Z_{0,c}-module, that is: Γ(Spec Z_{0,c} , R) = H_{0,c}e. We will view points of Spec Z_{0,c} as algebra homomorphisms χ : Z_{0,c} ⟶ ℂ. Then, for any point χ ∈ Spec Z_{0,c}, the geometric fiber of the sheaf R at χ equals: H_{0,c}e ⨂_{Z_{0,c}} χ. The left action of H_{0,c} on H_{0,c}e gives an H_{0,c}-module structure on each geometric fiber of R.

In the theorem below, we fix U ⊂ Spec Z_{0,c}, a Zariski-open affine subset, and use the subscript ‘U’ to denote ‘restriction to U’. In particular we write Z_{U} := ℂ[U] ⊃ Z_{0,c}, and H_{U} := Z_{U} ⨂_{Z_{0,c}} H_{0,c}, and also R_{U} := R|_{U}.

Theorem 7 Let U be an affine Zariski-open subset contained in the smooth locus of Spec Z_{0,c}. Then we have:

(i) The sheaf R_{U} is locally free, and the algebra H_{U} is isomorphic to End_{Z_{U}}(R_{U}).

(ii) The algebra H_{U} is Morita equivalent to Z_{U}, i.e. the categories of finitely generated H_{U} -modules and Z_{U} -modules are equivalent.

(iii) Any simple H_{U} -module is isomorphic to one of the form: H_{U} e ⨂_{Z_{U}} χ, for a certain character χ : Z_{U} → ℂ. In particular, the action of the center defines a bijection between the isomorphism classes of simple H_{U} -modules and points of U.

(vi) Any simple H_{U} -module has dimension |Γ| and is isomorphic, as a Γ-module, to the regular representation of Γ.

If Spec Z_{0,c} is itself smooth then claims (i) -(vi) hold for U = Spec Z_{0,c}, and H_{U} = H_{0,c}.

Following [Alv], on the group algebra ℂΓ, define an increasing filtration F_{•}(ℂΓ) by letting F_{k}(ℂΓ) , k ≥ 0, be the ℂ-linear span of the elements g ∈ Γ such that rk(Id−g) ≤ k. This filtration is obviously compatible with the algebra structure on ℂΓ. Let F_{•}(ZΓ) denote the induced filtration on ZΓ, the center of ℂΓ. Write gr_{•}^{F} (ZΓ) for the corresponding associated graded algebra.

Further, write HH^{•}(A) for the Hochschild cohomology of an associative algebra A, and H^{•}(−) for the ordinary singular cohomology of a topological space (with complex coefficients). Part (i) of the theorem below is a generalization of [AFLS] and [Alv], while part (ii), may be viewed as a ‘symplectic generalization’ of McKay correspondence, cf. [Re].

Theorem 8 For any c ∈ C we have:

(i) For all t ∈ ℂ^{∗}, except possibly a countable set, there is a natural graded algebra isomorphism: HH^{•}(H_{t,c}) ≃ gr_{•}^{F} (ZΓ).

(ii) If the variety Spec Z_{0,c} is smooth, then the Poisson structure on Z_{0,c} makes Spec Z_{0,c} a symplectic manifold; Furthermore, there is a graded algebra isomorphism: H^{•}(Spec Z_{0,c}) ≃ gr_{•}^{F} (ZΓ).

Corollary 9 If Spec Z_{0,c} is smooth, then dim H^{odd}(Spec Z_{0,c}) = 0, and for all i ≥ 0, we have: dim H^{2i}(Spec Z_{0,c}) = n(i), where

n(i) := number of conjugacy classes of g ∈ Γ such that rk(Id − g) = 2i . □

Recall the result of Batyrev [Ba, Thm. 4.8] saying that, for any crepant (hence, symplectic, see e.g. [Ve]) resolution: V̂/Γ ↠ V/Γ, one has: dim H^{2i}(V̂/Γ) = n(i). It has been conjectured by E. Vasserot [Va] (and also by the second author) that, for a resolution V̂/Γ as above, there is a graded algebra isomorphism: H^{•}(V̂/Γ) ≃ gr_{•}^{F} (ZΓ). This conjecture is related to Theorem 8 as follows. According to [Ka2], if the symplectic orbifold V/Γ has a smooth symplectic resolution of singularities, V̂/Γ, then this resolution can be deformed to a smooth affine symplectic manifold. We expect that the latter coincides with the variety Spec Z_{0,c}, for an appropriate value of the parameter c ∈ C, in particular, Spec Z_{0,c} is smooth for generic c ∈ C. Our expectation is based on the conjecture saying that any Poisson deformation of V/Γ, in particular the one constructed by Kaledin, is given by a certain variety Spec Z_{0,c}. If our expectations are correct, then any smooth symplectic resolution, V̂/Γ, has the same cohomology algebra as Spec Z_{0,c}. Thus, Theorem 8 would imply the conjecture in [Va]. In many concrete examples the space V̂/Γ has a natural hyper-Kähler structure such that Spec Z_{0,c} is obtained from V̂/Γ by ‘rotating’ the complex structure. Hence the two spaces are even diffeomorphic, and Theorem 8 applies. This is the case, for instance, for Γ = S_{n} acting diagonally on V = ℂ^{n} ⊕ ℂ^{n}. In that case, V̂/Γ = Hilb^{n}(ℂ^{2}) is the Hilbert scheme of n points on the plane, and the corresponding result on the algebra structure of H^{•}(Hilb^{n}(ℂ^{2})) has been recently proved in [LSo] and [Va] (independently). Their approach is totally different from ours.

Further, write K(X) for the algebraic K-theory of an algebraic variety X, that is, for the Grothendieck group formed by algebraic vector bundles on X. Now let X = Spec Z_{0,c}, and assume that it is smooth so that the sheaf R is locally free, by Theorem 7(i). We view R as an algebraic vector bundle on Spec Z_{0,c}. Given an irreducible Γ-representation E, we let R^{E} := E ⊗_{Γ} R = Hom_{Γ}(E^{∗}, R) denote the E-isotypic component of R, which is again an algebraic vector bundle on Spec Z_{0,c}. Proposition 10 If the variety Spec Z_{0,c} is smooth, then K(Spec Z_{0,c}) is a free abelian group with the basis {R^{E}}_{E∈Irreps(Γ)}. Moreover, the Chern character map gives an isomorphism: ℂ ⊗_{ℤ} K(Spec Z_{0,c}) ⟶^{∼} H^{ev}(Spec Z_{0,c}).

Wreath-product case. An infinite series of interesting examples of groups generated by symplectic reflections^{1} is provided by a so-called wreath-product construction. Fix a finite subgroup Γ ⊂ SL_{2}(ℂ) and an integer n > 1. We write L = ℂ^{2} for the tautological 2-dimensional Γ-module, form the vector space V = L^{⊕n} with the symplectic structure ω_{V} induced from the standard one on L = ℂ^{2}, and let S_{n}, the Symmetric group, act on V by permutations of the direct summands. The group Γ_{n} := S_{n} ⋉ (Γ × Γ × … × Γ) ⊂ Sp(V) acts naturally on V = L^{⊕n} and is called the wreath-product of S_{n} with Γ.

It is known that C(Γ_{n}), the space of class functions on the set of symplectic reflections in Γ_{n} can be identified with ZΓ, the center of the group algebra of Γ. Thus there is a symplectic reflection algebra, H_{0,c}(Γ_{n}), attached to the triple (V, ω_{V} , Γ_{n}), and any parameter

c ∈ ZΓ ≃ C(Γ_{n}).

On the other hand, let V = ℂΓ^{⊕n} be the direct sum of n copies of the regular representation of Γ. Following Nakajima [Na], to any c ∈ ZΓ one associates an affine algebraic variety of quiver data:

𝕄c_{Γ}(V) := {(∇, I, J) ∈ (Hom_{ℂ}(V, V ⊗_{ℂ} L) ⨁ V ⨁ V^{∗})^{Γ} | [∇, ∇] + I ⊗ J = c|_{V}} ,

(11)

where I ⊗ J ∈ End_{Γ}V, and [∇, ∇] ∈ (End_{Γ}V) ⨂ ⋀^{2}L ≃ End_{Γ}V. The natural action on (Hom_{ℂ}(V, V ⊗_{ℂ} L) ⨁ V ⨁ V^{∗})^{Γ} of the group G_{Γ}(V) of Γ-equivariant automorphisms of V preserves the subvariety 𝕄c_{Γ}(V). The quotient: 𝔐_{Γ,n,c} = 𝕄c_{Γ}(V)/G_{Γ}(V) is a particular case of the Nakajima quiver variety. In general, the quotient is understood in the sense of the Geometric Invariant Theory, but in our case the GIT quotient coincides with the naive one because of the following well-known result, see also [Na] Lemma 12 If c ∈ ZΓ is generic enough, then the induced G_{Γ}(V)-action on 𝕄c_{Γ}(V) is free. Furthermore, 𝔐_{Γ,n,c} is a smooth symplectic affine algebraic variety of dimension 2n.

1see [Coh] for a classification of finite groups generated by symplectic reflections.

Write Z_{0,c}(Γ_{n}) for the center of the algebra H_{0,c}(Γ_{n}). One of our main results is Theorem 13 If c ∈ ZΓ is generic enough, then Spec Z_{0,c}(Γ_{n}) is isomorphic to 𝔐_{Γ,n,c}, as a Poisson algebraic variety.

Corollary 14 If c ∈ ZΓ is generic enough, then we have:

(i) Spec Z_{0,c}(Γ_{n}) is a smooth symplectic variety.

(ii) Simple H_{0,c}(Γ_{n})-modules all have dimension n! · |Γ|^{n}, and are parametrized by points of the quiver variety 𝔐_{Γ,n,c}.

(iii) There is a canonical algebraic vector bundle R on 𝔐_{Γ,n,c} whose fibers carry the regular representation of the group Γ_{n}. The classes: \{R^{E}\}_{E∈Irrep({Γ \atop n})} form a basis of K(𝔐_{Γ,n,c}).

(vi) There is an algebra isomorphism: H^{•}(𝔐_{Γ,n,c}) ≃ gr_{•}^{F} ZΓ_{n}. □

Rational Cherednik algebra. As we have mentioned earlier, all indecomposable triples (V, ω, Γ) fall into two large groups: (1) those where V is an irreducible Γ-module, and (2) those where V = 𝔥 ⊕ 𝔥^{∗} is a direct sum of two Γ-stable irreducible Lagrangian subspaces. In the second case, the action on 𝔥 by any symplectic reflection g ∈ Γ has a pointwise fixed codimension 1 hyperplane in 𝔥, hence is a complex reflection in 𝔥. Thus, in case (2), saying that Γ is generated by symplectic reflections amounts to saying that Γ = W is a finite complex reflection group in a vector space 𝔥, acting on V = T^{∗}𝔥 ≃ 𝔥+𝔥^{∗} by induced symplectic automorphisms. Any (complex) reflection s ∈ W induces the corresponding symplectic reflection of 𝔥 + 𝔥^{∗}.

We are mostly interested in the special case where W is the Weyl group of a finite reduced root system R ⊂ 𝔥^{∗}. The set S of reflections in W forms either one or two conjugacy classes, depending on whether all roots α ∈ R are of the same length or not. Thus, giving a W -invariant function c : S → ℂ amounts to giving a map: α ↦ c_{α} ∈ ℂ, where c_{α} depends only on the length of α ∈ R.

Let H_{t,c} be the symplectic reflection algebra associated to a complex number t ∈ ℂ, a map c as above, and the vector space V = 𝔥 + 𝔥^{∗}. Explicitly, write α^{∨} ∈ 𝔥 for the coroot corresponding to a root α ∈ R, and s_{α} ∈ W for the reflection relative to α. Then, formula (1) shows that the algebra H_{t,c} is generated by the spaces 𝔥, 𝔥^{∗}, and the group W, subject to the following defining relations:

\begin{matrix}w · x · w^{−1} = w(x) , w · y · w^{−1} = w(y) , & ∀y ∈ 𝔥 , x ∈ 𝔥^{∗} , w ∈ W \\ [x_{1}, x_{2}] = 0 = [y_{1}, y_{2}] , & ∀y_{1}, y_{2} ∈ 𝔥, x_{1} , x_{2} ∈ 𝔥^{∗} \\ [y, x] = t · ⟨y, x⟩ − \frac{1}{2} · ∑_{α∈R} c_{α} · ⟨y, α⟩⟨α^{∨}, x⟩ · s_{α} , & ∀y ∈ 𝔥 , x ∈ 𝔥^{∗} , \end{matrix} (15)

(the factor \frac{1}{2} on the RHS of the last equation accounts to the fact that roots α and −α give rise to the same reflection).

We will refer to the algebra H_{t,c} above as a rational Cherednik algebra because it is a certain ‘rational’ degeneration of the double affine Hecke algebra studied by Cherednik [Ch]. In the case of type A_{1}, the algebra H_{κ} was discussed in detail in [ChM].

Olshanetsky-Perelomov [OM], have associated a Calogero-Moser type integrable system to any Weyl group W. We show that the space Spec Z(H_{0,c}) for the rational Cherednik algebra attached to W is a natural completion of the phase space of that integrable system. The affine algebraic variety Spec Z(H_{0,c}) is always an irreducible normal Gorenstein variety with a Poisson structure. We propose to call it the Calogero-Moser space associated to the Weyl group W, with parameter c. For Weyl groups of type: A_{n}, B_{n}, C_{n} and sufficiently general parameters c ∈ C, the variety Spec Z(H_{0,c}) is a smooth symplectic manifold, by Theorem 13. Relation to the double-affine Hecke algebra. The Cherednik algebra H_{t,c} may be thought of as the following two-step degeneration of the double affine Hecke algebra H defined in [Ch]:

H = H^{elliptic} ⇝ H^{trigonometric} ⇝ H^{rational} = H_{t,c} .

In more detail, let G be the Lie group corresponding to the Lie algebra 𝔤, and T a maximal torus of G. The algebra H contains the group algebras of two (dual) lattices: X = Hom(ℂ^{∗}, T) and Y = Hom(T, ℂ^{∗}). Degeneration from H to H^{trigonometric} amounts, effectively, to replacing the group algebra of the lattice X = Hom(ℂ^{∗}, T) by the Symmetric algebra of the vector space Hom_{ℂ}(ℂ, Lie T) = 𝔥, while the group algebra of the second lattice Y remains unaffected. Degeneration from H^{trigonometric} to H^{rational} = H_{t,c} is obtained similarly, by further replacing the group algebra of the lattice Y = Hom(T, ℂ^{∗}) by the Symmetric algebra of the vector space Hom_{ℂ}(Lie T, ℂ) = 𝔥^{∗}. Harish-Chandra homomorphism. It turns out that the spherical subalgebra, eH_{t,c}e, of the symplectic reflection algebra associated to the root system of type A_{n−1} can be described as a ‘quantum Hamiltonian reduction’ of the algebra D(𝔤𝔩_{n}) of polynomial differential operators on the Lie algebra 𝔤𝔩_{n}. A similar description conjecturally exists for the spherical subalgebra of the symplectic reflection algebra associated to any wreath-product Γ_{n}, in which case 𝔤𝔩_{n} gets replace by a (vector) space of representations of a Dynkin quiver (𝔤𝔩_{n} is the space of n-dimensional representations of the quiver with one vertex and one edge-loop).

In more detail. Fix an arbitrary complex reductive Lie algebra 𝔤. Let 𝔥 be a Cartan subalgebra of 𝔤, and W the Weyl group of (𝔤, 𝔥). Let D(𝔤)^{𝔤} be the algebra of ad 𝔤-invariant polynomial differential operators on 𝔤, and D(𝔥)^{W} the algebra of W -invariant polynomial differential operators on 𝔥. Write Δ_{𝔤}, resp. Δ_{𝔥}, for the second order Laplacian on 𝔤, resp. on 𝔥, associated to a nondegenerate invariant bilinear form on 𝔤. In 1964, Harish-Chandra [HC] defined an algebra homomorphism Φ : D(𝔤)^{𝔤} → D(𝔥)^{W}, that reduces to the restriction map: ℂ[𝔤]^{𝔤} → ℂ[𝔥]^{W} , f ↦ f|_{𝔥}, on zero order differential operators, and such that Φ(Δ_{𝔤}) = Δ_{𝔥}.

We construct a 1-parameter deformation of the Harish-Chandra homomorphism Φ in the special case 𝔤 = 𝔤𝔩_{n}, and show that this deformed homomorphism may (and should) be viewed as an isomorphism between the quantum Hamiltonian reduction of the algebra D(𝔤𝔩_{n}) and the spherical subalgebra eH_{t,c}e.

Now let 𝔥 = ℂ^{n} be the Cartan subalgebra of diagonal matrices in 𝔤 = 𝔤𝔩_{n}. Write x_{1}, …, x_{n} for coordinates in ℂ^{n}, and 𝔥^{reg} ⊂ ℂ^{n} for the open subset of points with pairwise distinct coordinates. We apply the formalism of radial parts of invariant differential operators to construct a family of algebra homomorphisms Φ_{k} : D(𝔤)^{𝔤} → D(𝔥^{reg} )^{W} depending on a parameter k ∈ ℂ, which specializes to the Harish-Chandra homomorphism Φ at k = 0. For each k ∈ ℂ, this homomorphism Φ_{k} reduces to the restriction map: ℂ[𝔤]^{𝔤} → ℂ[𝔥]^{W} , f ↦ f|_{𝔥}, on zero order differential operators. Further, we have: Φ_{k}(Δ_{𝔤}) = L_{k}, where

L_{k} = ∑_{j} \frac{∂^{2}}{∂x_{j}^{2}} − ∑_{i≠j} \frac{k(k + 1)}{(x_{i} − x_{j})^{2}} .  (16)

is the Calogero-Moser operator with rational potential, corresponding to the parameter k.

We now describe the image of the map Φ_{k}. Let C_{k} denote the centraliser of L_{k} in the algebra D(𝔥^{reg} )^{W} of W -invariant differential operators on 𝔥^{reg} . If k = 0, then it is classically known that C_{0} = S𝔥^{W} , is the algebra of W -invariant differential operators with constant coefficients. In general, it is known (see e.g. [O]) that for each k, there exists an algebra isomorphism σ_{k} : S𝔥^{W} ⟶^{∼} C_{k}, such that, for any homogeneous element u ∈ S𝔥^{W} of degree ℓ ≥ 0, the differential operator σ_{k}(u) has order ℓ, and its principal symbol equals u; moreover, σ_{k}(Δ_{𝔥}) = L_{k}. It turns out that the image of the homomorphism Φ_{k} equals B_{k}, the associative subalgebra in D(𝔥^{reg} )^{W} generated by C_{k} and by the polynomial subalgebra ℂ[𝔥]^{W} ⊂ D(𝔥)^{W} , of zero order W -invariant operators on 𝔥.

Thus, we summarize: Theorem 17 There exists a (flat) family of surjective algebra homomorphisms Φ_{k} : D(𝔤)^{𝔤} ↠ B_{k}, all of which reduce to the restriction map: f ↦ f|_{𝔥}, on zero order differential operators. This family specializes, at k = 0, to the Harish-Chandra homomorphism Φ; furthermore, Φ_{k}(Δ_{𝔤}) = L_{k}, for any k ∈ ℂ.

A drawback of Theorem 17 is that elements of the algebra B_{k} ⊂ D(𝔥^{reg} )^{W} may have quite complicated singularites at the divisor 𝔥 ∖ 𝔥^{reg} . Therefore, it is tricky to give an explicit description of B_{k}, as a subalgebra inside D(𝔥^{reg} )^{W}. Remarkably, the algebra B_{k} admits an imbedding into H_{κ}, the rational Cherednik algebra of type A_{n−1}. The image of B_{k} has a very simple description that we now explain.

If R = A_{n−1}, is the root system of 𝔤 = 𝔤𝔩_{n} then all roots are of the same length, and C = ℂ. We treat the parameter κ = (t, c) as a point on the Riemann sphere ℙ^{1} = (ℂ ⊕ C)/ℂ^{∗} with homogeneous coordinates (c : t), i.e., as a complex number κ = c/t ∈ ℂ ∪ {∞}; rescaling: (t, c) ↦ (r · t, r · c) , r ∈ ℂ^{∗}, makes no essential effect on the algebra H_{t,c}). Thus, in the A_{n−1}-case we make no distinction between the parameter: κ = c/t, and the complex parameter ‘k’ entering the deformed Harish-Chandra homomorphism Φ_{k}. Note that the limit κ → ∞ corresponds to t → 0.

For any Weyl group, Cherednik defined a faithful representation of the double affine Hecke algebra, given by the so-called Demazure-Lusztig-Dunkl operators. A version of Cherednik’s construction yields a faithful representation of H_{κ} in the vector space ℂ[𝔥^{reg} ]. It is easy to see that the restriction of this representation to the spherical subalgebra eH_{κ}e ⊂ H_{κ} keeps the subspace ℂ[𝔥^{reg} ]^{W} ⊂ ℂ[𝔥^{reg} ] stable, and the resulting eH_{κ}e-action on ℂ[𝔥^{reg} ]^{W} is given by W -invariant polynomial differential operators on 𝔥^{reg} . This gives an algebra homomorphism

Θκ^{spher} : eH_{κ}e ⟶ D(𝔥^{reg} )^{W} .  (18)

Thus, for 𝔤 = 𝔤𝔩_{n} and W = S_{n}, we get the following diagram:

\xymatrix{D(𝔤)^{𝔤} \ar[r]^{Φ_{κ}} & D(𝔥^{reg} )^{W} & eH_{κ}e \ar[l]_{Θ_{κ}^{spher}}}  (19)

The remarkable fact is, that the homomorphisms Φ_{κ} and Θκ^{spher} above have the same image, the subalgebra B_{κ} ⊂ D(𝔥^{reg} )^{S_{n}}. Further, the map Θκ^{spher} in (19) is a bijection onto its image. Inverting this map, we thus obtain an algebra homomorphism:

Φκ^{spher} = (Θκ^{spher})^{−1}^{∘} Φ_{κ} : D(𝔤𝔩_{n})^{𝔤𝔩^{n}} ⟶ eH_{κ}e .  (20)

This homomorphism provides a link between invariant differential operators on 𝔤𝔩_{n} and the Cherednik algebra H_{κ}. Furthermore, replacing the Lie algebra 𝔤 = 𝔤𝔩_{n} by the group G = GL_{n}, one can define a ‘trigonometric’ version of the deformed Harish-Chandra homomorphism (20):

Φκ^{trigonometric} : D(G)^{G} ⟶ e · H^{trigonometric} · e .  (21)

One might expect that replacing further D(G) by the so-called Heisenberg double of the corresponding quantum group one gets an elliptic analogue of the sperical Harish-Chandra homomorphism with image eHe. Calogero-Moser space. It is a well-known principle that the ‘classical’ limit of the algebra D(𝔤) is the commutative algebra of regular functions on T^{∗}𝔤, the cotangent bundle on 𝔤. Since T^{∗}𝔤 = 𝔤 ⊕ 𝔤^{∗} ≃ 𝔤 ⊕ 𝔤, the ‘classical’ limit of D(𝔤)^{𝔤} is the algebra ℂ[𝔤 ⊕ 𝔤]^{𝔤} of ad 𝔤-invariant polynomials on 𝔤 ⊕ 𝔤. Now, let 𝔤 = 𝔤𝔩_{n}. There are reasons to think of the limit: κ = (t, c) ⟶ ∞ = (0, 1) ∈ ℙ^{1}, as a ‘classical’ limit. Accordingly, we show that the family of Harish-Chandra homomorphisms Φκ^{spher} : D(𝔤)^{𝔤} ↠ eH_{κ}e, see (20), has a ‘classical’ limit, a surjective algebra homomorphism Φ∞^{spher} : ℂ[𝔤 ⊕ 𝔤]^{𝔤} ↠ eH_{∞}e. It turns out that this homomorphism is intimately related to the so-called Calogero-Moser space M_{n}, a symplectic algebraic manifold introduced by Kazhdan-Kostant-Sternberg [KKS], and studied further by G. Wilson [Wi]. This space, which is a particular case of the quiver variety M_{Γ,n,c}, for Γ = {1}, is defined as a categorical quotient: M_{n} = M//Ad GL_{n}, where

M := {(X, Y) ∈ 𝔤 × 𝔤 | [X, Y] + Id = rank 1 matrix} ,  (22)

is an affine subvariety in 𝔤 ⊕ 𝔤, a special case of the quiver data variety 𝕄_{Γ}^{c}(V). Thus, by definition, the coordinate ring of M_{n} equals ℂ[M_{n}] = ℂ[𝔤 ⊕ 𝔤]^{𝔤}/(I_{M})^{𝔤}, where I_{M} ⊂ ℂ[𝔤⊕𝔤] denotes the defining ideal of M, and the superscript ‘ 𝔤 ’ stands for ‘ad 𝔤-invariants’. A ‘commutative’ counterpart of homomorphism (20) reads: Theorem 23 The kernel of the homomorphism Φ∞^{spher} equals (I_{M})^{𝔤}; hence one has the following chain of Poisson algebra isomorphisms:

ℂ[M_{n}] = ℂ[𝔤 ⊕ 𝔤]^{𝔤}/Ker(Φ∞^{spher}) ⟶^{∼}^{Φ{spher \atop ∞}} eH_{∞}e ⟶^{∼}^{Satake^{−1}} Z(H_{∞}) .

This result provides, in the special case Γ = {1}, a transparent conceptual interpretation of the isomorphism of Theorem 13. Thus, for Γ_{n} = S_{n} and κ = ∞, Corollary 14 yields Theorem 24 If W = S_{n} is the Symmetric group and κ = ∞ = (0, c) , c ≠ 0, then

(i) The algebra H_{∞} = H_{0,c}(S_{n}) is Morita equivalent to the algebra: Z_{∞} ≃ ℂ[M_{n}].

(ii) Any simple H_{∞}-module is isomorphic to one of the form: H_{∞}e ⊗_{Z∞} χ, for a certain character χ : Z_{∞} → ℂ. Thus, the isomorphism classes of simple H_{∞}-modules are parametrized by points of the Calogero-Moser space M_{n}.

(iii) Any simple H_{∞}-module has dimension n! and is isomorphic, as an S_{n}-module, to the regular representation of S_{n}.

Berest and Wilson have introduced an (infinite-dimensional) group 𝔾 of the automorphisms of ℂ⟨x, y⟩, the free associative algebra on two generators, that keep the element xy − yx ∈ ℂ⟨x, y⟩ fixed. We show that the group 𝔾 acts naturally on D(𝔤𝔩_{n}) and on H_{κ} by algebra automorphisms, preserving subalgebras D(𝔤𝔩_{n})^{𝔤𝔩^{n}} and eH_{κ}e, respectively. Moreover, the map Φκ^{spher} : D(𝔤𝔩_{n})^{𝔤𝔩^{n}} ⟶ eH_{κ}e commutes with the 𝔾-action. Further, in [BW], Berest-Wilson defined a transitive action of 𝔾 on the Calogero-Moser space M_{n}. It follows from our results that the 𝔾-action on M_{n} can be canonically lifted to a 𝔾-action on the vector bundle R, see Theorem 7. It seems to us quite important to obtain a better geometric understanding of this vector bundle.

References

[AFLS] J. Alev, M.A. Farinati, T. Lambre, and A.L. Solotar: Homologie des invariants d’une algèbre de Weyl sous l’action d’un groupe fini. J. of Algebra 232 (2000), 564–577.

[Alv] M.S. Alvarez: Algebra structure on the Hochschild cohomology of the ring of invariants of a Weyl algebra under a finite group. Preprint 1999.

[Ba] V. Batyrev: Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs. J. Eur. Math. Soc. (JEMS) 1 (1999), 5–33.

[BW] Yu. Berest, G. Wilson: Automorphisms and ideals of the Weyl algebra. Math. Ann. 318(2000), 127–147.

[BMR] M. Broué, G. Malle, R. Rouquier: Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math. 500 (1998), 127–190.

[Ch] I. Cherednik: Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald operators , IMRN (Duke math.J.) 9 (1992), p.171-180.

[Ch2] I. Cherednik: A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras. Invent. Math. 106 (1991), 411–431.

[ChM] I. Cherednik, Y. Markov: Hankel transform via double Hecke algebra, math.QA/0004116.

[Coh] A.M. Cohen: Finite quaternionic reflection groups. J. Algebra 64 (1980), 293–324.

[Co] A. Connes: Noncommutative geometry. Academic Press, Inc., 1994.

[CBH] W. Crawley-Boevey, M. Holland: Noncommutative deformations of Kleinian singularities. Duke Math. J. 92 (1998), 605–635.

[D1] C. Dunkl: Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311 (1989), 167–183.

[D2] C. Dunkl: Operators commuting with Coxeter group actions on polynomials. Invariant theory and tableaux, 107–117, IMA Vol. Math. Appl., 19, Springer, New York, 1990.

[HC] Harish-Chandra: Invariant differential operators and distributions on a semisimple Lie algebra. Amer. J. Math, 86 (1964), 534-564.

[Ka1] D. Kaledin: McKay correspondence for symplectic quotient singularities. Invent. Math. (2001),

       math.AG/9907087.

[Ka2] D. Kaledin: Symplectic resolutions: deformations and birational maps. math.AG/0012008

[KKS] D. Kazhdan, B. Kostant, and S. Sternberg: Hamiltonian group actions and dynamical systems of Calogero type, Comm.Pure Apll. Math., 31(1978), 481-507.

[Kon] M. Kontsevich: Deformation quantization of Poisson manifolds, I. math.QA/9709040

[LSo] M. Lehn, C. Sorger: Symmetric groups and the cup product on the cohomology of Hilbert schemes. math.AG/0009131 (To appear in: Duke Math. Journ.).

[Na] H. Nakajima: Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76 (1994), 365–416.

[O] E.M. Opdam: Some applications of hypergeometric shift operators. Inv. Math., 98 (1988), 1–18.

[O2] E.M. Opdam: Lecture notes on Dunkl operators for Real and Complex reflection groups. MSJ Memoirs, Math. Soc. of Japan, 8 (2000).

[OP] M. A. Olshanetsky, A. M. Perelomov: Quantum integrable systems related to Lie algebras. Phys. Rep. 94 (1983), 313–404.

[Re] M. Reid: La correspondance de McKay. Séminaire Bourbaki, 867 (1999), math.AG/9911165.

[Va] E. Vasserot : Sur l’anneau de cohomologie du schèma de Hilbert de C^{2}. C.R. Acad. Sci. Paris, 332 (2001), 7–12.

[Ve] M. Verbitsky: Holomorphic symplectic geometry and orbifold singularities. math.AG/9903175.

[Wi] G. Wilson: Collisions of Calogero-Moser particles and an adelic Grassmannian, Inv. Math. 133

(1998), 1-41. P.E.: Department of Mathematics, Rm 2-165, MIT, 77 Mass. Ave, Cambridge, MA 02139

    etingof@math.mit.edu

V.G.: Department of Mathematics, University of Chicago, Chicago, IL 60637, USA;

    ginzburg@math.uchicago.edu