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Hilbert schemes, Hecke algebras and the Calogero-Sutherland system

We describe the ring structure of the cohomology of the Hilbert scheme of points for a smooth surface X. When the canonical class K_X = 0, this was done by Lehn and Sorger, extending earlier work when X = C^2. Their approach does not generalize. Instead we recast this problem as a question of finding integrals of motion for a Hamiltonian which describes ``intersection with the boundary''. To do so, we use the identification of the cohomology of the Hilbert schemes with a Fock space modelled on the lattice H(X). With this identification, Lehn computed the operator of intersection with the boundary. It is essentially the Calogero-Sutherland Hamiltonian. We then solve the problem of finding integrals of motion by using the Dunkl-Cherednik operators to find an explicit commuting family of differential operators; these operators represent cup product on the Hilbert scheme. We provide two characterizations of the Hilbert scheme multiplication operators; the first as an algebra of operators that can be inductively built from functions and the CS Hamiltonian, and the second in terms of the centralizer of the CS Hamiltonian inside an appropriate ring of differential operators.


Introduction
This paper describes the ring structure on the cohomology of the Hilbert scheme of points for a smooth algebraic surface. The main new idea in our solution of this problem is to consider the problem as a question in integrable systems. Having done this, we must also solve the integrable system, i.e produce integrals of motion. We do this by defining a variant of the Cherednik Hecke algebra depending on a finite dimensional algebra H. We then give several characterizations of the family of commuting differential operators this constructs.
Let us describe the problem we solve, and previous work on it.
Let X be an smooth algebraic surface, and Hilb n (X) the 2n-dimensional smooth algebraic variety parameterising length n subschemes of X-the n'th Hilbert scheme of points for X.
The papers [10,19] discovered that the correspondences between Hilbert schemes given by adding points constrained to lie on cycles in X organize into an action of an infinite dimensional Lie algebra. This gives a canonical identification F = Sym xH[x] ∼ = ⊕ n H * (Hilb n (X)) as modules for a Heisenberg Lie algebra, and hence in particular an isomorphism of graded vector spaces. (We always work in the category of super vector spaces, so that Sym V means the symmetric algebra of V ev tensored with the exterior algebra of V odd ). We write F = ⊕ n F n , where F n is the n'th-eigenspace of the energy operator ∂ = x ∂ ∂x .
The ring structure on cohomology now induces an additional map F ⊗ F → F , and the problem that this paper addresses is to describe this map-to determine the cohomology ring of the Hilbert scheme, in the Fock space coordinates.
This problem was first considered by Lehn [12], who also made significant progress to its solution. Each Hilbert scheme Hilb n (X) contains a natural divisor, the complement of the loci of distinct points, and hence a class in H 2 (Hilb n (X)). Cup product with this class, summed over all n, is thus an operator L : F → F , and [12] identifies this operator explicitly. In the Fock space coordinates it can be written as a 2nd order differential operator depending on the Frobenius algebra structure of H(X) and on the canonical class K ∈ H 2 (X).
The operator L is naturally associated with the first Chern class of the cotangent bundle of X. In a similar manner any cohomology class on X inductively determines a set of operators on F , and using the results of [12], Li, Quin and Wang show in a series of papers [15,16] that the subring these operators generate is the ring of multiplication operators on the Hilbert scheme. After [10,12], no new algebraic geometry is needed-these results determine the ring implicitly.
However, though much progress was made in in the above papers, the relations that these operators satisfy is not clear. In particular, there is no construction of the operators without already knowing the Hilbert scheme exists, and no presentation of the cohomology ring.
When K = 0, the situation was satisfactorily resolved in [14]: they construct an algebra A n whose underlying vector space is H({(x, g) ∈ X n × S n | gx = x}) and show that its S n -invariants are H(Hilb n (X)).This algebra is the Chen-Ruan orbifold cohomology ring of X n /S n , and provides evidence for a conjecture of Ruan on ring structures on crepant resolutions.
Unfortunately, this approach fails when K = 0. It seems that there is no flat family of algebras A n (K) equipped with an S n action such that A n (0) = A n and whose generic point has A n (K) Sn = H(Hilb n (X)).
New ideas are needed.
Instead, we begin with the following key observation. Cup product with an element a ∈ n F n defines a linear map on F , and, for reasonable a, a differential operator on F . This differential operator commutes with the operator L-after all, cohomology is a (super)-commutative ring. Hence there are as least as many differential operators which commute with L as the dimension of the space F .
Thinking of L as a Hamiltonian, this is precisely saying L defines an integrable system. It rephrases the problem of determining the ring structure on F as the problem of finding integrals of motion-determining the centralizer of L in an appropriate ring of differential operators on F , and showing that the centralizer is commutative and in bijection with F . However, as noticed by [3,8,18], L is a variant of a well known operator-it is a version of the bosonised Calogero-Sutherland Hamiltonian, familiar from exactly solvable models of statistical mechanics [1]. So the problem of describing the the ring structure on the Hilbert scheme becomes an algebraic problem, that of solving a deformed version of the Calogero-Sutherland system.
We do that in this paper. We begin by defining a ring of continuous differential operators on the Fock space, Diff H,K ′ F . This depends on both the ring structure on H(X) and on an additional parameter K ′ ∈ H ev (X). Section 2 of this paper is devoted to its properties. It is a limit of finitely generated algebras, and much smaller than the ring of all differential operators.
We also define an operator L K ′ ∈ Diff H,K ′ F ; when K = K ′ this is the operator L. For generic K ′ , define IM K ′ to be the centralizer in Diff H,K ′ F of L K ′ and of the energy operator ∂. As IM K ′ commutes with ∂, for each n it projects to give a subalgebra of End F n . Degenerating K ′ to K, we get an algebra IM K for any K.
Then our first description of the ring structure is Theorem 1.0.1. IM K is a commutative algebra, independent of any choices, and for any n the image of IM K in End F n is precisely the algebra of multiplication operators H(Hilb n (X)) ⊆ End F n .
In order to prove this theorem, we give an explicit construction of IM K ′ as a polynomial algebra. We begin by rewriting the Calogero-Sutherland Hamiltonian L as an inverse limit of r-particle generalized Calogero-Sutherland Hamiltonians. (To do this precisely forces us to work with augmented or non-unital  algebras).
We then show each r-particle system is integrable, following the methods of Dunkl, Cherednik and Opdam. This consists of writing an explicit polynomial algebra of commuting difference-differential operators. The symmetric group acts on these operators, and the invariants act purely as differential operators. The quadratic invariant is the CS Hamiltonian, and these operators are the entire centralizer of the Hamiltonian. Put differently, we define a version of the degenerate Cherednik double affine Hecke algebra depending on a Frobenius algbera H = H(X), and its spherical subalgebra is the desired algebra of integrals of motion.
This gives an explicit description of the integrals of motion for each finite r, as well as showing there are exactly enough of them. Taking the limit over r, we get a description of IM K in terms of degenerating families of Dunkl-Cherednik operators. In particular we recover the previous theorem.
Section 4 is a description of the generalized Cherednik algebra, and its behaviour in the inverse system and flatness properties with respect to twisting. This enables us to solve the Calogero-Sutherland Hamiltonian L attached to any Frobenius algebra H and parameter K, without needing the existence of the Hilbert scheme to produce "enough" commuting operators.
In order to compare the integrals of motion we construct with the algebra of operators constructed in [12,15,16], we need to characterize this latter algebra. It turns out that this has a simple description as differential operators built out of L and the (usual) algebra structure on the Fock space F . We encode this in the notion of locality in section 2.8, and in section 7 we prove that the algebra of differential operators local for L is just the desired algebra of multiplication maps. In section 5 we show that IM K are the local operators for L for any H and K, proving the main theorem.
It seems this is the first occurrence of integrable systems in this manner in algebraic geometry.

Notation.
All vector spaces are super vector spaces, that is they are Z/2-graded. By a structure of a commutative algebra on a super vector space, we mean a super-commutative algebra. Lie brackets are taken in the super sense, that is [α, β] = αβ − (−1) |α||β| βα for homogeneous α and β. Endomorphism algebras of super vector spaces have a natural Z/2-grading. If A is a Z/2-graded algebra, we write A × for the subgroup of even invertible elements.
If X 1 , . . . , X n are subsets of an algebra A we write X 1 , . . . , X n for the subalgebra of A that they generate.
We will often use filtered objects. A filtration on an object X in an abelian category is a sequence of Then there is a functor from filtered objects to exhaustive filtered objects which sends X to F ∞ X/F −∞ X.
We write

Differential operators on Fock space
Let A be a commutative ring. In this section we use the ring structure on A to define a sequence of ideals in the Fock space Sym A; i.e. to define a topology on Sym A. We then define the ring of continuous differential operators on Sym A in the standard way.
In fact, we use the equivalent language of pro-rings. The main result of this section is proposition 2.4.7, which identifies Diff Sym A.

Non-unital algebras
We will need to work with augmented algebras. Equivalently, with non-unital algebras.
Let CAlg U denote the category of unital commutative associative algebras, CAlg N denote the category of (possibly) non-unital commutative associative algebras. The forgetful functor CAlg U → CAlg N has a left adjoint + : where the underlying vector space of A + is C × A, and multiplication is defined by (λ, u) · (µ, v) = (λµ, λv + µu + uv).
We write (1, 0) = 1, and note there are two exact sequences of non-unital aglebras If A ∈ CAlg U , these exact sequences split, and there is an isomorphism of unital algebras . The essential image of + is the category of augmented unital algebras. Both CAlg U and CAlg N admit coproducts, denoted ⊗, ⊗ + respectively. We have, if A, B ∈ CAlg N that Note that the underlying vector space of A ⊗ + B is not the tensor product of the underlying vector spaces of A and B. There is also an action of CAlg U on CAlg N , denoted ⊗; we have (A ⊗ B) (Recall that the coproduct of A and B is the object in CAlg equipped with a morphism from A and a morphism from B; the images of these morphisms generate A ⊗ B and the only relation imposed is the images commute. In CAlg, the maps A → A ⊗ B, B → A ⊗ B are injective, as the above description shows.) We will write (CAlg, ⊗) to mean either one of the categories (CAlg U , ⊗), (CAlg N , ⊗ + ) when no confusion is possible.
Similarly, if ∂ ∈ Der(A, A) is a derivation of A, we write for the derivation of A ⊗n which satisfies ∂ i (a j ) = 0 if i = j and ∂ i (a i ) = (∂a) i .

Infinite symmetric products and symmetric algebras
For A ∈ CAlg N , let us denote the S n -invariants of A ⊗ n + by Sym n + A. This is again in CAlg N . We have (Sym n + A) + = Sym n (A + ), where Sym n refers to the usual symmetric power (in either vector spaces or unital algebras). The filtration 0 → C → A + → A → 0 induces a filtration F · of Sym n A + with Gr a F Sym n A + = Sym a A if a ≤ n; i.e. there is a natural inclusion of vector spaces Sym a A + ֒→ Sym n A + with image F a Sym n A + . This is not an algebra map; instead each Sym n A + is a filtered algebra: F a Sym n A + · F b Sym n A + ⊆ F a+b Sym n A + . We have an isomorphism of vector spaces Dually, the augmentation map 0 → A → A + → C → 0 induces surjective algebra maps and p nm (F a Sym n A + ) = F a Sym m A + if a ≤ m. So Sym n A + form an inverse system of filtered algebras in which each filtered piece eventually stabilises.
be the inverse limit taken in the category of filtered algebras.
We have F k Sym ∞ A + ∼ = F k Sym n A + = r≤k Sym r A for any n ≥ k.
as vector spaces, above, the coefficient of x a in x * · · · * x is t n−a times the number of ways of partitioning n into a non-empty subsets.
Recall that for any vector space V , the vector space Sym V = ⊕ Sym n V is the free commutative unital algebra generated by V . In particular Sym A is an algebra (non-negatively graded and hence exhaustively filtered).
The natural map A → A + = F 1 Sym n A + → Sym n A + lifts to a map A → Sym ∞ A + , and hence to an algebra homomorphism Sym A → Sym ∞ A + , which we denote P . Write P n : Sym A → Sym n A + for the quotient maps. The sequence of ideals ker P n endow Sym A with a seperated topology.
is an isomorphism of filtered algebras.
Proof. To show surjectivity, it suffices to show that Sym n A + is generated by for the composite of the symmetrisation map a 1 ⊗ · · · ⊗ a r → w∈Sr a w1 ⊗ · · · ⊗ a wr with the inclusion map (which is itself a symmetrisation). For r > n put δ r = 0. Then for a ∈ A, δ 1 (a) = P (a), and the elements 1, δ r (a 1 , . . . , a r ) for r > 0, a i ∈ A span Sym n A + . We have . From this it is clear that A generates Sym n A + . Injectivity is obvious.
Remark 2.3.4. If A ∈ CAlg U , then the isomorphism A + ∼ = C×A induces an isomorphism of unital algebras Sym n A + ∼ = C × A × · · · × Sym n A. This isomorphism is compatible with the projection maps p nm , but not the filtration.

Differential operators
In this section we define and identify differential operators on the filtered pro-ring Sym A.
Recall that a pro-object in a category C is a functor from N to C, where N is the category with objects {0, 1, 2, . . . } and a single morphism from n to m if n ≥ m. We write (X n , p nm : X n → X m ) for such a pro-object. If (X n ), (Y n ) are pro-objects, then Furthermore, if X n and Y n are filtered pro-objects then we define . Now define the "filtered" Hom as When each X n and Y n are exhaustively filtered this is the correct object; in general quotient this by F −∞ .
Recall that if R is a ring, and M an R-bimodule, then the differential part of M , denoted is the ring of differential operators on Spec R. When R is not noetherian this is too big a ring. Now let (A n , p nm ) be a commutative pro-ring. Then for n ≥ m Hom(A n , A m ) is an A n -bimodule, and we may take its differential part. Suppose that the maps p nm : A n → A m are surjective. Then for n ≥ m, the maps More generally, lim and Diff . pro End((A n )) = s Diff s pro End((A n )). If (A n , p nm ) is filtered, then the filtration on A n induces a filtration on Hom(A n , A m ), and thus on Diff s Hom(A n , A m ) and Diff s pro End((A n )). As always, we redefine the "filtered" differential operators to be the exhaustive part F ∞ Diff s pro End((A n )). Now, if (d n : A n → A n , n ≥ 0) is such that d m p nm = p nm d n when n ≥ m, then d n defines an element in End(lim ← A n ). The set of all such elements form a subalgebra. Hence if we define pro End((A n )) is a subalgebra of Diff . pro End((A n )). We have equality for s = 0, but not in general.
Let A ∈ CAlg N , and consider the pro-object (Sym n A + , p nm ).
Proof. We induct on s. The case s = 0 is clear. Let d ∈ Diff s Hom(Sym n A + , Sym m A + ). We show that dδ r (a 1 , . . . , a r ) = 0 for all r > m by descending induction on r. (Our notation is that of lemma 2.3.3.) For r > n this is clear. Apply d to equation 2.3.1, to get Hence to finish we need only show that as multiplication by k, so this is clear.
The above proposition shows that Diff 1 pro Hom((Sym n A + ), pro End((Sym n A + )).
pro End((Sym n A + )) from now on, and refer to these as the differential operators on the Fock space Sym A. As we have seen, they depend on the ring structure of A. The following sequence of propositions describe Diff Sym A. In particular, where for a ∈ A, we write a i for the image of a under the i'th coproduct map A → A ⊗+n , and for Θ ∈ A ⊗+n restricts to give an element, call it P s n (D), of Diff s End C (Sym n A + ), which satisfies the properties of the proposition.
It is clear that P 0 n = P n : A → Sym n A + , that for n ≥ m, p nm P s n (D) = P s m (D)p nm , that P s (D) ∈ F 1 Diff s pro End((Sym n A + )) always; and that if D(1 A ) = 0, then P s (D) ∈ F 0 Diff s Sym A.
So P is a map of C-Lie algebras. The enveloping algebra of the source consists of "non-linear differential operators".
The natural map p n : Diff Sym A → Diff Sym n A + admits a section, defined as follows.
If D ∈ Diff Sym n A + , then composing with the symmetrisation map e = 1 n! w∈Sn w, we get a map, , . . . , r} is a subset such that #I = n, let D I : A ⊗+r → A ⊗+r be the map which is D ⊗ Id, D in the I'th places. Finally, put Γ n (D) r = I D I ; the sum is over subsets I ⊆ {1, . . . , r} of size n.
It is immediate that Γ n (D) r is an S r -invariant differential operator; that Γ n (D) n = D, and that p rk Γ n (D) r = Γ n (D) r for all r ≥ k ≥ n. Hence we have defined an injection Γ n : Diff Sym n A + → Diff Sym A, such that p n Γ n = Id; in particular p n is surjective. Note that Γ n is not an algebra homomorphism, and that the image of Γ n is in F n Diff Sym A.
Proof. A differential operator of order s is determined by its values on products of ≤ s elements. As Sym A is generated by To finish the description of Diff Sym A, we need only describe Diff Sym n A + . If A ∈ CAlg U , then as Sym n A + ∼ = ⊕ 0≤i≤n Sym i A, Diff Sym n A + ∼ = ⊕ 0≤i≤n Diff End Sym i A. Hence Diff End Sym n A embdeds into Diff Sym n A + . WriteΓ n : Diff End Sym n A → Diff s Sym A for the composite of Γ n with the embedding. For exampleΓ 1 = P is the map defined in proposition 2.4.5.
Corollary 2.4.7. Let A ∈ CAlg U . There is an isomorphism of vector spaces ⊕ n≥0Γn : ⊕ n≥0 Diff s End Sym n A ∼ = Diff s Sym A.

Frobenius algebras
A non-unital Frobenius algebra H is an algebra H ∈ CAlg N equipped with a map ∆ : H → H ⊗ H such that ∆ is an H-bimodule map: ∆(ahb) = a∆(h)b. If H ∈ CAlg U , then ∆ is determined by ∆(1 H ) and we call H a weak Frobenius algebra. If in addition ∆ has a counit ǫ :

Relative differential operators
The following definition is somewhat ad hoc, but will do for our purposes.
Suppose A = H ⊗ R = H R , where H is a weak Frobenius algebra, and R is a localisation of C[x]. (We will only use C[x] or C[x, x −1 ] below). Let n ≥ 1, and δ disc = i<j (x i − x j ) ∈ R ⊗n be the discriminant. Write ∆ = ∆(1 H ). Now define the relative differential operators where End H ⊗n refers to H ⊗n -linear endomorphisms, and H ⊗n R [ ∆ij xi−xj | i < j] is the subring of the localisation H ⊗n R [ 1 δ disc ] generated by H ⊗n R and the elements ∆ij xi−xj . Clearly this is an algebra; if H = C and ∆ = 0 it is just Diff End Sym n R. This construction works whether we intepret H R to be in (CAlg U , ⊗) or (CAlg N , ⊗ + ); hence repeating the discussion of the previous section we can define

Twisting
For u ∈ A, define an algebra endomorphism For simplicity we assume throughout that u is even.
It follows that for u ∈ Z, Φ u (ker P n ) = ker P n , and moreover that Φ u is not continuous. (It is continuous if for all m > 1, there exists an n > m such that Φ u (ker P n ) ⊆ ker P m ). We so that if ∂(u) = 0, then Φ u P (∂)Φ −1 u = P (∂). Hence if ∂ i ∈ Der(A, A) and ∂ i u = 0 for i = 1, 2, ii)(The splitting principle.) Suppose K ∈ H is given. Then there is a Frobenius algebraH containing H as an index 2 subalgebra, and an element u ∈H such that u 2 − Ku − e = 0 inH.
Definition 2.7.3. Let (K λ , u λ ) be a degeneration direction. Define The notation is abusive; it omits the choice of degeneration direction. Note that the limit exists as a subalgebra of Diff End Sym H for general reasons, but that these general reasons give little control over the resulting algebra.

Locality
Let D A ⊆ Diff End(A) be a subalgebra of differential operators on A such that A ⊆ D A .
If E ′ and E ′′ are local with respect to B then so is E ′ , E ′′ . Hence there is a maximal algebra local with respect to B, which we denote Loc DA (B). So We will apply these notions to Diff H,K Sym H R and Diff End F .

The Calogero-Sutherland system
In this section we define a generalized Calogero-Sutherland type Hamiltonian L, which will turn out to be integrable. This is a second order differential operator acting on the Fock space Sym xH[x], where H is a Frobenius algebra. In section 6 we will take H to be the cohomology of a smooth surface, and then the integrals of motion for this Hamiltonian we will be precisely the operations of multiplication on the individual Hilbert schemes. In the next section we describe the integrals of motion for arbitrary H in terms of a Cherednik Hecke algebra.
It is clear that for a ≥ 0 these operators descend to give differential operators on F (H). The operator P (∂) is called the energy operator, and its eigenspaces the energy weight spaces. If H is finite dimensional then the eigenspaces of P (∂) on F (H) are finite dimensional; those on Sym H Γ are not.
Let K ∈ H. Then we define a second order differential operator L = L(H, K) ∈ Diff 2 End Sym H Γ by requiring that L.1 = 0 and [L, P (hx n )] = 2nP (hx n ∂) + n 2 P (Khx n ) + |n|∆ * P (hx n ) Again, it is clear that this descends to give a differential operator on F (H), the Calogero-Sutherland operator.
We retain this notation for the action of these operators on any invariant subquotient of Sym H Γ .
, and let u ∈ H × be any even invertible element. Then in the notation of section 2.7 In particular, L(H, We must show that L ′ = L ′′ . This is straightforward from the definitions. Begin by observing But we also have [L ′ , P (hx n )] = 2nP (uhx n ∂) + n 2 P ((u − eu −1 )hx n ) + |n|∆ * P (u −1 hx n ).
As L ′ 1 = L ′′ 1 = 0, we have L ′ = L ′′ . Finally, observe that as we have equality on the subset H × ⊆ H, it must be that the Φ u L ′′ Φ −1 u depends only on u − eu −1 . Hence for any degeneration direction (K λ , u λ ) its limit is just L(H, K) ∈ F 1 Diff 2 . if K = u − eu −1 for some invertible e, and in general let IM K = lim λ→0 IM u λ −eu −1 λ for a choice of degeneration direction (K λ , u λ ). As defined IM K seems to depend on the choice of degeneration direction. The notation is acceptable, because of the following theorem, which will be proved in section 4, and in theorem 5.0.5. Theorem 3.0.4. i) The algebras IM K depend only on K, as algebras, subalgebras of End Sym H Γ , and pro-finitely generated algebras.
ii) IM K forms a flat family of algebras as K varies. iii) IM K is commutative.

Hecke algebras
In this section we define a variant of the Dunkl-Cherednik operators. These operators act on H ⊗n Γ , where H is a weak Frobenius algebra. As in the usual case these operators commute, and together with the group algebra CS n form a generalization of the degenerate affine Hecke algebra of type A n . The algebra obtained by including the operations of multiplication by elements of H ⊗n Γ also closes; this is a generalization of the Cherednik degenerate double affine Hecke algebra.
Our exposition follows closely [11] (which considers the rational case). See also [6,20]. When H = C ∈ CAlg U , the results in sections 4.1-4.3 are contained in these papers.
In sections 4.1-4.3 all constructions work in either of the categories (CAlg U , ⊗) or (CAlg N , ⊗ + ). In section 4.4 we must specialise to the non-unital case in order to get augmentation maps.
If α = ε i − ε j define r ij = r α : Z n → Z n , l → l − α, l α to be the associated reflection, so that the group generated by the relections r α with α ∈ Φ is just S n .
Observe that S n acts on A ⊗n , and that for ∂ ∈ Der(A, A),

Dunkl-Cherednik operators
We now fix A to be The results in this section make sense if we consider H Γ to be in either (CAlg U , ⊗) or (CAlg N , ⊗ + ). Our notation is such that H ⊗n Γ is generated by the elements (hx a ) i = h i x a i , with 1 ≤ i ≤ n, h ∈ H, a ∈ Z. Further suppose that H is a weak Frobenius algebra, and k ∈ {i, j}.
Definition 4.2.1. Let l ∈ Z n , u ∈ H × . The Dunkl-Cherednik operator with parameter u is defined to be We write y i instead of y εi . Define an action of S n on the Dunkl operators by w y l = y wl .
ii) For f (y) ∈ C[y 1 , . . . , y n ], and α = ε i − ε i+1 , as operators from H ⊗n Γ to H ⊗n Γ . The most straightforward proof is by direct computation. We recall some of the main steps.
Explanding out ∇ α ∇ β this reduces the problem to checking it in the rank two root systems. Arguing as in [11] 2.2, the proposition reduces to the identity which we apply with z = x i /x j , w = x j /x k and i, j, k all distinct. We omit further details.
The proposition can rephrased: Define the Cherednik algebra (or degenerate double affine Hecke algebra) to be the vector space H n = H ⊗n Γ ⊗ CS n ⊗ C[y 1 , . . . , y n ] with the unique algebra structure that makes it a subalgebra of End C (H ⊗n Γ ) by which y l act as Dunkl operators, S n act as permutations, and H ⊗n Γ acts by multiplication. Namely, we require that C[y 1 , . . . , y n ], CS n and H ⊗n Γ are subalgebras, and that the relations of 4.2.2(ii) hold, and that [y l , Θ] = y l (Θ), Θ w = w w Θ for Θ ∈ H ⊗n Γ , l ∈ Z n , w ∈ S n . (For this to make sense, always regard CS n and C[y 1 , . . . , y n ] as unital algebras when taking tensor product with H ⊗n Γ ). Notice that CS n ⊗H[y] ⊗n forms a subalgebra of H n . This algebra has the same relation to the "degenerate affine Hecke" algebra of Drinfeld and Lusztig that H n has to the usual Cherednik algebra.
More generally we have then wf = f w for all w ∈ S n , so f preserves Sym n H Γ . To show this action is by differential operators, it suffices to show that H[y] ⊗n is in Diff Sym n HΓ Hom(Sym n H Γ , H ⊗n Γ ). But (hy) i = (hu∂) i ∈ Diff 1 Hom(Sym n H Γ , H ⊗n Γ ) and these elements generate H[y] ⊗n . This suffices, as it is immediate from the definition of Diff s R M that if D 1 , D 2 ∈ End(H ⊗n Γ ) are such that they restrict to differential operators D i ∈ Diff si Hom(Sym n H Γ , H ⊗n Γ ) then D 1 D 2 ∈ Diff s1+s2 Hom(Sym n H Γ , H ⊗n Γ ). We can rephrase the lemma: Write C[y 1 , . . . , y n ] ≤s for polynomials in y of total degree less than s; the degree of each y i is 1.
where S n acts on the n-fold tensor power in the obvious way.
This notation is compatible with all previous maps called P . So lemma 4.3.1(i) shows P (y) = P (∂), [P (hx a y), P (h ′ x m )] = mP (hh ′ x a+m ) and 4.3.1(ii) says Recall this means Let CS ⊆ Diff End Sym n H Γ be the subspace spanned by P (y 2 ) and P (hy), for h ∈ H, and set CS = Sym n H Γ , CS ⊆ Diff End Sym n H Γ . As the third term of 4.3.1 is clearly in Sym n H Γ , we have P (hx m y) ∈ CS for all m ∈ Z, h ∈ H. In fact:

CS-locality on the Fock space F
The definition of locality with respect to L (section 2.8) encodes the notion of differential operators which can be built out of functions and L. In this section we show that the symbols of local operators on F are contained in Sym H[p]; then we show that the Hecke algebra construction produces enough (all) the local operators.
We retain the notation of the previous sections, so H denotes a weak Frobenius algebra, K ∈ H, Filter Diff End F by order of differential operator, so that Gr Diff End F is a Poisson algebra with bracket denoted {, }, and Gr 0 Diff End F = F . Note that Gr i Diff End F is of uncountable dimension for i > 0. Write P (hx n p m ) for the symbol of P (hx n ∂ m ).
Observe that Gr . This implies part (ii) of the following lemma; part (i) is immediate from the definitions.  6 Hilbert schemes of points on surfaces.
Let X be a smooth, connected projective surface over C. We recall the basic results about the geometry and cohomology of the Hilbert scheme of points on X.
Definition 6.0.6. Denote by Hilb n (X) the moduli space of zero-dimensional subschemes Z ⊂ X satisfying Hilb n (X) is a projective variety if X is. What is special about the geometry of Hilbert schemes of points on surfaces are the following theorems: Theorem 6.0.7 (Fogarty [7]). Hilb n (X) is a smooth, connected, irreducible, projective variety.
Let S n X the symmetric product X n /S n . There is a natural Hilbert-Chow map Hilb n (X) → S n X which sends a subscheme Z ⊂ X to it's support, counted with multiplicity. This map is a resolution of singularities; its main property is: Theorem 6.0.8 (Briancon,[4]). If x ∈ X, the fiber of the Hilbert-Chow morphism over nx is an irreducible variety of dimension n − 1.
Both these theorems are false if dim X > 2. Since Hilb n (X) represents a functor, there is a universal family of subschemes Ξ n ⊂ X × Hilb n (X) flat and finite over Hilb n (X), of degree n. For each vector bundle V on X, Ξ n induces the tautological vector bundle V [n] = p 2 * p * 1 O Ξn on Hilb n (X), with rank V [n] = n rank V .

Fock space structure on the Hilbert scheme
We recall certain algebraic structures on the cohomology of the Hilbert schemes induced from correspon- This is a super vector space, graded by cohomology degree. It is also graded by length of subscheme: set F n = ⊕ i H i (Hilb n (X), C). The bilinear form on H * (Hilb n (X)) induces a non-degenerate, super-symmetric bilinear form on F . Define E n+m n,m ⊂ Hilb n (X) × Hilb m (X) × Hilb n+m (X) to be the closure of the locus of triples (a, b, c) of subschemes of X, satisfying: Putting all these together, we get maps These maps are adjoint : m † = c.
Theorem 6.1.1 (Grojnowski [10]). m and c give F the structure of a commutative, cocommutative graded Hopf algebra, in the super sense. The unit is given by the identity in the ring H * (Hilb 0 (X), C) ∼ = C.
An equivalent formulation was obtained in [19], which describes the structure of F as a module over the Heisenberg algebra.
A commutative, cocommutative graded Hopf algebra is naturally isomorphic to the free commutative algebra on the space of primitive elements, One can identify Prim n, * explicitly as the image of a map induced from a certain correspondence. Let and hence a canonical isomorphism of graded Hopf algebras In this isomorphism x is given degree 1, so this identifies F n with the energy weight space of weight n.
If we take X smooth, but not necessarily projective, we still have a canonical identification F ∼ = F (H * (X, C)). However, now this isomorphism is only as graded coalgebras. Dually, if we intepret H * (, C) as compactly supported cohomology, we again have such an isomorphism; but now it is of graded algebras.

Lehn's theorem
Cup product on H * (Hilb n (X), C) = F n induces a ring structure ⋆ : F n ⊗ F n → F n . This is distinct from the Fock space structure induced by correspondances. We would like to describe it.
Let T * X be the cotangent bundle of X, and T * [n] X be the associated tautological bundle on Hilb n (X), a bundle of rank 2n. Write H = H * (X, C), and K = c 1 (T * X ) for the canonical class. H is a weak Frobenius algebra; if X is projective it is even a Frobenius algebra. Note moreover that e = c 2 (T * X ), and so an element u such that u 2 − Ku − e = 0 is precisely a Chern root of T * X . We define the Calogero-Sutherland operator L(H, K) ∈ End F . The main result of [12] is the computation of cup product with the boundary of the Hilbert scheme in terms of the Fock space coordinates of [10,19]: Theorem 6.2.1 (Lehn [12]). The linear map F → F defined on F n as is the Calogero-Sutherland operator L(H, K). This is theorems 3.10 and 4.2 of [12]. The following theorem is due to Lehn [12] for the subalgebra H * alg (X, C) ⊂ H * (X, C) of algebraic cohomology, and was extended to all of H * (X, C) by Li, Qin and Wang [15,16]. Define CS K = L, P (h∂) | h ∈ H . Write D hilb for the subalgebra of Diff End F generated by E i (h) for h ∈ H, i ≥ 0. Then CS K ⊆ D hilb , and the image of D hilb in End F n coincides with the algebra of left multiplication operators, i.e. the image of the natural map H(X, C) → End F n , a → a * .

Hecke algebras and Hilbert schemes
We now apply the results of sections 2-5 in the context of section 6. This gives a description of the ring structure on H * (Hilb n (X), C) in the Fock space coordinates. In particular, this gives an explicit algebraic construction of the ring D hilb , and shows it depends only on the weak Frobenius algebra H = H(X, C) and the class K ∈ H. In particular, the image of IM K in End F n coincides with the algebra of left multiplication operators, i.e. the image of the natural map H(X, C) → End F n , a → a * .
Proof. We have already proved (ii) holds for any weak Frobenius algebra H, and K ∈ H as theorem 5.0.5. We prove (i).
Let us compute the symbol s of E i (h). The conditions in theorem 6.2.2 give {s, P (h ′ x)} = Ad P (p 2 ) i P (hh ′ x) = 2 i P (hh ′ xp i ), is Poisson self-centralising in Gr Diff End F ; it follows that D hilb is also the algebra of all differential operators on F which preserve each F n , and whose restriction to each F n is cup product with some class. (As this algebra is obviously commutative, and contains E i .) In this paper we have set up a formalism precisely linking the combinatorics of intersection theory on the Hilbert scheme with integrable systems and certain generalizations of Cherednik algebras.
It is now straightforward to compute all geometric information on the Hilbert scheme in terms of the well understood combinatorics of Jack polynomials and/or Cherednik algebras. Elaborations and further generalizations will appear elsewhere.