aa r X i v : . [ m a t h . R T ] J a n PROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS
IVAN LOSEV
Abstract.
In this survey we describe an interplay between Procesi bundles on sym-plectic resolutions of quotient singularities and Symplectic reflection algebras. Procesibundles were constructed by Haiman and, in a greater generality, by Bezrukavnikovand Kaledin. Symplectic reflection algebras are deformations of skew-group algebrasdefined in complete generality by Etingof and Ginzburg. We construct and classify Pro-cesi bundles, prove an isomorphism between spherical Symplectic reflection algebras,give a proof of wreath Macdonald positivity and of localization theorems for cyclotomicRational Cherednik algebras. Introduction
Procesi bundles: Hilbert scheme case.
A Procesi bundle is a vector bundleof rank n ! on the Hilbert scheme Hilb n ( C ) whose existence was predicted by Procesiand proved by Haiman, [Hai1]. This bundle was used by Haiman to prove a famous n ! conjecture in Combinatorics that, in turn, settles another famous conjecture: Schurpositivity of Macdonald polynomials.1.1.1. n! theorem. Consider the Vandermond determinant ∆( x ), where we write x for( x , . . . , x n ), it is given by ∆( x ) = det( x j − i ) ni,j =1 . Consider the space ∂ ∆ spanned byall partial derivatives of ∆. This space is graded and carries an action of the symmetricgroup S n (by permuting the variables x , . . . , x n ). A deeper fact is that dim ∂ ∆ = n ! (and ∂ ∆ ∼ = C S n as an S n -module), in fact, ∂ ∆ coincides with the space of the S n -harmonicpolynomials, i.e., all polynomials annihilated by all elements of C [ ∂ ] S n without constantterm.One can ask if there is a two-variable generalization of that fact. We have severaltwo-variable versions of ∆, one for each Young diagram λ with n boxes. Namely, let( a , b ) , . . . , ( a n , b n ) be the coordinates of the boxes in λ , e.g., λ = (3 ,
2) gives pairs(0 , , (1 , , (2 , , (0 , , (1 , (0 ,
0) (1 ,
0) (2 , ,
1) (1 , Then set ∆ λ ( x, y ) := det( x a j i y b j i ) ni,j =1 so that, for λ = ( n ), we get ∆ λ ( x, y ) = ∆( x ), for λ = (1 n ), we get ∆ λ ( x, y ) = ∆( y ), while, for λ = (2 , λ ( x, y ) = x y + x y + x y − x y − x y − x y . Theorem 1.1 (Haiman’s n! theorem) . The space ∂ ∆ λ spanned by the partial derivativesof ∆ λ is isomorphic to C S n as an S n -module (where S n acts by permuting the pairs ( x , y ) , . . . , ( x n , y n ) ) and, in particular, has dimension n ! . MSC 2010: 14E16, 53D20, 53D55 (Primary) 05E05 , 16G20, 16G99, 16S36, 17B63, 20F55 (Secondary).
This is a beautiful result with an elementary statement and an extremely involvedproof, [Hai1].1.1.2.
Macdonald positivity.
Before describing some ideas of the proof that are relevantto the present survey, let us describe an application to
Macdonald polynomials , partic-ularly important and interesting symmetric polynomials with coefficients in Q ( q, t ). Itwill be more convenient for us to speak about representations of S n rather than aboutsymmetric polynomials (they are related via taking the Frobenius character) and to dealwith Haiman’s modified Macdonald polynomials. Definition 1.2.
The modified Macdonald polynomial ˜ H λ is the Frobenius character of abigraded S n -module P λ := L i,j ∈ Z P λ [ i, j ] subject to the following three conditions(a) The class of P λ ⊗ P ni =0 ( − i V i C n [1 ,
0] is expressed via the irreducibles V µ with µ > λ (in the K of the bigraded S n -modules).(b) P λ ⊗ P ni =0 ( − i V i C n [0 ,
1] is expressed via the irreducibles V µ with µ > λ t .(c) The trivial module V ( n ) occurs in P λ once and in degree (0 , µ > λ is the usual dominance order on the set of Young diagrams (meaning that P ki =1 µ i > P ki =1 λ i ) and λ t denotes the transpose of λ .It is not clear from this definition that the representations P λ exist (the statement onthe level of K is easier but also non-trivial, this was known before Haiman’s work). Theorem 1.3 (Haiman’s Macdonald positivity theorem) . A bigraded S n -module P λ exists(and is unique) for any λ . Moreover, P λ coincides with ∂ ∆ λ , where ∂ ∆ λ is is given thestructure of a bigraded S n -module as the quotient of C [ ∂ x , ∂ y ] (via f f ∆ λ ). Hilbert schemes and Procesi bundles.
The proofs of the two theorems above givenin [Hai1] are based on the geometry of the Hilbert schemes Hilb n ( C ) of points on C .A basic reference for Hilbert schemes of points on smooth surfaces is [Nak4]. As a set,Hilb n ( C ) consists of the ideals J ⊂ C [ x, y ] of codimension n . It turns out that Hilb n ( C )is a smooth algebraic variety of dimension 2 n . It admits a morphism (called the Hilbert-Chow map) to the variety Sym n ( C ) of the unordered n -tuples of points in C : to anideal J one assigns its support, where points are counted with multiplicities. Of course,Sym n ( C ) is nothing else but the quotient space ( C ) ⊕ n / S n , the affine algebraic varietywhose algebra of functions is the invariant algebra C [ x, y ] S n . The Hilbert-Chow map is aresolution of singularities.Note that the two-dimensional torus ( C × ) acts on Hilb n ( C ) and on Sym n ( C ), theaction is induced from the following action on C : ( t, s ) . ( a, b ) := ( t − a, s − b ). The fixedpoints of this action on Hilb n ( C ) correspond to the monomial ideals (=ideals generatedby monomials) in C [ x, y ], they are in a natural one-to-one correspondence with Youngdiagrams (as before we fill a Young diagram with monomials and take the ideal spannedby all monomials that do not appear in the diagram). Let z λ denote the fixed pointcorresponding to a Young seminar λ .Following Haiman, consider the isospectral Hilbert scheme I n , the reduced Cartesianproduct C n × Sym n ( C ) Hilb n ( C ), let η : I n → Hilb n ( C ) be the natural morphism. Itis finite of generic degree n !. The main technical result of Haiman, [Hai1], is that I n isCohen-Macaulay and Gorenstein. So P := η ∗ O I n is a rank n ! vector bundle on Hilb n ( C )(the Procesi bundle). By the construction, each fiber of this bundle carries an algebrastructure that is a quotient of C [ x, y ]. Let us write P λ for the fiber of P in z λ , this is an ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 3 algebra that carries a natural bi-grading because the bundle P is ( C × ) -equivariant bythe construction. On the other hand, ∂ ∆ λ is a quotient of C [ ∂ x , ∂ y ] by an ideal and so isalso an algebra. The latter algebra is bigraded. Haiman has shown that P λ ∼ = ∂ ∆ λ , anisomorphism of bigraded algebras. This finishes the proof of Theorem 1.1.Let us proceed to Theorem 1.3. The class in (a) of Definition 1.2 is that of the fiber at z λ of the Koszul complex(1) P ← h x ⊗ P ← Λ h x ⊗ P ← . . . , where h x is the span of x , . . . , x n viewed as endomorphisms of P . Haiman has shown that I n is flat over Spec( C [ x ]) (with morphism I n → Spec( C [ x, y ]) → Spec( C [ x ])). It followsthat (1) is a resolution of P / h x P . Now (a) follows from the claim that, for any Youngdiagram λ , the support of the isotypic V λ -component in P / h x P contains only points z µ with µ λ . This was checked by Haiman. Part (b) is analogous, while (c) follows directlyfrom the construction.There are several other proofs of Theorem 1.3 available. Two of them use the geometryof Hilbert schemes and Procesi bundle, [G4, BF]. We will discuss (a somewhat modified)approach from [BF] in detail in Section 5.1.2. Quotient singularities and symplectic resolutions.
Setting.
Let V be a finite dimensional vector space over C equipped with a sym-plectic form Ω ∈ V V ∗ . Let Γ be a finite subgroup of Sp( V ). The invariant algebra C [ V ] Γ is a graded Poisson algebra (as a subalgebra of C [ V ]) and the corresponding quo-tient V /
Γ = Spec( C [ V ] Γ ) is a singular Poisson affine variety that comes with a C × -actioninduced from the action on V by dilations: t.v := t − v .1.2.2. Symplectic resolutions.
One can ask if there is a resolution of singularities of
V /
Γthat is nicely compatible with the Poisson structure (and with the C × -action). Thiscompatibility is formalized in the notion of a (conical) symplectic resolution. Definition 1.4.
Let X be a singular normal affine Poisson variety such that the regularlocus X reg is symplectic. We say that a variety X equipped with a morphism ρ : X → X is a symplectic resolution of X if X is symplectic (with form ω ), ρ is a resolution ofsingularities and ρ : ρ − ( X reg ) → X reg is a symplectomorphism. Definition 1.5.
Further, suppose that X is equipped with a C × -action such that • the corresponding grading C [ X ] = L i ∈ Z C [ X ] i is positive, meaning that C [ X ] i = { } for i < C [ X ] = C , • and the Poisson bracket on C [ X ] has degree − d for some fixed d ∈ Z > : { C [ X ] i , C [ X ] j } ⊂ C [ X ] i + j − d for all i, j .We say that a symplectic resolution X is conical if it is equipped with a C × -action making ρ equivariant.The variety X = V /
Γ is normal and carries a natural C × -action (by dilations) as inDefinition 1.5 with d = 2. Also note that the C × -action on X automatically satisfies t.ω = t − d ω . Finally, note that, under assumptions of Definition 1.4, we have C [ X ] = C [ X ]. IVAN LOSEV
Symplectic resolutions for quotient singularities.
In the previous subsection, wehave already seen an example of ( V, Γ) such that
V /
Γ admits a conical symplectic reso-lution: V = ( C ) ⊕ n , Γ = S n , in this case one can take X = Hilb n ( C ) together with theHilbert-Chow morphism, see [Nak4, Section 1].There are other examples as well. Let Γ be a finite subgroup of SL ( C ), such sub-groups are classified (up to conjugacy) by Dynkin diagrams of ADE types. Say, thecyclic subgroup Z / ( ℓ + 1) Z (embedded into SL ( C ) via n diag( η n , η − n ) with η :=exp(2 π √− / ( ℓ + 1))) corresponds to the diagram A ℓ . The quotient singularity C / Γ admits a distinguished minimal resolution to be denoted by ^C / Γ . This resolution isconical symplectic, see, e.g., [Nak4, Section 4.1].The examples of S n and Γ can be “joined” together. Consider the group Γ n := S n ⋉ Γ n .It acts on V n := ( C ) ⊕ n : the symmetric group permutes the summands, while each copyof Γ acts on its own summand. The quotient singularities V n / Γ n admit symplecticresolutions. For example, one can take X := Hilb n ( ^C / Γ ). But there are other conicalsymplectic resolutions of V n / Γ n , conjecturally, they are all constructed as Nakajima quivervarieties, we will recall the construction of these varieties in 3.1.4.To finish this section, let us point out that, presently, two more pairs ( V, Γ) such that
V /
Γ admits a symplectic resolutions are known, see [Be, BS]. In this paper, we are notinterested in these cases.1.3.
Procesi bundles: general case.
Smash-product algebra.
One nice feature of quotient singularities
V /
Γ is that theyalways have a nice resolution of singularities which is, however, noncommutative algebraicrather than algebro-geometric: the smash-product algebra C [ V ] C [ V ] C [ V ] ⊗ C Γ, and the product on C [ V ] f ⊗ γ ) · ( f ⊗ γ ) = f γ ( f ) ⊗ γ γ , f , f ∈ C [ V ] , γ , γ ∈ Γ , where γ ( f ) denotes the image of f under the action of γ . The definition is arrangedin such a way that a C [ V ] C [ V ]-module.Note that the algebra C [ V ] f of degree n , thedegree of f ⊗ γ is n .Let us explain what we mean when we say that C [ V ] V /
Γ. Note that C [ V ] Γ can be recovered from C [ V ] C [ V ] Γ ֒ → C [ V ] f f ⊗
1. The image lies inthe center (this is easy) and actually coincides with the center (a bit harder). Second,consider the element e ∈ C Γ , e := | Γ | − P γ ∈ Γ γ , the averaging idempotent. Consider thesubspace e ( C [ V ] e ⊂ C [ V ] e is aunit with respect to the multiplication there. So e ( C [ V ] e is an algebra, to be calledthe spherical subalgebra of C [ V ] C [ V ] Γ , an isomorphism is givenby f ef .Thanks to the realization of C [ V ] Γ as a spherical subalgebra, we can consider the functor M eM : C [ V ] → C [ V ] Γ -mod (an analog of the morphism ρ ). Note that thealgebra C [ V ] C [ V ] does) and so is “smooth”.The algebra C [ V ] C [ V ] Γ which can be thought as an analog of ρ beingproper. Also, after replacing C [ V ] , C [ V ] Γ with sheaves O V reg , O V reg / Γ on V reg / Γ, ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 5 where(2) V reg := { v ∈ V | Γ v = { }} , the functor M eM becomes a category equivalence. This is an analog of ρ beingbirational.1.3.2. Procesi bundle: an axiomatic description.
Now let X be a conical symplectic res-olution of V /
Γ. We want to relate X to C [ V ] Definition 1.6.
A Procesi bundle P on X is a C × -equivariant vector bundle on X together with an isomorphism End O X ( P ) ∼ −→ C [ V ] C [ X ] = C [ V ] Γ such that Ext i ( P , P ) = 0 for i > O X ( P ) ∼ −→ C [ V ] P . Theinvariant sheaf e P is a vector bundle of rank 1. We say that P is normalized if e P = O X (as a C × -equivariant vector bundle). We can normalize an arbitrary Procesi bundle bytensoring it with ( e P ) ∗ . Below we only consider normalized Procesi bundles.In particular, Haiman’s Procesi bundle on X = Hilb n ( C ) fits the definition, this isessentially a part of [Hai2, Theorem 5.3.2] (and is normalized). The existence of a Procesibundle on a general X was proved by Bezrukavnikov and Kaledin in [BK2]. We will seethat the number of different Procesi bundles on a symplectic resolution of C n / Γ n equals2 | W | if n >
1, where W is the Weyl group of the Dynkin diagram corresponding to Γ .For example, when Γ = Z /ℓ Z , we get W = S ℓ and so the number of different Procesibundles is 2 ℓ !.1.4. Symplectic reflection algebras.
Definition.
Symplectic reflection algebras were introduced by Etingof and Ginzburgin [EG]. Those are filtered deformations of C [ V ] γ with rk( γ − V ) = 2. Notethat the rank has to be even: the image of γ − V is a symplectic subspace of V . By S we denote the set of all symplectic reflections in Γ, it is a union of conjugacy classes, S = ⊔ ri =1 S i . Now pick t ∈ C and c = ( c , . . . , c r ) ∈ C r . We define the algebra H t,c as thequotient of T ( V ) u ⊗ v − v ⊗ u = t Ω( u, v ) + r X i =1 c i X s ∈ S i Ω( π s u, π s v ) s, u, v ∈ V. Here we write π s for the projection V ։ im( s − V ) corresponding to the decomposition V = im( s − V ) ⊕ ker( s − V ).As Etingof and Ginzburg checked in [EG], the algebra H t,c satisfies the PBW property:if we filter H t,c by setting deg Γ = 0 , deg V = 1, then gr H t,c = C [ V ] V with V ∗ by means of Ω so that C [ V ] ∼ = S ( V )). Moreover, we will see that H t,c satisfiesa certain universality property so this deformation of C [ V ] Connection to Procesi bundles.
It may seem that Symplectic reflection algebrasand Procesi bundles are not related. This is not so. It turns out that the algebra H t,c isthe endomorphism algebra of a suitable understood deformation of a Procesi bundle P .This connection is beneficial for studying both. On the Procesi side, it allows to classifyProcesi bundles, [L4], and prove the Macdonald positivity in the case of groups Γ n withΓ = Z /ℓ Z , [BF]. On the symplectic reflection side, it allows to relate the algebras H t,c to IVAN LOSEV quantized Nakajima quiver varieties, see [EGGO, L3] and references therein, which thenallows to study the representation theory of H t,c ([BL]) and to prove versions of Beilinson-Bernstein localization theorems, [GL, L5]. Connections between Procesi bundles andSymplectic reflection algebras is a subject of this survey.1.5. Notation and conventions.
Let us list some notation used in the paper.
Quantizations and deformations . We use the following conventions for quantizations.For a Poisson algebra A , we write A ~ for its formal quantization. When A is graded,we write A for its filtered quantization. The notation D ~ is usually used for a formalquantization of a variety, while D usually denotes a filtered quantization.When X is a conical symplectic resolution of singularities, we write ˜ X for its universalconical deformation (over H DR ( X )) and ˜ D ~ stands for the canonical quantization of ˜ X . Symplectic reflection groups and algebras . We write Γ for a finite subgroup of SL ( C )and Γ n for the semidirect product S n ⋉ Γ n . This semi-direct product acts on V n := C n .In the case when Γ = { } , we usually write V n for T ∗ C n − , where C n − is the reflectionrepresentation of S n .For a group Γ acting on a space V by linear symplectomorphisms, by S we denote theset of symplectic reflections in Γ. By e we denote the averaging idempotent of Γ. By H we denote the universal symplectic reflection algebra of ( V, Γ). Its specializations aredenoted by H t,c . Quotients and reductions . Let G be a group acting on a variety X . If G is finite and X is quasi-projective, then the quotient is denoted by X/G (note that this quotient mayfail to exist when X is not quasi-projective). If G is reductive and X is affine, then X//G stands for the categorical quotient. A GIT quotient of X under the G -action with stabilitycondition θ is denoted by X// θ G .When X is Poisson, and the G -action is Hamiltonian, we write X/// λ G for µ − ( λ ) //G and X/// θλ G for µ − ( λ ) // θ G . Miscellaneous notation . b ⊗ the completed tensor product of complete topological vector spaces/modules.( a , . . . , a k ) the two-sided ideal in an associative algebra generated by elements a , . . . , a k . A ∧ χ the completion of a commutative (or “almost commutative”) algebra A with respect to the maximal ideal of a point χ ∈ Spec( A ). A ( V ) the Weyl algebra of a symplectic vector space V . D ( X ) the algebra of differential operators on a smooth variety X . F q the finite field with q elements.gr A the associated graded vector space of a filtered vector space A . H iDR ( X ) the i th De Rham cohomology of X with coefficients in C . O X the structure sheaf of a scheme X . R ~ ( A ) := L i ∈ Z ~ i A i :the Rees C [ ~ ]-module of a filtered vector space A . S n the symmetric group in n letters. S ( V ) the symmetric algebra of a vector space V .Sp( V ) the symplectic linear group of a symplectic vector space V .Γ( S ) global sections of a sheaf S . ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 7
Acknowledgements : This survey is a greatly expanded version of lectures I gave atNorthwestern in May 2012. I would like to thank Roman Bezrukavnikov and Iain Gordonfor numerous stimulating discussions. My work was supported by the NSF under GrantDMS-1161584. 2.
Quantizations
In this section we review the quantization formalism. In Section 2.1 we discuss quan-tizations of Poisson algebras. There are two formalisms here: filtered quantizations andformal quantizations. We introduce both of them, discuss a relation between them andthen give examples.Then, in Section 2.2, we proceed to quantizations of non-necessarily affine Poissonalgebraic varieties. Here we quantize the structure sheaf. We explain that to quantize anaffine variety is the same thing as to quantize its algebra of functions. Then we mentiona theorem of Bezrukavnikov and Kaledin classifying quantizations of symplectic varietiesunder certain cohomology vanishing conditions.After that we proceed to modules over quantizations. We define coherent and quasi-coherent sheaves of modules and outline their basic properties. For a coherent sheaf ofmodules, we define its support. Then we discuss global section and localization functorsand their derived analogs.We finish this system by discussing Frobenius constant quantizations in positive char-acteristic.2.1.
Algebra level.
Here we will review formalisms of quantizations of Poisson algebras.Let A be a Poisson algebra (commutative, associative and with a unit).2.1.1. Formal quantizations.
First, let us discuss formal quantizations. By a formal quan-tization of A we mean an associative C [[ ~ ]]-algebra A ~ equipped with an algebra isomor-phism π : A ~ / ( ~ ) ∼ −→ A such that(i) A ~ ∼ = A [[ ~ ]] as a C [[ ~ ]]-module and this isomorphism intertwines π and the naturalprojection A [[ ~ ]] → A .(ii) We have π ( ~ [ a, b ]) ≡ { π ( a ) , π ( b ) } (note that π ([ a, b ]) = [ π ( a ) , π ( b )] = 0 and so ~ [ a, b ] makes sense).Condition (i) can be stated equivalently as follows: A ~ is flat over C [[ ~ ]] and is completeand separated in the ~ -adic topology.2.1.2. Filtered quantizations.
Second, we will need the formalism of filtered quantizations.Suppose that A is equipped with an algebra grading, A = L i ∈ Z A i , that is compatiblewith {· , ·} in the following way: { A i , A j } ⊂ A i + j − .First, we consider the case when the grading on A is non-negative: A i = { } for i < A one means a Z > -filtered algebra A = S i > A i together with a graded algebra isomorphism π : gr A ∼ −→ A such that, for a ∈ A i , b ∈ A j ,one has { π ( a + A i − ) , π ( b + A j − ) } = π ([ a, b ] + A i + j − ) (note that [ a, b ] ∈ A i + j − because gr A is commutative).2.1.3. Relation between the two formalisms.
Let us explain a connection between the twoformalisms (that will also motivate the definition of a filtered quantization in the case whenthe grading on A has negative components). Take a filtered quantization A of A . Formthe Rees algebra R ~ ( A ) := L i > A i ~ i that is equipped with a graded algebra structure as IVAN LOSEV a subalgebra in A [ ~ ]. We have natural identifications R ~ ( A ) / ( ~ ) ∼ = A, R ~ ( A ) / ( ~ − ∼ = A .The ~ -adic completion R ~ ( A ) ∧ ~ := lim ←− n → + ∞ R ~ ( A ) / ( ~ n ) satisfies (i) and (ii) and so is aformal quantization of A . Moreover, it comes with a C × -action by algebra automorphismssuch that t. ~ = t ~ , t ∈ C × : the action is given by t. P + ∞ i =0 a i ~ i := P + ∞ i =0 t i a i ~ i . Clearly,the induced action on A coincides with the action coming from the grading. Conversely,suppose we have a formal quantization A ~ of A equipped with a C × -action by algebraautomorphisms such that t. ~ = t ~ and the epimorphism π is C × -equivariant. Assume,further, that the action is pro-rational meaning that it is rational on all quotients A ~ / ( ~ n ).Consider the subspace A ~ ,fin ⊂ A ~ consisting of all C × -finite elements, i.e., those elementsthat are contained in some finite dimensional C × -stable subspace. This is a C × -stable C [ ~ ]-subalgebra of A ~ . It is easy to see that π induces an isomorphism A ~ ,fin / ( ~ ) ∼ = A .Then A := A ~ ,fin / ( ~ −
1) is a filtered quantization.2.1.4.
Filtered quantizations, general case.
Let us proceed to the case when the gradingon A is not necessarily non-negative. We can still consider a formal quantization A ~ witha C × -action as above, the subalgebra A ~ ,fin ⊂ A ~ and the quotient A := A ~ ,fin / ( ~ − Z -filtration rather than a Z > -filtration) but, moreover, the filtration on A hasa special property: it is complete and separated meaning that a natural homomorphism A → lim ←− n →−∞ A / A n is an isomorphism. By a filtered quantization of A we now meana Z -filtered algebra A , where the filtration is complete and separated, together with anisomorphism π : gr A ∼ −→ A of graded algebras such that { π ( a + A i − ) , π ( b + A j − ) } = π ([ a, b ] + A i + j − ).Our conclusion is that the following two formalisms are equivalent: filtered quantiza-tions and formal quantizations with a C × -action. To get from a filtered quantization A to a formal one, one takes R ~ ( A ) ∧ ~ . To get from a formal quantization A ~ to a filteredone, one takes A ~ ,fin / ( ~ − Examples.
Let us proceed to examples. In examples, one usually gets Z > -filteredquantizations, more general Z -filtered or formal quantizations arise in various construc-tions (such as (micro)localization or completion). Example 2.1.
Let g be a Lie algebra. Then, by the PBW theorem, the universal en-veloping algebra U ( g ) is a filtered quantization of S ( g ). Example 2.2.
Let Y be an affine algebraic variety. The algebra D ( Y ) of linear differentialoperators on Y (together with the filtration by the order of differential operators) is afiltered quantization of C [ T ∗ Y ]. Remark 2.3.
Often one needs to deal with a more general compatibility conditionbetween the grading and the bracket: { A i , A j } ⊂ A i + j − d for some fixed d >
0. Inthis case, one can modify the definitions of formal and filtered quantizations. Namely,in the definition of a formal quantization one can require that [ A ~ , A ~ ] ⊂ ~ d A ~ and π ( ~ d [ a, b ]) = { π ( a ) , π ( b ) } . The definition of a filtered quantization can be modified simi-larly. Example 2.4.
Let V be a symplectic vector space and Γ ∈ Sp( V ) be a finite group.Consider A = S ( V ) Γ with Poisson bracket {· , ·} restricted from S ( V ). In the notation ofRemark 2.3, d = 2. As was essentially checked in [EG], the spherical subalgebra eH ,c e ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 9 (with a filtration restricted from H ,c ) is a quantization of S ( V ) Γ for any parameter c When Γ = { V } , we recover the usual Weyl algebra, A ( V ), of V .To check that eH ,c e is a quantization carefully we note that the proof of Theorem 1.6in loc.cit. shows that the bracket on S ( V ) Γ coming from the filtered deformation eH ,c e coincides with a {· , ·} , where a is a nonzero number independent of c . Then we notice thatfor c = 0 we get eH ,c e = A ( V ) Γ and so a = 1.In fact, in the previous example we often can also achieve d = 1. Namely, if − V ∈ Γ,then all degrees in S ( V ) Γ are even and so we can consider the grading A = L i > A i with A i consisting of all homogeneous elements with usual degree 2 i . We introduce a filtrationon eH ,c e in a similar way (this filtration is not restricted from H ,c ). Then we get afiltered quantization according to our original definition. When Γ = Γ n , we only have − V Γ if Γ = Z /ℓ Z for odd ℓ . For Γ = Z /ℓ Z (and any ℓ ), V splits as h ⊕ h ∗ , where h = C n . We can grade S ( V ) by setting deg h ∗ = 0 , deg h = 1 and take the induced gradingon S ( V ) Γ and the induced filtration on H ,c .2.2. Sheaf level.
Above, we were dealing with Poisson algebras or, basically equivalently,with affine Poisson algebraic varieties. Now we are going to consider general Poissonvarieties (or schemes). Recall that by a Poisson variety one means a variety X such thatthe structure sheaf O X is equipped with a Poisson bracket (meaning that all algebras ofsections are Poisson and the restriction homomorphisms respect the Poisson brackets).In this case a quantization of X will be a (formal or filtered) quantization of O X in thesense explained below in this section.2.2.1. Formal quantizations.
We start with a formal setting. A quantization D ~ of X is asheaf of C [[ ~ ]]-algebras on X together with an isomorphism π : D ~ / ( ~ ) ∼ −→ O X such that(a) D ~ is flat over C [[ ~ ]] (equivalently, there are no nonzero local sections annihilatedby ~ ) and complete and separated in the ~ -adic topology (meaning that D ~ ∼ −→ lim ←− n → + ∞ D ~ / ( ~ n )).(b) π ( ~ [ a, b ]) = { π ( a ) , π ( b ) } for any local sections a, b of D ~ .2.2.2. Motivation: star-products.
The origins of this definition are in the deformationquantization introduced in [BFFLS]. Let us adopt this definition to our situation. Let A be a Poisson algebra. By a star-product on A one means a bilinear map ∗ : A ⊗ A → A [[ ~ ]]subject to the following conditions:(1) The C [[ ~ ]]-bilinear extension of ∗ to A [[ ~ ]] is associative and 1 ∈ A is a unit.(2) a ∗ b ≡ ab mod ~ A [[ ~ ]], a ∗ b − b ∗ a ≡ ~ { a, b } mod ~ A [[ ~ ]].Of course, A [[ ~ ]] together with ∗ is a formal quantization of A in the sense of the previoussection. Conversely, any formal quantization A ~ is isomorphic to ( A [[ ~ ]] , ∗ ).Traditionally, one imposes an additional restriction on ∗ : the locality axiom that re-quires that the coefficients D i in the ~ -adic expansion of ∗ ( a ∗ b = P ∞ i =0 D i ( a, b ) ~ i ) arebidifferential operators. If ∗ is local, then it naturally extends to any localization A [ a − ].So, if A = C [ X ] for X affine, then a local star-product defines a quantization of O X .Let us provide an example of a local star-product. Consider A = C [ x, y ] with standardPoisson bracket: { x i , x j } = { y i , y j } , { y i , x j } = δ ij . Then set(4) f ∗ g = m ◦ exp( ~ n X i =1 ∂ y i ⊗ ∂ x i ) f ⊗ g, where µ : A ⊗ A → A is the usual commutative product. For example, we have x i ∗ x j = x i x j , y i ∗ y j = y i y j , x i ∗ y j = x i y j , y j ∗ x i = x i y j + ~ δ ij . In this case, A [ ~ ] is closed withrespect to ∗ and is identified with R ~ ( D ( C n )).2.2.3. Algebra vs sheaf setting in the affine case.
It turns out that any formal quantization A ~ of C [ X ] for an affine variety X defines a quantization of X . The reason is that wecan localize elements of C [ X ] in A ~ . The construction is as follows. Pick f ∈ C [ X ] andlift it to ˆ f ∈ A ~ . The operator ad ˆ f is nilpotent in A ~ / ( ~ n ) for any n and so the set { ˆ f n } ⊂ A ~ / ( ~ n ) satisfies the Ore conditions, hence the localization A ~ / ( ~ n )[ ˆ f − ] makessense. It is easy to see that these localizations do not depend on the choice of the lift ˆ f and form an inverse system. We set A ~ [ f − ] := lim ←− n → + ∞ A ~ / ( ~ n )[ ˆ f − ]. Exercise 2.5.
Check that there is a unique sheaf D ~ in the Zariski topology on X suchthat D ~ ( X f ) = A ~ [ f − ] for any f ∈ C [ X ] and that this sheaf is a quantization of X .So we see that there is a natural bijection between the quantizations of X and of C [ X ](to get from a quantization of X to that of C [ X ] we just take the global sections). Thanksto this, we can view a quantization of a general variety X as glued from affine pieces.2.2.4. Filtered quantizations.
Let us proceed to the filtered setting. Suppose that X isequipped with a C × -action such that the Poisson bracket has degree −
1. Obviously, for anarbitrary open U ⊂ X , the algebra C [ U ] does not need to be graded. However, it is gradedwhen U is C × -stable. By a conical topology on X we mean the topology, where “open”means Zariski open and C × -stable. One can ask whether this topology is sufficiently rich,for example, whether any point has an open affine neighborhood. Theorem 2.6 (Sumihiro) . Suppose X is normal. Then any point in X has an open affineneighborhood in the conical topology. Below we always assume that X is normal. Note that O X is a sheaf of graded algebras inthe conical topology. By a filtered quantization of X we mean a sheaf D of filtered algebras(in the conical topology on X ) equipped with an isomorphism π : gr D ∼ −→ O X of gradedalgebras such that the filtration on D is complete and separated and π is compatible withthe Poisson brackets as in 2.1.2.We still have a one-to-one correspondence between filtered quantizations and formalquantizations with C × -actions. This works just as in 2.1.3 (note that D ~ ,fin makes senseas a sheaf in conical topology).2.2.5. Quantization in families.
Let X be a smooth scheme over a scheme S . It still makessense to speak about closed and non-degenerate forms in Ω ( X/ S ). By a symplectic S -scheme we mean a smooth S -scheme X together with a closed non-degenerate form ω S ∈ Ω ( X/ S ). Note that from ω one can recover an O S -linear Poisson bracket on X .By a formal quantization D ~ of X we mean a sheaf of O S -algebras on X satisfyingconditions (a),(b) in 2.2.1.Note that the definition above still makes sense when S is a formal scheme and X is aformal S -scheme.2.2.6. Classification theorem.
Let us finish this section with a classification theorem dueto Bezrukavnikov and Kaledin, [BK1] (with a ramification given in [L3]).
ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 11
Theorem 2.7.
Let X be a smooth symplectic variety. Suppose H ( X, O X ) = H ( X, O X ) =0 (this holds when X is affine, for example). Then the formal quantizations of X are pa-rameterized by H DR ( X, C )[[ ~ ]] . If X has a C × -action compatible with the bracket (wherewe have d = 1 ), then the filtered quantizations are in one-to-one correspondence with H DR ( X, C ) . Even without the cohomology vanishing assumption, there is a so called period map
Per from the set
Quant ( X ) of formal quantizations of X (considered up to an isomorphism)to H DR ( X )[[ ~ ]]. When the vanishing condition holds, this map is a bijection. The clas-sification of filtered quantizations follows from the observation that once a quantizationadmits a C × -action by automorphisms, its period lies in H DR ( X ) ⊂ H DR ( X )[[ ~ ]] (and ifthe vanishing holds, the converse is also true), see [L3, 2.3].Assume until the end of the section that the vanishing condition holds.A formal quantization D ~ having a C × -action by automorphisms and satisfying Per( D ~ ) =0 has a nice property: it is even . When X is affine this means that the quantiza-tion can be realized by a star-product f ∗ g = P ∞ i =0 D i ( f, g ) ~ i with deg D i = − i and D i ( f, g ) = ( − i D i ( g, f ). For general X , being even means that there is an antiautomor-phism ̺ of D ~ that commutes with the C × -action, is the identity modulo ~ , and maps ~ to − ~ .Let us finish this subsection with the discussion of the universal quantization. Thevariety X has a universal symplectic deformation b X over the formal disc S that is theformal neighborhood of 0 in H DR ( X ) (provided H i ( X, O X ) = 0 for i = 1 , X is obtainedfrom b X by pull-back. Further, there is a canonical quantization b D ~ of b X/ S . All quantiza-tions of X are obtained by pulling back b D ~ . More precisely, we can view b D ~ as a sheaf of C [[ H DR ( X ) , ~ ]]-algebras on X (via the sheaf-theoretic pull-back) and then we can obtainquantizations of X by base change to C [[ ~ ]].In the case when X , in addition, has a C × -action rescaling the symplectic form, we canconsider the universal C × -equivariant deformation ˜ X over H DR ( X ) as well as its canonicalquantization ˜ D ~ .2.3. Modules over quantizations.
Let X be a Poisson variety (or scheme). We aregoing to define coherent and quasi-coherent modules over filtered and formal quantizationsof X (to be denoted by D and D ~ , respectively).2.3.1. Coherent modules over formal quantizations.
By definition, a sheaf M ~ of D ~ -modules on X is called coherent if M ~ / ~ M ~ is a coherent O X -module and M ~ is completeand separated in ~ -adic topology. Note that the condition of being complete and separatedin the ~ -adic topology is local. So being coherent is a local condition (as in Algebraicgeometry).Let X be affine and let A ~ := Γ( D ~ ). Let N ~ be a finitely generated A ~ -module. Thenit is easy to see that N ~ is complete and separated in the ~ -adic topology. It follows that D ~ ⊗ A ~ N ~ is a coherent D ~ -module. Conversely, for a coherent D ~ -module M ~ , the globalsections Γ( M ~ ) is a finitely generated A ~ -module. Lemma 2.8.
Let X be affine. Then the functors D ~ ⊗ A ~ • and Γ( • ) are mutually quasi-inverse equivalences between the categories of coherent D ~ -modules and finitely generated A ~ -modules. Proof.
Note that these functors define compatible equivalences between the categories ofcoherent D ~ / ( ~ n )-modules and of finitely generated A ~ / ( ~ n )-modules for any n (which isproved in the same way as the classical statement for n = 1). Then we use that all objectswe consider are complete and separated in the ~ -adic topology. (cid:3) From this lemma we easily see that a subsheaf and a quotient sheaf of a coherent D ~ -module is coherent itself. So the category Coh( D ~ ) of coherent D ~ -modules is an abeliancategory.2.3.2. Quasi-coherent modules over formal quantizations.
By a quasi-coherent D ~ -modulewe mean a direct limit of coherent D ~ -modules. Lemma 2.8 implies that, when X is affine,the category of quasi-coherent D ~ -modules is equivalent to the category of Γ( D ~ )-modules.Analogously to the classical algebro-geometric result, the category QCoh( D ~ ) of quasi-coherent D ~ -modules has enough injective objects. Note that the natural functor from D b (Coh( D ~ )) to the full subcategory in D b (QCoh( D ~ )) of all complexes with coherenthomology is a category equivalence. This is because any quasi-coherent complex is aunion of coherent subcomplexes, as in the usual Algebro-geometric situation.2.3.3. Modules over filtered quantizations.
Let us proceed to modules over filtered quan-tizations. Let M be a sheaf of D -modules. We say that M is coherent if it can beequipped with a global complete and separated filtration compatible with that on D andsuch that gr M is a coherent sheaf on X (such a filtration is usually called good ). The ~ -adic completion of the Rees sheaf R ~ ( M ) is then a C × -equivariant coherent D ~ -module.Conversely, if we take a C × -equivariant coherent D ~ -module M ~ , take the C × -finite part M ~ ,fin , then M ~ ,fin / ( ~ −
1) is a coherent D -modules. Lemma 2.9.
Consider the full subcategory
Coh C × ( D ~ ) tor consisting of all modules thatare torsion over C [[ ~ ]] . Then taking quotient by ~ − gives rise to an equivalence Coh C × ( D ~ ) / Coh C × ( D ~ ) tor ∼ −→ Coh( D ) .Proof. Let us produce a quasi-inverse functor. Of course, the R ~ ( D )-module R ~ ( M )depends on the choice of a good filtration. Let F , F ′ be two good filtrations. Then onecan find positive integers d , d such that F i − d M ⊂ F ′ i M ⊂ F i + d M the inclusion ofsubsheaves (of vector spaces) in M (it is enough to check this claim for local sections overopen subsets from an affine cover, where it is easy). It follows that modulo ~ -torsion thesheaf R ~ ( M ) is independent of the choice of a good filtration. Our quasi-inverse functorsends M to the ~ -adic completion of R ~ ( M ). To check that this is indeed a quasi-inversefunctor is standard. (cid:3) Supports.
For a coherent D ~ -module M ~ we have the notion of support. By defi-nition, Supp( M ~ ) := Supp( M ~ / ~ M ~ ), this is a closed subvariety in X .Now let M ∈
Coh( D ). Then we can take a good filtration on M and set Supp( M ) :=Supp(gr M ). By the argument in the proof of Lemma 2.9, the support of M is well-defined, i.e., it does not depend on the choice of a good filtration.2.3.5. Global sections and localization.
Let D be a filtered quantization of X . We havenatural functors Coh( D ) → Γ( D ) -mod of taking global sections (to be denoted by Γ) aswell as a functor in the opposite direction Loc : Γ( D ) -mod → Coh( D ) , M
7→ D ⊗ Γ( D ) M .Let us discuss a situation when these functors behave particularly nicely. Namely, let X be a conical symplectic resolution of singularities of an affine variety X . Note that, ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 13 by the Grauert-Riemenschneider theorem, the higher cohomology of O X vanish. This hasthe following corollary (the proof is left to the reader). Lemma 2.10.
We have H i ( D ) = 0 for i > . Moreover, Γ( D ) is a quantization of X . Thanks to this lemma, it makes sense to consider derived functors R Γ : D (Coh( D )) → D (Γ( D ) -mod) and L Loc : D (Γ( D ) -mod) → D (Coh( D )). In fact, R Γ is given by theˇCech complex and so restricts to bounded (to the left and to the right) derived categories.The functor L Loc restricts to D − ’s. Lemma 2.10 implies that R Γ ◦ L Loc is the identityon D − (Γ( D ) -mod). Furthermore, if Γ( D ) has finite homological dimension, then L Locmaps D b (Γ( D ) -mod) to D b (Coh( D )) and is left inverse to R Γ. It is likely (and is provedin many cases, see, e.g., [MN]) that R Γ and L Loc are mutually quasi-inverse equivalencesin this case.2.4.
Frobenius constant quantizations.
Above, we were dealing with the case whenthe ground field is C . Everything works the same for any algebraically closed field ofcharacteristic 0. In this section we are going to work over an algebraically closed field F of positive characteristic.The notions of filtered and formal quantizations still make sense, both for algebrasand for varieties. But in positive characteristic we have an important special class ofquantizations: Frobenius constant ones.2.4.1. Basic example.
Let us start our discussion with an example of a quantization: theWeyl algebra A ( V ), where V is a symplectic F -vector space. A new feature is that thisalgebra is finite over its center. Namely, for v ∈ V ⊂ A ( V ), the element v p ∈ A ( V ) liesin the center. We have a semi-linear map ι : V → A ( V ) given by v v p on v ∈ V with central image that extends to a ring homomorphism S ( V ) → A ( V ). The semi-linearity condition is ι ( av ) = Fr( a ) ι ( v ), where Fr : F → F is the Frobenius automorphism.Let V (1) denote the F -vector space identified with V as an abelian group but with newmultiplication by scalars: a.v = Fr − ( a ) v . So ι becomes an algebra homomorphism whenviewed as a map S ( V (1) ) → A ( V ), its image is usually called the p -center, in our case itcoincides with the whole center. Another important feature of this example is that A ( V )is an Azumaya algebra over V (1) , i.e., A ( V ) is a vector bundle over Spec( S ( V (1) )) and all(geometric) fibers are matrix algebras (of rank p dim V/ ).2.4.2. Definition.
The notion of a Frobenius constant quantization generalizes the exam-ple in 2.4.1. We will give the definition in the filtered setting and only for symplecticvarieties – we will only need it in this case. Let X be a smooth symplectic F -varietyequipped with an F × -action rescaling the symplectic form (by the character t t d ). Let X (1) be the F -variety that is identified with X as a scheme over Spec( Z ) but with twistedmultiplication by scalars in the structure sheaf just as in 2.4.1. We have a natural mor-phism Fr : X → X (1) of F -varieties and hence we have a sheaf Fr ∗ ( O X ) on X (1) . This isa coherent sheaf of algebras and a vector bundle of rank p dim X . Definition 2.11.
A Frobenius constant quantization is a filtered sheaf D of Azumayaalgebras on X (1) together with an isomorphism gr D ∼ −→ Fr ∗ O X of graded algebras (inconical topology) that satisfies our usual compatibility condition on Poisson brackets.It is not difficult to show that a Frobenius constant quantization gives rise to a filteredquantization of X . But, as we will see 3.3.3, not every filtered quantization arises thisway. Differential operators.
Let us give another example that should be thought as aglobal analog of 2.4.1. Let Y be a smooth F -variety. Consider the sheaf D Y of differentialoperators on Y . Let ξ be a vector field on an open subset Y ′ ⊂ Y . Define a vector field ξ [ p ] as follows. For every open affine subvariety Y ⊂ Y ′ , we can regard ξ as a derivationof F [ Y ]. The map ξ p : F [ Y ] → F [ Y ] is again a derivation. The corresponding vectorfield on Y ′ (that is easily seen to be well-defined) is what we denote by ξ [ p ] . It is easyto see that f p , for a function f on Y , and ξ p − ξ [ p ] , for a vector field ξ (here ξ p is takenwith respect to the product on D Y ), are central. The maps f f p , ξ ξ p − ξ [ p ] giverise to a sheaf of algebras homomorphism π ∗ O ( T ∗ Y ) (1) → Fr ∗ D Y , where we write π for theprojection ( T ∗ Y ) (1) = T ∗ ( Y (1) ) → Y (1) . The sheaf D Y then becomes a Frobenius constantquantization of T ∗ Y .To finish this section, let us mention that, under some restrictions on X , there is aclassification of Frobenius constant quantizations, see [BK3].3. Hamiltonian reductions
In this section we recall the notions of the classical and quantum Hamiltonian reduction.The classical Hamiltonian reduction produces a new Poisson variety from an existingPoisson variety with suitable symmetries. The quantum Hamiltonian reduction does thesame on the level of quantizations.We start by discussing classical Hamiltonian reductions, Section 3.1. First, we recallHamiltonian actions and moment maps. Then we define classical Hamiltonian reductionsin the settings of categorical quotients and of GIT quotients. We then proceed to theconstruction and basic properties of Nakajima quiver varieties that are our main examplesof Hamiltonian reductions. Next, we explain how quotient singularities V n / Γ n are realizedas quiver varieties. Finally, we construct symplectic resolutions of singularities for V n / Γ n and establish, following Namikawa, some isomorphisms between some of these resolutions.In Section 3.2 we proceed to quantum Hamiltonian reductions. We define them on thelevel of algebras and on the level of sheaves and compare the two levels. After that westate one of the main results of this survey: an isomorphism between spherical SRA forwreath-product groups and quantum Hamiltonian reductions. We finish this section bydiscussing a quantum version of Namikawa’s Weyl group action.Section 3.3 deals with Hamiltonian reductions for Frobenius constant quantization. Wefirst recall some basic results on GIT in positive characteristic. Then we discuss Nakajimaquiver varieties in sufficiently large positive characteristic. Finally, we prove, followingBezrukavnikov, Finkelberg and Ginzburg, that the quantum Hamiltonian reduction of aFrobenius constant quantization at an integral value of the quantum comoment map isFrobenius constant.3.1. Classical Hamiltonian reduction.
Hamiltonian group actions.
Let X be a Poisson variety (over an algebraically closedfield) and let G be an algebraic group acting on X . The action induces a Lie algebra ho-momorphism g → Vect( X ), the image of ξ ∈ g under this homomorphism will be denotedby ξ X . We say that the G -action on X is Hamiltonian , if there is a G -equivariant linearmap g → C [ X ] , ξ H ξ , such that { H ξ , ·} = ξ X . Note that this map is automatically a Liealgebra homomorphism. This map is called the comoment map , the dual map µ : X → g ∗ is the moment map .Let us provide two examples of Hamiltonian actions. ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 15
Example 3.1.
Let Y be a smooth variety, G act on Y . Then X := T ∗ Y carries a natural G -action. This action is Hamiltonian with H ξ = ξ Y (viewed as a function on X ). Example 3.2.
Let V be a vector space (with symplectic form Ω) and let G act on V bylinear symplectomorphisms. The action is Hamiltonian with H ξ ( v ) = Ω( ξv, v ).Below we will need a standard property of Hamiltonian actions. Lemma 3.3.
Let x ∈ X . Then im d x µ ⊂ g ∗ coincides with the annihilator of g x :=Lie( G x ) . In particular, µ is a submersion at x if and only if G x is finite. Hamiltonian reduction.
Let A be a Poisson algebra and g be a Lie algebra equippedwith a Lie algebra homomorphism g → A, ξ H ξ . Consider the ideal I := A { H ξ , ξ ∈ g } .The adjoint action of g on A preserves this ideal so we can take the invariants A/// g :=( A/I ) g . This algebra comes with a natural Poisson bracket: { a + I, b + I } := { a, b } + I (but A/I has no Poisson bracket!).This construction has several ramifications. First, let λ : g → C be a character (i.e.,a function vanishing on [ g , g ]). Then we can set A/// λ g := ( A/A { H ξ − h λ, ξ i} ) g . Alsowe can set A/// g := ( A/A { H ξ , ξ ∈ [ g , g ] } ) g . The latter is a Poisson S ( g / [ g , g ])-algebrawhose specialization at λ ∈ ( g / [ g , g ]) ∗ coincides with A/// λ g provided that the g -actionon A/A { H ξ , ξ ∈ [ g , g ] } is completely reducible.Let us proceed to a geometric incarnation of this construction. Suppose the base fieldis of characteristic 0. To ensure a good behavior of quotients assume that G is a reductivegroup. Let X be an affine Poisson variety equipped with a Hamiltonian G -action. Then wecan take A := C [ X ] together with the comoment map ξ H ξ . We set A/// G := ( A/I ) G ,this algebra coincides with A /// g when G is connected. It is finitely generated by theHilbert theorem, here we use that G is reductive. The variety (or scheme) Spec( A/// G )is nothing else but the categorical quotient X/// G := µ − (0) //G .Here is a corollary of Lemma 3.3. Corollary 3.4.
Suppose that X is smooth and symplectic and that the G -action on µ − (0) is free. Then X/// G is smooth and symplectic of dimension dim X − G .Proof. The variety µ − (0) is smooth by Lemma 3.3. That the quotient is smooth ofrequired dimension is a straightforward corollary of the Luna slice theorem, see, e.g., [PV,Section 6.3].The form on X/// G can be recovered as follows. Let Ω denote the form on X , ι : µ − (0) ֒ → X denote the inclusion map and π : µ − (0) → X/// G be the projection. Thenthere is a unique 2-form Ω red on X/// G such that π ∗ Ω red = ι ∗ Ω and this is the form weneed. (cid:3)
GIT Hamiltonian reduction.
We will be mostly interested in Hamiltonian reduc-tions for linear actions G y V . The assumptions of Corollary 3.4 are not satisfied in thiscase. However, if one uses GIT quotients instead of the usual categorical quotients, onecan often get a smooth symplectic variety that will be a resolution of the usual reduction V /// G .Let us recall the construction of a GIT quotient. Let G be a reductive algebraic groupacting on an affine algebraic variety X . Fix a character θ : G → C × . We use the additivenotation for the multiplication of characters. Then consider the space C [ X ] G,nθ of nθ -semiinvariants: C [ X ] G,nθ := { f ∈ C [ X ] | g.f := θ ( g ) n f } (recall that g.f ( x ) := f ( g − x )).Consider the graded algebra L n > C [ X ] G,nθ , where deg C [ X ] G,nθ := n . Then we set X// θ G := Proj( L n > C [ X ] G,nθ ), this is a projective variety over
X//G . Note that we nolonger have a morphism X → X// θ G . Instead, consider the open subset of θ -semistable points X θ − ss , a point x ∈ X is called semistable if there is f ∈ C [ X ] G,nθ for n > f ( x ) = 0. We clearly have a natural morphism X θ − ss → X// θ G that makes the followingdiagram commutative XX θ − ss X//GX// θ G ❄ ❄✲✲ ⊆ The variety
X// θ G is glued from the varieties of the form X f //G , where f ∈ C [ X ] G,nθ with some n >
0. The intersection of X f //G, X g //G inside X// θ G is identified with X fg //G , where the inclusions X fg //G ֒ → X f //G, X g //G are induced from the inclusions X fg ֒ → X f , X g by passing to the quotients.In the setting of 3.1.2, we set X/// θ G := µ − (0) θ − ss //G . This is a Poisson variety(the bracket comes from gluing together the brackets on the open subvarieties X f /// G )equipped with a projective morphism X/// θ G → X/// G of Poisson varieties. If X issmooth and symplectic, and the G -action on µ − (0) θ − ss is free, then X/// θ G is smoothand symplectic of dimension dim X − G . The symplectic form on X/// θ G is recoveredsimilarly to the case of X/// G considered above.3.1.4. Nakajima quiver varieties:construction.
Now we are going to introduce an impor-tant special class of varieties constructed by means of Hamiltonian reduction: Nakajimaquiver varieties, introduced in [Nak1], see also [Nak3].By a quiver, we mean an oriented graph. Formally, it can be presented as a quadruple Q = ( Q , Q , t, h ), where Q , Q are finite sets of vertices and arrows, respectively, and t, h : Q → Q are maps that to an arrow a assign its tale and head.Let us proceed to (framed) representations of Q . Fix two elements v, w ∈ Z Q > and set V i := C v i , W i := C w i , i ∈ Q . Consider the space R (= R ( Q, v, w )) := M a ∈ Q Hom C ( V t ( a ) , V h ( a ) ) ⊕ M i ∈ Q Hom C ( W i , V i ) . An element of R can be thought as a collection of linear maps, one for each arrow,between the corresponding vector spaces, together with collections of vectors in each V i .This description suggests a group of symmetry of R : we set G := Q i ∈ Q GL( V i ), thisgroup acts by changing bases in the spaces V i .A character of G is of the form g = ( g i ) i ∈ Q Q i ∈ Q det( g i ) θ i , where θ = ( θ i ) i ∈ Q ∈ Z Q .We will identify the character group of G with Z Q .A Nakajima quiver variety M θλ ( v, w ) is, by definition, the reduction T ∗ R/// θλ G . Here λ is a character of g , it can be thought as an element of C Q via λ ( x ) := P i ∈ Q λ i tr( x i ).The moment map µ : T ∗ R → L i ∈ Q End( V i ) = g ( ∼ = g ∗ ) is explicitly given as follows:( x a , x a ∗ , i k , j k ) a ∈ Q ,k ∈ Q X a ∈ Q ( x a x a ∗ − x a ∗ x a ) − X k ∈ Q j k i k , where x a ∈ Hom( V t ( a ) , V h ( a ) ) , x a ∗ ∈ Hom( V h ( a ) , V t ( a ) ) , i k ∈ Hom( V k , W k ) , j k ∈ Hom( W k , V k ). ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 17
We also would like to remark that the quiver variety is independent of the choice of anorientation of Q . Indeed, let Q ′ be a quiver obtained from Q by changing the orientationof a single arrow a and let R ′ be the corresponding representation space. Then we have anisomorphism T ∗ R ∼ = T ∗ R ′ that sends x a to x ′ a ∗ , x a ∗ to − x ′ a and does not change the othercomponents. This is a G -equivariant symplectomorphism that intertwines the momentmaps and hence inducing a symplectomorphism of the corresponding Nakajima quivervarieties.When λ = 0, we have a C × -action on M θ ( v, w ) that rescales the Poisson structure.For example, one can take the action induced by the dilation action on T ∗ R , that is, t.v := t − v, t ∈ C × , v ∈ T ∗ R to be called the dilation action as well. Then the Poissonbracket on M θ ( v, w ) has degree −
2. We can also have an action such that the Poissonbracket has degree − t. ( r, r ∗ ) := ( r, t − r ∗ ) , r ∈ R, r ∗ ∈ R .3.1.5. Nakajima quiver varieties: structural results.
Let us explain some structural re-sults regarding the quiver varieties and the corresponding moment maps. We will needalgebro-geometric properties of µ − ( λ ) and of M λ ( v, w ) due to Crawley-Boevey and alsoa criterium for the freeness of the G -action on µ − ( λ ) θ − ss due to Nakajima. Theorem 3.5 (Crawley-Boevey, [CB2]) . The scheme M λ ( v, w ) is reduced and normal. We now want to provide a criterium for µ : T ∗ R → g ∗ to be flat proved in [CB1]. Definethe symmetrized Tits form C Q × C Q → C :( v , v ) := X a ∈ Q ( v t ( a ) v h ( a ) + v h ( a ) v t ( a ) ) − X i ∈ Q v i v i and quadratic maps p, p w : C Q → C by p ( v ) := 1 −
12 ( v, v ) , p w ( v ) := w · v −
12 ( v, v ) . Theorem 3.6 (Crawley-Boevey, [CB1]) . The following two conditions are equivalent: (i) µ is flat. (ii) p w ( v ) > p w ( v ) + P ki =1 p ( v i ) for any decomposition v = v + v + . . . + v k with v i ∈ Z Q > for i = 1 , . . . , k . Theorem 3.7 (Crawley-Boevey, [CB1]) . Suppose that, for a proper decomposition v = v + v + . . . + v k , we have p w ( v ) > p w ( v ) + P ki =1 p ( v i ) . Then µ − (0) is irreducible anda generic G -orbit there is closed and free. Let us proceed to a criterium for the action of G on µ − ( λ ) θ − ss to be free. We canview Q as a Dynkin diagram and form the corresponding Kac-Moody algebra g ( Q ). Then C Q gets identified with the dual of the Cartan of g ( Q ) in such a way that the coordinatevector ǫ i , i ∈ Q , becomes a simple root. Then, [Nak1], the action of G on µ − ( λ ) θ − ss isfree if and only if there are no roots v ′ of g ( Q ) such that v ′ v (component-wise) and v ′ · θ = v ′ · λ = 0.The equations v ′ · θ = 0, where v ′ is a root satisfying v ′ v, v ′ · λ = 0 split the characterlattice into the union of cones. It is a classical fact from GIT, that when θ, θ ′ are genericand inside one cone, we have µ − ( λ ) θ − ss = µ − ( λ ) θ ′ − ss . So M θλ ( v, w ) = M θ ′ λ ( v, w ). n ( C ) and C n / S n as quiver varieties. Let Q be a quiver with a single vertexand a single loop (a.k.a. the Jordan quiver). We are going to show that Hilb n ( C ) isidentified with M − ( n,
1) and C n / S n is identified with M ( n,
1) (and the Hilbert-Chowmap from 1.1.3 becomes the natural morphism M − ( n, → M ( n,
1) from 3.1.3).An identification M − ( n, ∼ = Hilb n ( C ) is an easier part. We have R = End( C n ) ⊕ C n .Using the trace pairing, we identify R ∗ with End( C n ) ⊕ C n ∗ so that T ∗ R = End( C n ) ⊕ ⊕ C n ⊕ C n ∗ . We write ( A, B, i, j ) for a typical point of T ∗ R . Identifying g with g ∗ again usingthe trace pairing, we can write the moment map µ : T ∗ R → g as µ ( A, B, i, j ) = [
A, B ]+ ij .Using the Hilbert-Mumford theorem from Invariant theory, see, e.g., [PV, 5.3], oneshows that ( T ∗ R ) θ − ss = { ( A, B, i, j ) | C h A, B i i = C n } . Then it is a nice Linear Al-gebra exercise to show that if [ A, B ] + ij = 0 and C h A, B i i = C n , then j = 0. So µ − (0) θ − ss //G = { ( A, B, i ) | [ A, B ] = 0 , C [ A, B ] i = C n } /G that recovers the classical de-scription of Hilb n ( C ), see [Nak4, Theorem 1.14].An identification M ( n, ∼ = C n / S n is more subtle. An easy part is to constructa morphism ι : C n / S n → M ( n, x, y ) ∈ C n to (diag( x ) , diag( y ) , , ∈ µ − (0) and this induces a morphism of quotients. Then one checks that ι is a closedembedding. For this, one uses a classical result of Weyl to see that polynomials of theform ι ∗ F m,n , where F m,n ( A, B, i, j ) := tr( A n B m ) generate the algebra C [ x, y ] S n . It remainsto prove that ι is surjective. This is based on an even nicer linear algebra fact: A, B ∈ End( C n ) with rk[ A, B ] Lemma 3.8.
The isomorphism M ( n, ∼ = C n / S n intertwines the Poisson brackets.Proof. Consider the principal open subsets R reg = { ( A, i ) | A has distinct e-values } , C n,reg := { ( x , . . . , x n ) | x i = x j , for i = j } . Note that under the above embedding C n ֒ → T ∗ R , we have T ∗ C n,reg ֒ → T ∗ R reg . More-over, the pull-back of the symplectic form from T ∗ R reg to T ∗ C n,reg coincides with the nat-ural symplectic form on the latter. Using the description of the symplectic form on the re-duction, we conclude that the induced morphism of quotients T ∗ C n,reg / S n → T ∗ R reg /// G is a symplectomorphism. But T ∗ R reg /// G embeds as an open subset into M ( n,
1) andthe symplectomorphism above is the restriction of the isomorphism C n / S n ∼ −→ M ( n, T ∗ C n,reg / S n . The claim of the lemma follows. (cid:3) McKay correspondence.
Let Γ be a finite subgroup of SL ( C ). It turns out thatthe singular Poisson variety V n / Γ n (where recall V n = C n ) and its symplectic resolutionsalso can be realized as Nakajima quiver varieties.The first step in this isomorphism is the McKay correspondence: a way to label the finitesubgroups of SL ( C ) by Dynkin diagrams. Let Γ be a finite subgroup of SL ( C ) and let N , . . . , N r be the irreducible representations of Γ , where N is the trivial representation.Let us define the McKay graph of Γ : its vertices are 0 , , . . . , r and the number of edges(we consider a non-oriented graph) between i and j is dim Hom Γ ( C ⊗ N i , N j ), note thatthis is well-defined because C is a self-dual representation of Γ and so the number ofedges between i and j is the same as between j and i . McKay proved the following facts:(i) The resulting graph is an extended Dynkin graph of types A, D, E and 0 is theextending vertex.(ii) The vector (dim N i ) ri =0 is the indecomposable imaginary root δ of the correspond-ing affine Kac-Moody algebra. ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 19 C / Γ as a quiver variety. Let Q be the McKay graph of Γ with an arbitraryorientation. Then there is an isomorphism M ( δ, ∼ = C / Γ .Let us explain how this is established following [CBH, Section 8]. For this, we will needthe representation varieties.Let A be a finitely generated associative algebra and V be a vector space. Then the set X := Hom( A, End( V )) of algebra homomorphisms is an algebraic variety. More precisely,if A is the quotient of C h x , . . . , x n i by relations F α ( x , . . . , x n ), where α runs over anindexing set I , then X = { ( A , . . . , A n ) ∈ End( V ) | F α ( A , . . . , A n ) = 0 , α ∈ I } . Thegroup G := GL( V ) naturally acts on X and so we can form the quotient X//G (calledthe representation variety). Recall that, in general, the points of
X//G correspond tothe closed G -orbits on X , in our case an orbit is closed if its element is a semisimplerepresentation.This construction has various ramifications. For example, we can consider a semisimplefinite dimensional subalgebra A ⊂ A and an A -module V . This leads to the variety X of A -linear homomomorphisms A → End( V ) acted on by the group G of A -linearautomorphisms of V . In this situation we still can speak about representation varieties.We will realize M ( δ, , C / Γ as the representation varieties of this kind and then showthat the algebras involved are Morita equivalent, this will yield an isomorphism of interest.Let us start with C / Γ . Set A := C [ x, y ] , A := C Γ ⊂ A and V := C Γ , a regularrepresentation. Then one can show that C / Γ is the representation variety for this triple.Let us proceed to M ( δ, Q be the double quiver of Q . It is obtained from Q byadding the inverse arrow to each arrow in Q . Formally, ¯ Q = Q , ¯ Q = Q ⊔ Q ∗ , where Q ∗ is in bijection with Q , a a ∗ , in such a way that t ( a ∗ ) = h ( a ) , h ( a ∗ ) = t ( a ). Then formthe path algebra C ¯ Q of ¯ Q , it has a basis consisting of the paths in ¯ Q , the multiplication isgiven by concatenation (if two paths cannot be concatenated, the product is zero). Thisalgebra is graded by the length of a path, where, by convention, the degree 0 paths arejust vertices so the corresponding graded component C ¯ Q is C Q .Let us consider the quotient Π ( Q ) of C ¯ Q called the preprojective algebra. It is givenby the following relation: X a ∈ Q [ a, a ∗ ] = 0 . Note that C Q naturally embeds into Π ( Q ). It is easy to see that M ( δ,
0) is the repre-sentation variety for the triple (Π ( Q ) , C Q , L i ∈ Q C δ i ).It turns out that there is an idempotent f ∈ C Γ such that f ( C [ x, y ] ) f ∼ = Π ( Q ).Namely, take primitive idempotents f i , i = 0 , . . . , r, in the matrix summands of C Γ .Set f := P i ∈ Q f i . Obviously, f ( C Γ ) f ∼ = C Q . Further, the construction of Q im-plies that f (Span( x, y ) ⊗ C Γ ) f ∼ = C ¯ Q . These identifications induce an isomorphism f ( C h x, y i ) f ∼ = C ¯ Q . Under this isomorphism, the ideal f ( xy − yx ) f becomes ( P a ∈ Q [ a, a ∗ ]),see [CBH, Section 2]. Also note that the C Q -module L i ∈ Q C δ i is nothing else but f C Γ .Finally, note that f defines a Morita equivalence between C [ x, y ] , Π ( Q ). An isomor-phism C / Γ ∼ = M ( δ,
0) now follows from the next lemma, whose proof is left to thereader.
Lemma 3.9.
Let A ⊂ A and V be as above and let f ∈ A be an idempotent giving aMorita equivalence. Then the representation varieties for ( A, A , V ) and ( f Af, f A f, f V ) are naturally isomorphic. Note that the algebras C [ x, y ] and Π ( Q ) are graded and an isomorphism Π ( Q ) ∼ = C [ x, y ] preserves the grading. From here one easily deduces that the isomorphism C / Γ ∼ = M ( δ,
0) is equivariant with respect to the dilation C × -actions.3.1.9. V n / Γ n as a quiver variety. Let us proceed now to the case of an arbitrary n . Let ǫ ∈ C Q be the coordinate vector at the extending vertex. Proposition 3.10.
We have a C × -equivariant isomorphism M ( nδ, ǫ ) ∼ = V n / Γ n (ofPoisson schemes).Proof. We have a diagonal embedding T ∗ R ( Q, δ, ⊕ n → T ∗ R ( Q, nδ, ǫ ), compare to 3.1.6,that restricts to µ − (0) n ֒ → µ − (0), where µ stands for the moment map T ∗ R ( Q, δ, → gl ( δ ) ∗ . This gives rise to a S n -invariant morphism M ( δ, n → M ( nδ, ǫ ) and hence toa morphism ι : C n / Γ n = ( C / Γ ) n / S n → M ( nδ, ǫ ). One can show that this morphismis bijective. Also it is C × -equivariant, where the C × -actions on C n / Γ n , M ( nδ, ǫ ) areinduced from the dilation actions on C n , T ∗ R ( Q, nδ, ǫ ). It follows that ι is finite. ByTheorem 3.5, M ( nδ, ǫ ) is normal and this implies that ι is an isomorphism.We can make the isomorphism ι Poisson if we rescale it using the C × -actions. This isa consequence of the following lemma. (cid:3) Lemma 3.11. [EG, Lemma 2.23]
Let V be a symplectic vector space and Γ ⊂ Sp( V ) be a finite subgroup such that V is symplectically irreducible, i.e., there are no propersymplectic Γ -stable subspace in V . Then there are no nonzero brackets (=skew-symmetricbi-derivations) of degree < − on C [ V ] Γ . Further, the space of brackets of degree − isone-dimensional. One can ask why we use M ( nδ, ǫ ) instead of M ( nδ,
0) in the proposition. The reasonis that the moment map for T ∗ R ( nδ, ǫ ) is flat, this can be checked using Theorem 3.6.3.1.10. Symplectic resolutions of V n / Γ n . Here we will study symplectic resolutions of V n / Γ n constructed as non-affine Nakajima quiver varieties for generic stability conditions θ . Let us consider the case n = 1 first. Let ¯ G denote the quotient of G = GL( δ ) modulothe one-dimensional torus T const := { ( x id C δi ) ri =0 , x ∈ C × } . Note that the G -action on R := R ( Q, δ,
0) factors through ¯ G . Analogously to Nakajima’s result explained in 3.1.5,the group ¯ G acts freely on µ − (0) θ − ss if and only if θ · α = 0 for every Dynkin root of Q (these are the roots α ∈ C Q with α = 0). For such θ , we get a conical symplecticresolution M θ ( δ, → M ( δ, ^C / Γ : there arejust no other symplectic resolutions.Let us proceed to the case n >
1. We get a projective morphism M θ ( nδ, ǫ ) →M ( nδ, ǫ ). Theorem 3.7 no longer holds, in fact, µ − (0) has n + 1 irreducible componentsby [GG1, Section 3.2]. Still, M θ ( nδ, ǫ ) → M ( nδ, ǫ ) is a resolution of singularities. Onejust needs to check that the fiber over a generic point in M ( nδ, ǫ ) consists of a singlepoint. A generic closed G -orbit in µ − (0) has a point of the form r ⊕ . . . ⊕ r n , where r , . . . , r n are pair-wise non-isomorphic simple representations of Π ( Q ) of dimension δ .Then one can analyze the structure of the G -action near that orbit using a symplecticslice theorem, see, for example, [CB2, Section 4] or 4.3.3 below. This analysis showsthat there is a unique semistable G -orbit containing Gr in its closure. So we see that M θ ( nδ, ǫ ) → M ( nδ, ǫ ) is a conical symplectic resolution. ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 21
Isomorphic resolutions.
Now let us discuss how many resolutions we get. Thestability condition θ is generic if θ · δ = 0 and θ · v = 0 for v of the form v = α + mδ ,where α is a Dynkin root and | m | < n . So we get resolutions labeled by the open conesin the complement to these hyperplanes in R n . However, some of these resolutions areisomorphic: there is an action of W × Z / Z on Z Q such that, for θ, θ ′ lying in one orbit,the resolutions M θ ( nδ, ǫ ) → M ( nδ, ǫ ) , M θ ′ ( nδ, ǫ ) → M ( nδ, ǫ ) are isomorphic (here W denotes the Weyl group of the Dynkin diagram obtained from Q by removing thevertex 0). This is a special case of a construction due to Namikawa, [Nam], that we aregoing to explain now.Let X → X be an arbitrary symplectic resolution. The variety X has finitely manysymplectic leaves, [K]. Let L , . . . , L k be the leaves of codimension 2. Take formal slices S ∧ , . . . , S ∧ k through L , . . . , L k . The slices are formal neighborhoods of 0 in Kleinian singu-larities S , . . . , S k . From these Kleinian singularities one produces Weyl groups ˜ W , . . . , ˜ W k (of the same types as the singularities) acting on the spaces H ( S k , C ) identified withtheir reflection representations ˜ h i . The fundamental group π ( L i ) acts on the irreduciblecomponents of the exceptional divisor in S i . Hence it also acts on ˜ W i (by diagram au-tomorphisms) and on ˜ h i . Set W i := ˜ W π ( L i ) i , h i := ˜ h L i i so that W i is a crystallographicreflection group and h i is its reflection representation. There is a natural restriction map H ( X ) → h := L i h i . Namikawa proved that this map is surjective. Furthermore, hehas constructed a W := Q i W i -action on H DR ( X ) that makes the map equivariant and istrivial on the kernel.Let us return to our situation. The symplectic leaves in V /
Γ are in one-to-one corre-spondence with conjugacy classes of stabilizers of points in V . The leaf correspondingto Γ ′ ⊂ Γ is the image of V Γ ′ ,reg := { v ∈ V | Γ v = Γ ′ } under the quotient morphism π : V → V /
Γ. The leaf is identified with V Γ ′ ,reg /N Γ (Γ ′ ). So, in the case when V = V n andΓ = Γ n , we get two leaves of codimension 2 (provided Γ = { } , in that case we get justone leaf of codimension 2). One of them, say L , corresponds to Γ ⊂ Γ n (the stabilizerof a point of the form (0 , p , . . . , p n − ), where p , . . . , p n − are pairwise different points of C ). The other, say L , corresponds to S (the stabilizer of ( p , p , p , . . . , p n − )). Thefundamental group actions from the previous paragraph are easily seen to be trivial. So weget W = W, W = Z / Z . Further, H ( X ) = C Q and h = { ( x i ) i ∈ Q | x · δ = 0 } , h = C δ .The group W acts on C δ by ±
1, while h is identified with the Cartan space for W via( x i ) i ∈ Q P ri =1 x i ω ∨ i , where we write ω ∨ i for the fundamental coweights.Let us remark that the W -action can be recovered by using the quiver variety settingas well, see [M] and [L3] for more detail.3.2. Quantum Hamiltonian reduction.
Here we will explain a quantum counterpartof the constructions of the previous section.3.2.1.
Quantum Hamiltonian reduction: algebra level.
Let A be an associative algebra, g a Lie algebra and Φ : g → A be a Lie algebra homomorphism. Then, for a character λ of g , set I λ := A{ x − h λ, x i , x ∈ g } , this is a left ideal in A that is stable under the adjointaction of g . We set A /// λ g := ( A / I λ ) g . This space has a natural associative product givenby ( a + I λ )( b + I λ ) := ab + I λ . With this product, A /// λ g becomes naturally isomorphicto End A ( A / I λ ) opp , an element a + I λ gets mapped to the unique endomorphism sending1 + I λ to a + I λ . We also have a universal variant of quantum Hamiltonian reduction: A /// g := ( A / A Φ([ g , g ])) g . Now suppose A is a filtered quantization of C [ X ], where X is an an affine Poissonvariety (we assume that the bracket on C [ X ] has degree − G acts on X in a Hamiltonian way and the functions µ ∗ ( ξ ) have degree 1 for all ξ ∈ g . By aquantization of the Hamiltonian G -action on C [ X ] we mean a rational G -action on A together with a G -equivariant map Φ : g → A such that(i) the filtration on A is G -stable and the isomorphism gr A ∼ = C [ X ] is G -equivariant,(ii) Φ( ξ ) lies in A and coincides with µ ∗ ( ξ ) modulo A ,(iii) and [Φ( ξ ) , · ] = ξ A , where ξ A is the derivation of A coming from the G -action.Note that gr I λ ⊃ I := C [ X ] µ ∗ ( g ) and so we have a surjective homomorphism C [ X/// G ] ։ gr A /// λ G . We want to get a sufficient condition for gr I λ = I for all λ . Lemma 3.12.
Let ξ , . . . , ξ n be a basis in g . Suppose µ ∗ ( ξ ) , . . . , µ ∗ ( ξ n ) form a regularsequence. Then gr I λ = I for any λ .Proof. The proof is based on the observation that the 1st homology in the Koszul complexassociated to µ ∗ ( ξ ) , . . . , µ ∗ ( ξ n ) is zero. In other words, if f , . . . , f n ∈ C [ X ] are such that P ni =1 f i µ ∗ ( ξ i ), then there are f ij ∈ C [ X ] with f ij = − f ji and f i = P nj =1 f ij µ ∗ ( ξ j ). Detailsof the proof are left to the reader. (cid:3) So if G is reductive and the assumptions of Lemma 3.12 hold, then A /// λ g is a filteredquantization of C [ X/// G ].We can also give the definition of a quantization of a Hamiltonian action in the settingof formal quantizations. One should modify (i)-(iii) as follows. In (i) one requires the G -action to be C [[ ~ ]]-linear and the isomorphism A ~ / ~ A ~ ∼ = C [ X ] has to be G -equivariant. In(ii), one requires that Φ( ξ ) coincides with µ ∗ ( ξ ) modulo ~ . In (iii) one requires ~ [Φ( ξ ) , · ] = ξ A ~ . We then can consider reductions of the form A ~ /// λ ( ~ ) G , where λ ( ~ ) is an elementin ( g ∗ G )[[ ~ ]]. If G is reductive, and the elements µ ∗ ( ξ i ) − h λ (0) , ξ i i , i = 1 , . . . , n , form aregular sequence in C [ X ], then A ~ /// λ ( ~ ) G is a formal quantization of C [ X/// λ (0) G ].3.2.2. Quantum Hamiltonian reduction: sheaf level.
Let X be a smooth affine symplecticalgebraic variety equipped with a Hamiltonian action of G and let θ be a character of G . Assume that, for a basis ξ , . . . , ξ n of g , the elements µ ∗ ( ξ ) , . . . , µ ∗ ( ξ n ) form a regularsequence at points of µ − (0) θ − ss . Let D ~ be a formal quantization of O X . Our goal is todefine a (formal) quantization D ~ /// θλ ( ~ ) G of X/// θ G (so λ (0) = 0).Recall that it is enough to define the following data:(1) For an open affine covering X/// θλ G := S i Y i , the algebras of sections Γ( Y i , D ~ /// θλ ( ~ ) G )that quantize Y i ,(2) and identifications Γ( Y i , D ~ /// θλ ( ~ ) G ) Y i ∩ Y j ∼ = Γ( Y j , D ~ /// θλ ( ~ ) G ) Y i ∩ Y j satisfying cocy-cle conditions.Recall that we can choose an open covering by setting Y i := X f i /// G , where polyno-mials f i ∈ C [ X ] G,n i θ are such that X θ − ss = S i X f i . Then we set Γ( Y i , D ~ /// θλ ( ~ ) G ) :=Γ( X f i , D ~ ) /// λ ( ~ ) G . The sections of the corresponding sheaf on Y i ∩ Y j are easily seen tobe Γ( X f i ∩ X f j , D ~ ) /// λ ( ~ ) G and this yields the gluing maps.Now let us discuss the period map mentioned in 2.2.6. Suppose that the G -action on µ − (0) θ − ss is free so that X/// θ G is smooth and symplectic. In this case we have a periodmap associated to the quantization of D ~ /// θλ ( ~ ) G . Assume, for simplicity, that λ ( ~ ) := λ ~ for λ ∈ g ∗ G – this is the most interesting case, for example, it is the only case thatappears when we work with the filtered setting. Further, assume that D ~ is canonical, ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 23 i.e., has period 0. Recall that this means the existence of a parity anti-automorphism,let us denote it by ̺ . Finally, assume that Φ is symmeterized , meaning that ̺ ◦ Φ = Φ,this can be achieved by modifying Φ. Then the period of D ~ /// θλ ~ G equals to the Chernclass associated to λ (if λ integrates to a character of G , then it defines the line bundleon X/// θ G , in general, we extend the notion of a Chern class by linearity). This wasessentially checked in [L3, Sections 3.2,5.4].3.2.3. Algebra vs sheaf level.
We need to relate the sheaf D ~ /// θλ ( ~ ) G to the algebra D ~ /// λ ( ~ ) G .What one could expect is that the algebra is the global sections (or even better, the de-rived global sections) of the sheaf. Let us provide some sufficient conditions for this tohold. Proposition 3.13.
Assume, for simplicity, that λ (0) = 0 . Further, suppose that thefollowing holds. (1) The moment map µ is flat. (2) X/// G is a normal reduced scheme. (3) X/// θ G → X/// G is a resolution of singularities.Then R Γ( D ~ /// θλ ( ~ ) G ) ∼ = D ~ /// λ ( ~ ) G .Proof. By (3) and the Grauert-Riemmenschneider theorem, the higher cohomology of O X/// θ G vanish. This implies that the higher cohomology of D ~ /// θλ ( ~ ) G vanish. Moreover,Γ( D ~ /// θλ ( ~ ) G ) / ( ~ ) ∼ = C [ X/// θ G ]. By (2) and (3), the right hand side is naturally identi-fied with C [ X/// G ]. By (1), ( D /// λ ( ~ ) G ) / ( ~ ) = C [ X/// G ]. Besides, we have a naturalhomomorphism D /// λ ( ~ ) G → Γ( D ~ /// θλ ( ~ ) G ). Modulo ~ , this homomorphism is the iden-tity. The source algebra is complete and separated in the ~ -adic topology, and the targetalgebra is flat over C [[ ~ ]]. It follows that the homomorphism D /// λ ( ~ ) G → Γ( D ~ /// θλ ( ~ ) G )is an isomorphism. (cid:3) Isomorphism theorem.
Recall a C × -equivariant isomorphism C n / Γ n ∼ = M θ ( nδ, ǫ )of Poisson varieties. The left hand side admits a family of quantizations, eH ,c e , and sodoes the right hand side, there quantizations are the quantum Hamiltonian reductions D ( R ) /// λ G , where we use the symmetrized quantum comoment map Φ( ξ ) = ( ξ R + ξ R ∗ ).In fact, these two families are the same. Let us state a precise result to be proved inSection 4.3 (using Procesi bundles). We write c for1 | Γ | X γ ∈ Γ \{ } c ( γ ) γ , where c ( γ ) := c i for γ ∈ S i (recall that S is the conjugacy class of a reflection in S n ⊂ Γ n and S , . . . , S r are conjugacy classes of elements of Γ ⊂ Γ n ). Theorem 3.14.
We have a filtered algebra isomorphism eH ,c e ∼ = D ( R ) /// λ G that is theidentity on the level of associated graded algebras (we consider the filtration on D ( R ) /// λ G induced from the Bernstein filtration on D ( R ) , where deg R = deg R ∗ = 1 ). Here λ := P ri =0 λ i tr i is recovered from c by the following formulas: (5) λ i := tr N i c , i = 1 , . . . , r, λ := tr N c −
12 ( c + 1) , where in the n = 1 case one needs to put c = 1 . For n = 1, this theorem was proved by Holland in [Ho]. The case of Γ = { } washandled in [EG, GG1] ([EG] proved a weaker statement and then in [GG1] the proofwas completed). The case of cyclic Γ was done in [O, G2]. In [EGGO] the proof wascompleted: they considered the case when Q is a bi-partive graph. Let us note thatin these papers formulas look different from (5): they use the quantum comoment mapΦ( ξ ) = ξ R . A uniform and more conceptual proof was given in [L3] using Procesi bundles.Theorem 3.14 is of crucial importance for the representation theory of the algebras H ,c . It turns out that the representation theory of the algebras D ( R ) /// λ G (actually,of sheaves D R /// θλ G ) is easier to study. The main ingredient here is the geometry ofthe quiver varieties M θ ( v, ǫ ). Using this, in [BL], the author and Bezrukavnikov haveproved a conjecture of Etingof, [E], on the number of the finite dimensional irreduciblerepresentations of H ,c .3.2.5. Automorphisms.
Here we are going to explain a quantum version of Namikawa’sconstruction recalled in 3.1.11. In the complete generality this construction was given in[BPW, Section 3.3].Let X be a conical symplectic resolution of X . Let ˜ X be its universal deformationover H DR ( X ) and let ˜ D ~ be the canonical quantization of ˜ X . Let ˜ A ~ denote the C × -finite part of Γ( ˜ D ~ ). Then Namikawa’s Weyl group W acts on ˜ A ~ by graded C [ ~ ]-algebraautomorphisms preserving H DR ( X ) ∗ . Moreover, the action on H DR ( X ) ∗ is as explainedin 3.1.11.3.3. Quantum Hamiltonian reduction for Frobenius constant quantizations.
Inthis section, we will consider the situation in characteristic p . Our main result is that aquantum GIT Hamitlonian reduction under a free Hamiltonian action is again Frobeniusconstant.3.3.1. GIT in characteristic p . The definition of a reductive group (one with trivial unipo-tent radical) makes sense in all characteristics. A crucial difficulty of dealing with reduc-tive groups in positive characteristic is that their rational representations are no longercompletely reducible, in general. The groups for which the complete reducibility holdsare called linearly reductive . Tori are still linearly reductive independently of the charac-teristic. We need to deal with GIT for reductive groups (such as products of GL’s) andso we need to explain how this works in positive characteristic.It turns out that reductive groups satisfy a weaker condition than being linearly re-ductive, they are geometrically reductive . This was conjectured by Mumford and provedby Haboush, [Hab]. To state the condition of being geometrically reductive, let us re-formulate the linear reductivity first: a group G is called linearly reductive, if, for anylinear G -action on a vector space V and any fixed point v ∈ V , there is f ∈ ( V ∗ ) G with f ( v ) = 0. A group G is called geometrically reductive if instead of f ∈ ( V ∗ ) G , one canfind f ∈ S r ( V ∗ ) G (for some r >
0) with f ( v ) = 0.This condition is enough for many applications. For example, if X is an affine algebraicvariety acted on by a reductive (and hence geometrically reductive) group G , then F [ X ] G is finitely generated. So we can consider the quotient morphism X → X//G . Thismorphism is surjective and separates the closed orbits. Moreover, if X ′ ⊂ X is a G -stablesubvariety, then the natural morphism X ′ //G → X//G is injective with closed image.The claim about the properties of the quotient morphism in the previous paragraphcan be deduced from the following lemma, [MFK, Lemma A.1.2].
ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 25
Lemma 3.15.
Let G be a geometrically reductive group acting on a finitely generatedcommutative F -algebra R rationally and by algebra automorphisms. Let I ⊂ R be a G -stable ideal and f ∈ ( R/I ) G . Then there is n such that f p n lies in the image of R G in ( R/I ) G . In characteristic p , we can still speak about unstable and semistable points for reductivegroup actions on vector spaces, about GIT quotients, etc.Another very useful and powerful result of Invariant theory in characteristic 0 is Luna’s´etale slice theorem, see, e.g., [PV, 6.3]. There is a version of this theorem in characteristic p due to Bardsley and Richardson, see [BR]. We will need a consequence of this theoremdealing with free actions.Recall that, in characteristic 0, an action of an algebraic group G on a variety X iscalled free if the stabilizers of all points are trivial. In characteristic p one should givethis definition more carefully: the stabilizer may be a nontrivial finite group scheme witha single point. An example is provided by the left action of G on G (1) , we will discussa closely related question in the next subsection. We have the following three equivalentdefinitions of a free action. • For every x ∈ X , the stabilizer G x equals { } as a group scheme. • For every x ∈ X , the orbit map G → X corresponding to x is an isomorphism ofalgebraic varieties. • For every x ∈ X , G x coincides with { } as a set and the stabilizer of x in g istrivial.The following is a weak version of the slice theorem that we need. Lemma 3.16.
Let X be a smooth affine variety equipped with a free action of a reductivealgebraic group G . Then the quotient morphism X → X/G is a principal G -bundle in´etale topology. Quiver varieties.
Let us now discuss Nakajima quiver varieties in characteristic p ≫
0. We have a finite localization R of Z with the following properties:(1) R together with the G -action and µ are defined over R .(2) µ − (0) θ − ss and the G -bundle µ − (0) θ − ss → µ − (0) θ − ss /G are defined over R .For an R -algebra R ′ , let R R ′ , G R ′ , µ R ′ etc. denote the R ′ -forms of the correspondingobjects. Let us write X R for an R -form of µ − (0) θ − ss /G . After a finite localization of R , wecan achieve that X R is a symplectic scheme over Spec( R ) with C ⊗ R Γ( X R , O X R ) ∼ −→ C [ X C ]and H i ( X R , O X R ) = 0 for i > R ′ , we can take F := F p when p is large enough. So we get a symplectic F -variety M θ ( nδ, F that is naturally identified with T ∗ R F /// θ G F as well as with Spec( F ) × Spec( R ) X R . For p ≫
0, we get F [ X F ] = F ⊗ R Γ( X R , O X R ) and H i ( X F , O X F ) = 0.We can take a finite algebraic extension of R and assume that the Γ n -module C n isdefined over R . Now we claim that (again for p ≫ M θ ( nδ, F is a symplectic resolutionof F n / Γ n . This follows from the claim that both Γ( X R , O X R ) , R [ x, y ] Γ n are R -forms of C [ x, y ] Γ n so they coincide after some finite localization of R .3.3.3. Quantum Hamiltonian reduction.
Now suppose that R is a symplectic vector spaceover F , G is a reductive group over F acting on R and θ is a character of G . We supposethat G acts freely on µ − (0) θ − ss . We are going to define a Frobenius constant quantiza-tion D R /// θλ G of T ∗ R/// θ G , where λ ∈ Hom( G, F × ) ⊗ Z F p ֒ → g ∗ G . The associated filtered quantization of T ∗ R/// θ G will be a quantization obtained by quantum Hamiltonian re-duction, see 3.2.2. We note that for λ Hom( G, F × ) we do not get a Frobenius constant quantization of T ∗ R/// θ G .Consider the Frobenius twist G (1) . It is a group and the morphism Fr : G → G (1) is agroup epimorphism. Its kernel (a.k.a. the Frobenius kernel) G is a finite group schemewhose Lie algebra coincides with g .The action of G on R induces an action of G (1) on R (1) . The G (1) -action on T ∗ R (1) isHamiltonian with moment map µ (1) : T ∗ R (1) → g (1) ∗ induced by µ . Consider the sheaf D R /// θλ G (a subquotient of D R ) on T ∗ R (1) θ − ss . One can show, see [BFG, Section 3.6],that it is supported on (cid:0) µ (1) (cid:1) − (0), here we use that λ ∈ Hom( G, F × ) ⊗ Z F p . Moreover,it is a G (1) -equivariant Azumaya algebra on (cid:0) µ (1) (cid:1) − (0). The descent of this algebra to( T ∗ R/// θ G ) (1) = T ∗ R (1) /// θ G (1) is an Azumaya algebra with a filtration induced fromthat on D R . We have a natural homomorphism gr( D R /// θλ G ) → Fr ∗ O T ∗ R/// θ G . To showthat it is an isomorphism one uses that the action of G is free (that yields the requiredcohomology vanishing). This isomorphism implies gr( D R /// θλ G ) ∼ −→ Fr ∗ O T ∗ R/// θ G . So D R /// θλ G is indeed a Frobenius constant quantization.Note that if λ Hom( G, F × ) ⊗ Z F p , then D R /// θλ G is supported on a nonzero fiber of µ (1) , see [BFG, Section 3.6] for details, and so D R /// λ G is no longer a Frobenius constantquantization of X/// θ G .4. Existence and classification of Procesi bundles
In this section we construct and classify Procesi bundles on X = M θ ( nδ, ǫ ) and alsoprove Theorem 3.14.In Section 4.1 we construct a Procesi bundle on X . The case n = 1 is relatively easy, itwas done in [KaVa]. For n >
1, we follow [BK2]. A key step here is to construct a specialFrobenius constant quantization of X F , where F is an algebraically closed field of largeenough positive characteristic. This quantization provides a suitable version of derivedMcKay equivalence and using this equivalence we can produce a Procesi bundle over F .Then we lift it to characteristic 0.In Section 4.2 we prove that Symplectic reflection algebras satisfy PBW property and,in some sense, the family of SRA H t,c is universal with this property. The proof is basedon computing relevant graded components in the Hochschild cohomology of SV D ( R ) /// λ G is isomorphic to some eH ,c e . Then the task is to show that thecorrespondence between the parameters λ and the parameters c is as in Theorem 3.14.We first do this for n = 1. Then we reduce the case of n > n = 1 by studyingcompletions of the algebras involved. This allows to show that the map between theparameters is conjugate to that in Theorem 3.14 up to a conjugation under an action ofthe group W × Z / Z , where W is the Weyl group of the finite part of the quiver Q . Butfrom 3.2.5 we know that this action lifts to an action on the universal reduction D ( R ) ///G by automorphisms. This completes the proof of Theorem 3.14.Then, in Section 4.4, we classify Procesi bundles. Namely, we show that, when n > | W | different Procesi bundles on X . For this, we use Theorem 3.14 to producethis number of bundles. And then we use techniques used in the proof to show that thenumber cannot exceed 2 | W | . Further, we show that each X carries a distinguished Procesibundle. ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 27
Construction of Procesi bundles.
Baby case: n = 1 . In this case it is easy to construct a vector bundle of requiredrank on X . Namely, for i = 0 , . . . , r , let U i be the G -module C δ i and let U i be thecorresponding vector bundle on X . We set P := L ri =0 U δ i i . It follows from results ofKapranov and Vasserot, [KaVa], that this bundle satisfies the axioms of a Procesi bundle.4.1.2. Procesi bundles and derived McKay equivalence.
Before we proceed to constructingProcesi bundles in general, let us explain their connection to derived Mckay equivalences,i.e., equivalences D b (Coh X ) ∼ −→ D b ( K [ V n ] n ), here K stands for the base field. Proposition 4.1.
Let P be a Procesi bundle on X . Then the functor R Hom O X ( P , • ) isa derived equivalence D b (Coh X ) → D b ( K [ V n ] . The proof is based on the following more general result (Calabi-Yau trick) of (in thisform) Bezrukavnikov and Kaledin.
Proposition 4.2. [BK2, Proposition 2.2]
Let X be a smooth variety, projective over anaffine variety, with trivial canonical class. Furthermore, let A be an Azumaya algebraover X such that Γ( A ) has finite homological dimension and H i ( X, A ) = 0 for i > .Then the functor R Γ : D b (Coh( X, A )) → D b (Γ( A ) -mod) is an equivalence. Proposition 4.1 follows from Proposition 4.2 with A = E nd ( P ).Now suppose that we have a derived equivalence ι : D b (Coh( X )) ∼ −→ D b ( K [ V ] n -mod).Assume P ′ := ι − ( K [ V ] n ) is a vector bundle. Then End O X ( P ′ ) = K [ V ] i ( P ′ , P ′ ) = 0 for i >
0. So P ′ is, basically, a Procesi bundle (it also needs to be K × -equivariant, but we will see below that this always can be achieved). In fact, this isroughly, how the construction of a Procesi bundle will work, although it is more involvedand technical.4.1.3. Quantization of X . Here and in 4.1.4 everything is going to be over an algebraicallyclosed field F of characteristic p ≫
0. The first step in the construction of a Procesi bundleis to produce a Frobenius constant quantization of X with special properties. Proposition 4.3.
There is a Frobenius constant quantization D of X such that Γ( D ) = A ( V n ) Γ n (an isomorphism of filtered algebras over F [ X (1) ] = F [ V (1) n ] Γ n ). Note that this proposition can be thought as a special case of the characteristic p versionof Theorem 3.14. Here Γ( D ) is an analog of D ( R ) /// λ G (indeed, the latter is the algebra ofglobal sections of some filtered quantization of X C , see Proposition 3.13), while A ( V n ) Γ n is the characteristic p analog of eH , e .In fact, the following is true. Lemma 4.4.
Theorem 3.14 (for c = 0 ) implies Proposition 4.3.Proof. First, let us see that we get an isomorphism Γ( D ) ∼ = A ( V n ) Γ n of filtered algebrasthat is the identity on the associated graded algebra. Set D := D ( R ) /// θλ G , where λ isthe parameter corresponding to c = 0.The algebra A ( V n, C ) Γ n is finitely generated and so an isomorphism in Theorem 3.14 isdefined over some finitely generated subring R of C . We can enlarge R and assume thatwe are in the situation described in 3.3.2. We can form filtered quantizations D ′ C , D ′ R , D ′ F of X C , X R , X . Both D ′ C , D ′ F are obtained as suitable completions of base changes of D ′ R (completions are necessary because of our condition on the filtration in the definition ofa filtered quantization, see 2.2.4). In particular, D ( R C ) /// λ G C = (Γ( D ) =) C ⊗ R Γ( D ′ R ),while Γ( D ) = (Γ( D ′ F ) =) F ⊗ R Γ( D ′ R ).So we can reduce an isomorphism from Theorem 3.14 (for c = 0) mod p ≫ D ) ∼ = A ( V n ) Γ n . What remains to show is that this isomorphism is F [ V (1) n ] Γ n -linear. The first step here is to show that F [ V (1) n ] Γ n is the center of A ( V n ) Γ n . Itis enough to check that F [ V (1) n ] Γ n coincides with the center of the Poisson algebra F [ V n ] Γ n .Here we just note that the Poisson center of F [ V n ] Γ n is finite and birational over F [ V (1) n ] Γ n and use that the latter algebra is normal. So the isomorphism Γ( D ) ∼ = A ( V n ) Γ n inducesan automorphism of F [ V (1) n ] Γ n . This isomorphism preserves the filtration and is trivial onthe level of associated graded algebras.The second step is to show that the algebra F [ V (1) n ] Γ n has no nontrivial automorphisms ϕ with such properties. Let us define a derivation ψ of F [ V (1) n ] Γ n that should be thoughtas ln ϕ . The degrees of generators of F [ V (1) n ] Γ n are bounded from above for all p ≫ ψ we note that ϕ − ψ := ln ϕ makes senseas long as p is sufficiently large. The derivation ψ lifts to F [ V (1) n ] because the quotientmorphism V (1) n → V (1) n / Γ n is ramified in codimension bigger than 1. Since it decreasesdegrees, we see that ψ has the form ∂ v for some v ∈ F n (1) . But, if Γ = { } , the vector v cannot be Γ n -equivariant and so ∂ v does not preserve F [ V (1) n ] Γ n . When Γ = { } , thereis a Γ n -invariant vector. However, in this case we can modify our construction: considerthe reflection representation h of S n instead of the permutation representation C n . Weneed to replace R with sl n ⊕ C n . Theorem 3.14 gets modified accordingly. (cid:3) However, the easiest way to prove Theorem 3.14 is by using Procesi bundles (at least fornon-cyclic Γ or general c , the case c = 0 may be easier). So we need some roundaboutway to construct D . In [BK2] the question of existence of D was reduced to n = 1.More precisely, let V sr denote the set of all v ∈ V n such that dim V Γ v >
2. Let uswrite X := ρ − ( V srn / Γ n ). This is an open subset in X with codim X X \ X >
1. First,Bezrukavnikov and Kaledin produce a Frobenius constant quantization D of X withΓ( D ) = A ( V n ) Γ n . This requires the existence of such a quantization in the case when n = 1. The latter case can be handled using Theorem 3.14 proved in this case byHolland (that can be alternatively proved using the existence of a Procesi bundle in thecase n = 1). When D is constructed, Bezrukavnikov and Kaledin use the inequalitycodim X \ X > D uniquely extends to a Frobenius constant quantization D of X , automatically with Γ( D ) = A ( V n ) Γ n .4.1.4. Construction of a Procesi bundle: characteristic p . Let D be as in the previoussubsection. We will produce a Procesi bundle on X (1) starting from D . Since X (1) ∼ = X (an isomorphism of F -varieties), this will automatically establish a Procesi bundle on X .The isomorphism X (1) ∼ = X follows from the observation that X is defined over F p andFr is an isomorphism of F fixing F p .By Proposition 4.2, we have a derived equivalence D b (Coh( X (1) , D )) ∼ −→ D b ( A ( V n ) Γ n -mod).Also we have an abelian equivalence A ( V n ) Γ n -mod ∼ −→ A ( V n ) n -mod = A ( V n ) -mod Γ n . ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 29
Composing the two equivalences, we get(6) D b (Coh( X, D )) ∼ −→ D b ( A ( V n ) -mod Γ n ) , while what we need is a derived McKay equivalence(7) D b (Coh X (1) ) ∼ −→ D b ( F [ V (1) n ] -mod Γ n ) . Recall that D is an Azumaya algebra on X , while A ( V n ) is a Γ n -equivariant Azumayaalgebra on V (1) n . If we had a splitting and a Γ n -equivariant splitting, respectively, we wouldget (7) from (6). However, this is obviously not the case: A ( V n ) admits no splitting atall.This can be fixed by passing to completions at 0. Namely, let X (1) ∧ denote the formalneighborhood of ( ρ (1) ) − (0) in X (1) . It was checked in [BK2, Section 6.3] that the restric-tion of D to X (1) ∧ splits. Also it was checked that the restriction of A ( V n ) to the formalneighborhood of 0 in F n (1) ∧ admits a Γ n -equivariant splitting. So, we get an equivalence ι : D b (Coh( X (1) ∧ )) ∼ −→ D b ( F [ V (1) n ] ∧ n -mod)that makes the following diagram commutative (all arrows are equivalences of triangulatedcategories and all arrows but R Γ come from abelian equivalences): D b (Coh( X (1) ∧ )) D b (Coh( X (1) ∧ , D )) D b ( A ( V n ) ∧ Γ n -mod) D b ( A ( V n ) ∧ n -mod) D b ( F [ V (1) n ] ∧ n -mod) ✻ B ∗ ⊗ • ✲ R Γ ✲ ✻✲ Here B denotes a splitting bundle for the restriction of D to X (1) ∧ .Set P ′ := ι − ( F [ V (1) n ] ∧ n ). We claim that P ′ is a vector bundle on X (1) ∧ . Indeed,the image of F [ V (1) n ] ∧ n in A ( V n ) ∧ Γ n -mod is a projective generator and so is a directsummand in the sum of several copies of A ( V n ) ∧ Γ n . But R Γ − ( A ( V n ) ∧ Γ n ) = B ∗ . So P ′ is a direct summand in a vector bundle (the sum of several copies of B ∗ ) and hence is avector bundle itself.So we get a vector bundle P ′ on X (1) ∧ that satisfies End( P ′ ) ∼ = F [ V (1) n ] ∧ n , Ext i ( P ′ , P ′ ) =0 for i >
0. The latter vanishing implies that P ′ is equivariant with respect to the F × -action on X (1) ∧ , see [V]. From here it follows that P ′ can be extended to X (1) (this is be-cause F × contracts X (1) to the zero fiber, see [BK2, Section 2.3]). Moreover, we can modifythe equivariant structure on P ′ and achieve that the isomorphism End( P ′ ) ∼ = F [ V (1) n ] ∧ n is F × -equivariant, see [L4, Section 3.1]. It follows that P is a Procesi bundle.4.1.5. Construction of a Procesi bundle: lifting to characteristic . Recall the R -scheme X R from 3.3.2. We may assume R is regular. Taking an algebraic extension of R , weget a maximal ideal m such that there is a Procesi bundle P F on X F , where F is analgebraic closure of F := R / m . We may assume that P F is defined over F , let P F bethe corresponding form. Let R ∧ be the m -adic completion of R . Since Ext i ( P F , P F ) = 0for i = 1 ,
2, we see that P F uniquely deforms to a G m -equivariant vector bundle on theformal neighborhood of X F in X R ∧ (see [BK2, Section 2.3]).Let us show that the G m -finite part of End( P R ∧ ) is R ∧ [ V n ] n . Consider the formalneighborhood Z of X reg F in X reg R ∧ . Note that Ext ( P F | X reg , P F | X reg ) = 0, see, for example, [BL, Appendix]. So the restriction of P R ∧ to Z coincides with η ∗ O ( R ∧ n ) reg , where η denotesthe quotient morphism R ∧ n → R ∧ n / Γ n . This implies the claim about endomorphisms.Since P R ∧ is G m -equivariant and the G m -action is contracting, it extends from a formalneighborhood of X F in X R to X R ∧ . So we get a Procesi bundle on X K , where K =Frac( R ∧ ). But being a finite extension of the p -adic field, K embeds into C and so weget a Procesi bundle on X .4.2. Symplectic reflection algebras.
Flatness and universality.
Let V be a symplectic vector space with form Ω andΓ ⊂ Sp( V ) be a finite group of symplectomorphisms. We write S for the set of symplecticreflections in Γ, it is a union of conjugacy classes: S = S ⊔ S ⊔ . . . ⊔ S r . We pickindependent variables t, c , . . . , c r .Recall the universal Symplectic reflection algebra H , the quotient of T ( V ) t, c , . . . , c r ]by the relations (3). Let us write c univ for the vector space with basis t, c , . . . , c r so that H is a graded S ( c univ )-algebra. Theorem 4.5.
The algebra H is a free graded S ( c univ ) -module. Moreover, assume that Γ is symplectically irreducible. Then H is universal with this property in the followingsense. Let c ′ be a vector space and H ′ be a graded S ( c ′ ) -algebra (with deg c ′ = 2 ) that isa free graded S ( c ′ ) -module and H ′ / ( c ′ ) = S ( V ) . Then there is a unique linear map ν : c univ → c ′ and unique isomorphism S ( c ′ ) ⊗ S ( c univ ) H ∼ −→ H ′ of graded S ( c ′ ) -algebras thatinduces the identity isomorphism of S ( V ) n . When Γ = { } , then the action of the group Γ n on V n = C n is symplectically irre-ducible. When Γ = { } , the module C n over Γ n is not symplectically irreducible, so wereplace C n with V n = h ⊕ h ∗ , where h is the reflection representation of S n . Note thatwe did the same in 4.1.3.4.2.2. Hochschild cohomology.
Before we prove this theorem we will need to get some in-formation about Hochshild cohomology of S ( V ) S ( V ) A be a graded algebra. We want to describe graded deformations of A . TheHochschild cohomology group HH i ( A ) inherits the grading from A , let HH i ( A ) j denotethe j th graded component. The general deformation theory implies the following. Lemma 4.6.
Assume that dim HH ( A ) − < ∞ and HH i ( A ) j = 0 for i + j < . Set P univ := ( HH ( A ) − ) ∗ . Then there is a free graded S ( P univ ) -algebra A univ (with deg P univ =2 ) such that A univ / ( P univ ) = A that is a universal graded deformation of A in the samesense as in Theorem 4.5. What we are going to do is to compute the relevant graded components of HH • ( SV P univ is more subtle.First, we use the fact that HH i ( A, M ) = Ext iA ⊗ A opp ( A, M ) (where M is an A -bimodule)to see that(8) HH i ( S ( V ) , S ( V ) i ( S ( V ) , S ( V ) Γ . We have a Γ-action on HH i ( S ( V ) , S ( V ) S ( V )-bimodules S ( V ) , S ( V ) S ( V ) L γ ∈ Γ S ( V ) γ of S ( V )-bimodules, where S ( V ) γ isidentified with S ( V ) as a left S ( V )-module and the right action is given by f · a := f γ ( a ). ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 31
Let us compute HH i ( S ( V ) , S ( V ) γ ) in degrees we are interested in: j < − i and also j = − i = 2. We have γ = diag( γ , . . . , γ n ), where we view γ i as elements of cyclicgroups acting on C . Then we have an isomorphism of bigraded spaces(9) M i HH i ( S ( V ) , S ( V ) γ ) ∼ = n O ℓ =1 M i HH i ( C [ x ] , C [ x ] γ ℓ ) . For an arbitrary γ ℓ , we have HH i ( C [ x ] , C [ x ] γ ℓ ) = 0 when i >
1. When γ ℓ = 1, we haveHH ( C [ x ] , C [ x ]) = C [ x ] and HH ( C [ x ] , C [ x ]) = C [ x ] { } , where { } indicates the gradingshift by 1 so that HH ( C [ x ] , C [ x ]) is a free module generated in degree −
1. When γ ℓ = 1,then HH ( C [ x ] , C [ x ] γ ℓ ) = 0 and HH ( C [ x ] , C [ x ]) = C { } .This computation easily implies that HH i ( S ( V ) , S ( V ) j = 0 when i + j <
0. Nowlet explain how to compute (cid:0) HH ( S ( V ) , S ( V ) − (cid:1) Γ . If HH ( S ( V ) , S ( V ) γ ) − = 0, theneither γ = 1 or γ is a symplectic reflection. When γ = 1, then HH ( S ( V ) , S ( V ) γ ) − = V V . When γ is a symplectic reflection, then HH ( S ( V ) , S ( V ) γ ) − = C . An element γ ∈ Γ maps S ( V ) γ to S ( V )( γ γγ − ). The action of Γ on HH ( S ( V ) , S ( V ) γ ) − = V V is a natural one. When γ is a symplectic reflection, then the action of Z Γ ( γ )on HH ( S ( V ) , S ( V ) γ ) − = C is trivial. From here we deduce thatdim HH ( S ( V ) , S ( V ) − = r + 2 , as claimed.4.2.3. Proof of Theorem 4.5.
Let us write H univ for the universal deformation, we needto prove that H univ ∼ −→ H .First of all, note that degree 0 and 1 components of H univ are the same as in S ( V ) , V ֒ → H univ . It is easy to see that c univ , V, Γ generate H univ . This gives rise to an epimorphism S ( c univ ) ⊗ T ( V ) ։ H univ . Further, for u, v ∈ V ⊂ H univ , we have [ u, v ] ∈ ( c univ ). The degree 2 of ( c univ ) is c univ ⊗ C Γ. So we get[ u, v ] = κ ( u, v ) in H univ , where κ is a map V V → c univ ⊗ C Γ. A computation done in[EG, Section 2] shows that, since H univ is free over S ( c univ ), we get κ = t Ω + r X i =0 c i X s ∈ S i Ω s ( u, v ) s. This completes the proof of Theorem 4.5.4.3.
Proof of the isomorphism theorem.
We will prove an isomorphism of e H e andthe universal Hamiltonian reduction A := A ~ ( T ∗ R ) ///G , where A ~ ( T ∗ R ) is the Reesalgebra of D ( R ) (with modified grading so that deg T ∗ R = 1 , deg ~ = 2). Here we take R := R ( Q, nδ, ǫ ) for n > R := R ( Q, δ,
0) for n = 1. In the case when n >
1, we take G := GL( nδ ). For n = 1, for G , we take the quotient of GL( δ ) by the one-dimensionalcentral subgroup of constant elements.Both e H e, A are graded algebras. The algebra e H e is over S ( c univ ) with deg c univ = 2.The algebra A is over S ( c red ), where c red := g / [ g , g ] ⊕ C~ . We will prove that there is agraded algebra isomorphism e H e ∼ −→ A that maps c univ to c red and induces the identityautomorphism e H e/ ( c univ ) = C [ V n ] Γ n = A / ( c red ). Further, we will explain why thecorresponding isomorphism c univ ∼ = c red maps ~ to t and gives (5) on the hyperplanes t = 1 and ~ = 1. In other words, the isomorphism ν : c univ → c red is the inverse of thefollowing map ~ t, ǫ i | Γ | tr N i ˜ c , i = 0 , ǫ | Γ | tr N ˜ c −
12 ( c + t ) , ( n > ~ t, ǫ i | Γ | tr N i ˜ c , i = 0 , ǫ | Γ | tr N ˜ c − t, ( n = 1)(10)Here the notation is as follows. We write ˜ c := t + P ri =1 c i P γ ∈ S i γ and ǫ i ∈ g / [ g , g ] isspecified by tr i ǫ j := δ ij .4.3.1. Application of a Procesi bundle.
An isomorphism e H e ∼ = A is produced as follows.The algebra e H e does not have good universality properties (although it is expectedto be semi-universal), it is H that does. We will produce a graded S ( c red )-algebra ˜ A deforming C [ V n ] n with e ˜ A e = A . This will give rise to a linear map ν : c univ → c red and to an isomorphism S ( c red ) ⊗ S ( c univ ) H ∼ = ˜ A and hence also to an isomorphism S ( c red ) ⊗ S ( c univ ) e H e ∼ = A . The algebra ˜ A will be constructed from a Procesi bundle P on X = M θ ( nδ, ǫ ).First, let us produce a sheaf version of A . Consider the variety e X := T ∗ R/// θ G , thisis a deformation of X over g ∗ G . Then we consider its deformation quantization obtainedby Hamiltonian reduction, the sheaf e D ~ := A ~ ( T ∗ R ) ∧ ~ /// θ G . The algebra A coincideswith the C × -finite part of Γ( e D ~ ). Now let us take a Procesi bundle P on X . SinceExt i ( P , P ) = 0, the bundle P deforms to a unique C × -equivariant vector bundle on theformal neighborhood of X in e X . But the C × -action contracts e X to X . So P extends to aunique C × -equivariant bundle e P on e X . The extension e P again satisfies the Ext-vanishingconditions and so further extends to a unique C × -equivariant right e D ~ -module e P ~ .Consider the endomorphism algebra End e D opp ~ ( e P ~ ). Modulo ( c red ), this algebra coincideswith End O X ( P ) = C [ V n ] n . Let ˜ A be the C × -finite part of End e D opp ~ ( e P ~ ). It is theendomorphism algebra of the right e D ~ ,fin -module e P ~ ,fin . The algebra ˜ A is a graded S ( c red )-algebra with ˜ A / ( c red ) = C [ V n ] n , where c red lives in degree 2. We conclude thatthere is a unique map ν P : c univ → c red with ˜ A ∼ = S ( c red ) ⊗ S ( c univ ) H . Then, automatically,we have(11) A (= e ˜ A e ) ∼ = S ( c red ) ⊗ S ( c univ ) e H e. We will study the linear maps ν : c univ → c red such that (11) holds. We will see that(a) any such ν is an isomorphism,(b) that there are | W | options for ν when n = 1 and 2 | W | options else,(c) and that one can choose ν as in (10).(c) will complete the proof of Theorem 3.14, while (b) will be used to classify the Procesibundles.First of all, let us point out that ν ( t ) = ~ . Indeed, the Poisson bracket on C [ M ( nδ, ǫ )]induced by the deformation A equals ~ {· , ·} , where {· , ·} is the standard bracket givenby the Hamiltonian reduction (more precisely, if we specialize to ( ~ ′ , λ ) ∈ C ⊕ g ∗ G , thenthe bracket induced by the corresponding filtered deformation is ~ ′ {· , ·} ). Similarly, thebracket on C [ V n ] Γ n induced by e H e coincides with t {· , ·} , see Example 2.4. Since theisomorphism M ( nδ, ǫ ) ∼ = V n / Γ n is Poisson, the equality ν ( t ) = ~ follows. ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 33
Case n = 1 . We start by proving (a)-(c) for n = 1.Let us prove (c). First of all, recall that X can be constructed as the moduli space ofthe C [ x, y ] -modules isomorphic to C Γ as Γ -modules that admit a cyclic vector. Theuniversal bundle on X is a Procesi bundle. Moreover, from [CBH, Section 8], it followsthat e X is the moduli space of the H / ( t )-modules isomorphic to C Γ and admitting acyclic vector. The corresponding isomorphism c red / C~ ∼ = c univ / C t is induced from ν .To show that ν then is given by (10) we consider the loci of parameters λ and c wherethe homological dimensions of A ,λ := A ( T ∗ R ) /// λ G, eH ,c e are infinite. Both are givenby the union of hyperplanes of the form λ · β = 0, where β runs over the set of the roots of Q \ { } (when we speak of the parameter λ for the algebra eH ,c e we mean the parametercomputed in Theorem 3.14). The claim for eH ,c e follows from [CBH, Theorem 0.4], andthat on A ,λ then follows from [L3, Section 5] (from an isomorphism of A ,λ with a centralreduction of a suitable W-algebra) or from [Bo].The same considerations as in the previous paragraph imply (a). To prove (b) one nowneeds to describe the group A of the automorphisms of A ( ∼ = e H e ) satisfying the following: • they preserve the grading, • they preserve c red as a subset of A , • they are the identity modulo c red .We have a natural homomorphism A → GL( c red ) that is easily seen to be injective. Fromthe isomorphism with a W-algebra mentioned above, one sees that W ⊂ A (recall thatthe W -action on g ∗ G was described in 3.1.11). With some more work, see [L3, Proposition6.4.5], one shows that actually W = A . This implies (b).4.3.3. Completions.
The case of a general n is reduced to n = 1 using suitable completionsof the algebras A , H . Let us explain what completions we use as well as general resultson their structure.First, let us describe completions of algebras of the form A := A ~ ( V ) ///G , where V is a symplectic vector space and G is a reductive group acting on V by symplec-tomorphisms. Let b ∈ V /// G . The point b defines a maximal ideal m ⊂ A . Sowe can form the b -adic completion A ∧ b := lim ←− n → + ∞ A / m n . Let v ∈ V be a pointwith closed G -orbit mapping to b . Let us write A ~ ( V ) ∧ Gv for the completion of A ~ ( V )with respect to the ideal of Gv . Then it is easy to see that A ∧ b ∼ = A ~ ( V ) ∧ Gv ///G .The algebra A ~ ( V ) ∧ Gv can be described using a suitable version of the slice theorem.More precisely, it follows, for example, from [CB2, Section 4] that the formal neigh-borhood V ∧ Gv is equivariantly symplectomorphic to the neighborhood of the base G/K in ( T ∗ G × U ) /// K , where K := G v , U := ( T v Gv ) ⊥ /T v Gv . This statement quantizes: A ~ ( V ) ∧ Gv ∼ = ( D ~ ( G ) ⊗ C [ ~ ] A ~ ( U )) /// K , this can be proved similarly to [L1, Theorem2.3.1]. From here one deduces that A ∧ b ∼ = C [[ g ∗ G ]] b ⊗ C [[ k ∗ K ]] ( A ~ ( U ) ∧ ///K ) , where a homomorphism C [[ k ∗ K ]] → C [[ g ∗ G ]] is induced from the restriction map g ∗ G → k ∗ K .On the other hand, take a symplectic vector space V ′ and a finite subgroup Γ ⊂ Sp( V ).From these data we can form the symplectic reflection algebra H . Pick b ∈ V ′ / Γ. We canproduce the completion H ∧ b : the point b defines a natural maximal ideal in C [ V ′ ] H and complete with respect to that preimage. The algebra H ∧ b canalso be described in terms of a “smaller” algebra of the same type, [L2, Theorem 1.2.1]. More precisely, let Γ be the stabilizer corresponding to b and let H stand for the SRAcorresponding to the pair (Γ , V ′ ), an algebra over S ( c univ ). Then H ∧ b ∼ = Z (Γ , Γ , H ∧ ),where Z (Γ , Γ , • ) is the centralizer algebra from [BE, 3.2], it is isomorphic to Mat | Γ / Γ | ( • ).A consequence we need is that e H ∧ b e ∼ = e H ∧ e . The algebra H can be described asfollows. Let us write V + for a unique Γ-stable complement to V ′ Γ in V ′ . Consider theSRA H + over S ( c univ ), where c univ is the parameter space for Γ. The inclusion Γ ֒ → Γgives rise to a natural map c univ → c univ . Then H = A t ( V ′ Γ ) ⊗ C [ t ] ( S ( c univ ) ⊗ S ( c univ ) H ).4.3.4. Completions at leaves of codimension 2.
We are going to use the completions of A and e H e at points lying in the codimension 2 symplectic leaves. Recall from 3.1.11 thatwhen n > = { } , we have two such leaves. One corresponds to Γ = Γ ⊂ Γ n ,the other to S ⊂ Γ n . Let H , H be the corresponding SRA’s. The correspondingparameter spaces are c univ = Span( c , . . . , c r , t ) and c univ = Span( c , t ). When Γ = { } ,we have just one leaf of codimension 2, it corresponds to S .Now let us describe the completions on the Hamiltonian reduction side. Let v , v beelements from closed G -orbits in µ − (0) ∈ T ∗ R whose images b , b in M ( nδ, ǫ ) , V n / Γ n liein the two leaves. We can take the points v , v as follows. We have a natural embedding µ − (0) n ֒ → µ − (0) from the proof of Proposition 3.10. Take pairwise different elements v , . . . , v n ∈ µ − (0) with closed GL( δ )-orbits. Then we can take v = ( v , . . . , v n − , ∈ T ∗ R ( nδ, ⊂ T ∗ R and v = ( v , . . . , v n − , v n − , v n − ).Let us describe the completion A ∧ b . We have K (= G v ) = ( C × ) n − × GL( δ ). So thespace k ∗ K coincides C n − ⊕ C Q . The restriction map C Q = g ∗ G → k ∗ K = C n − ⊕ C Q sends λ to ( λ · δ, . . . , λ · δ, λ ). The symplectic part U of the normal space T ∗ R/T v Gv splits into the direct sum of the trivial module C n − , of the ( C × ) n − -module ( T ∗ C ) ⊕ n − ,and of the GL( δ )-module T ∗ R ( δ, ǫ ). So A ~ ( U ) ///K ∼ = C [ z , . . . , z n − ] ⊗ A ~ ( C n − ) ⊗ C [ ~ ] A ~ ( T ∗ R ( δ, ǫ )) /// GL( δ ) , where z , . . . , z n − are homogeneous elements of degree 2, the images of the natural basisin Lie( C × ( n − ) under the comoment map.Let us write GL( δ ) for the quotient of GL( δ ) by the one-dimensional torus of constant el-ements. Set g ∗ G := g ∗ G / C δ , clearly, g ∗ G = gl ( δ ) ∗ GL( δ ) . Set A := A ~ ( T ∗ R ( δ, /// GL( δ ).It is easy to see that A ~ ( T ∗ R ( δ, ǫ )) /// GL( δ ) = C [ g ∗ G ] ⊗ C [ g ∗ G ] A . From here and thedescription of the map k ∗ K → g ∗ G given above, we deduce that C [ g ∗ G ] ⊗ C [ k ∗ K ] A ~ ( U ) ///K ∼ = A ~ ( C n − ) ⊗ C [ ~ ] ( C [ g ∗ G ] ⊗ C [ g ∗ G ] A ) . It follows that(12) A ∧ b ∼ = A ∧ ~ ( C n − ) b ⊗ C [[ ~ ]] ( C [[ c ∗ red ]] b ⊗ C [[ c ∗ red ]] A ∧ ) , where we write c red for { λ ∈ C Q | λ · δ = 0 } ⊕ C~ .Let us now deal with A ∧ b . We have K (= G v ) = ( C × ) n − × GL(2). The map g ∗ G → k ∗ K sends λ to the n − λ · δ . The symplecticpart U of the normal space T ∗ R/T v Gv is the sum of the trivial module C n − , the( C × ) n − -module ( T ∗ C ) ⊕ and the GL(2)-module T ∗ ( sl ⊕ C ). Let c red denote the span of P i ∈ Q δ i ǫ i and ~ . Set A := A ~ ( T ∗ ( sl ⊕ C )) /// GL(2), we can view it as an algebra over S ( c red ) (where a natural generator of gl / [ gl , gl ] corresponds to P i ∈ Q δ i ǫ i ). As above,we have C [ g ∗ G ] ⊗ C [ k ∗ K ] A ~ ( U ) ///K ∼ = S ( c red ) ⊗ S ( c red ) ( A ~ ( C n − ) ⊗ C [ ~ ] A ) ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 35 and we get the following description of A ∧ b :(13) A ∧ b ∼ = A ∧ ~ ( C n − ) b ⊗ C [[ ~ ]] ( C [[ c ∗ red ]] b ⊗ C [[ c ∗ red ]] A ∧ ) . Reduction to n = 1 . Using (12) we see that (11) yields an isomorphism of comple-tions A ∧ b ∼ = e H ∧ b e and hence an isomorphism A ∧ ~ ( C n − ) b ⊗ C [[ ~ ]] ( C [[ c ∗ red ]] b ⊗ C [[ c ∗ red ]] A ∧ ) ∼ = A ∧ ~ ( C n − ) b ⊗ C [[ ~ ]] ( C [[ c ∗ red ]] ⊗ C [[ c ∗ univ ]] C [[ c ∗ univ ]] ⊗ C [[ c ∗ univ ]] e H ∧ e ) . It was checked in [L3, Section 6.5] that this isomorphism restricts to S ( c red ) ⊗ S ( c red ) A ∼ = S ( c red ) ⊗ S ( c univ ) e H e that preserves the grading and is the identity modulo ( c red ). From here it is easy todeduce that ν maps c univ to c red and restricts to one of W -conjugates of the map in (10)for n = 1.Let us proceed to the second leaf. Similarly to 4.3.2, one can show that A ~ ( T ∗ ( sl ⊕ C )) /// GL(2) ∼ = e H e , where the isomorphism sends the element P i ∈ Q δ i ǫ i to ± ( c + t ) /
2. It follows that ν maps c univ to c red and induces one of two maps in the previoussentence. It follows that ν is an isomorphism that is W × Z / Z -conjugate to the mapgiven by (10) for n >
1. Since W × Z / Z -action comes from automorphisms, that preservethe grading, map c red to c red , and are the identity modulo ( c red ) (see 3.2.5), claims (b)and (c) follow. This completes the proof of Theorem 3.14.4.4. Classification of Procesi bundles.
Here we are going to prove that the numberof different Procesi bundles on X equals 2 | W | for n > | W | for n = 1.4.4.1. Upper bound.
Recall that a Procesi bundle P on X defines a linear isomorphism ν P : c univ → c red . We claim that if ν P = ν P , then P ∼ = P . Indeed, we have(14) Γ( ˜ P ~ ,fin ) = End( ˜ P ~ ,fin ) e ∼ = End( ˜ P ~ ,fin ) e = Γ( ˜ P ~ ,fin )(an isomorphism of graded right H -modules). Note that H ( ˜ X, ˜ P i ) = 0 because ˜ P i is adirect summand of E nd ( P i ) and the latter sheaf has no higher cohomology. It follows thatΓ( ˜ P i ~ ) / ~ Γ( ˜ P i ~ ) ∼ −→ Γ( ˜ P i ). Taking the quotient of (14) by ~ , we get an isomorphism Γ( ˜ P ) ∼ =Γ( ˜ P ) of graded C [ ˜ X ]-modules. We claim that this implies that the vector bundles ˜ P , ˜ P are C × -equivariantly isomorphic. Indeed, consider the resolution of singularities morphism˜ ρ : ˜ X → ˜ X . This morphism is birational over any p ∈ c ∗ red . Moreover, for a Zariski generic p , the morphism ρ p is an isomorphism, indeed, µ − ( p ) θ − ss = µ − ( p ). It follows that therestrictions of bundles ˜ P , ˜ P to some Zariski open subset in ˜ X with codimension ofcomplement bigger than 1 are isomorphic. It follows that ˜ P ∼ = ˜ P and hence P ∼ = P .We have seen above that ν P can only be one of 2 | W | (for n >
1) or | W | (for n = 1)maps. This implies the upper bound on the number of Procesi bundles.4.4.2. Lower bound.
Let us show that there are 2 | W | different Procesi bundles in the caseof n >
1. Recall that one can construct a Procesi bundle P D once one has a Frobeniusconstant quantization D of X F with Γ( D ) = A ( V n, F ) Γ n . Note that the action of W × Z / Z on A is defined over some algebraic extension of Z . So, as before, it can be reducedmodulo q for q = p ℓ , p ≫
0. Let D λ be the Frobenius constant quantization obtained byHamiltonian reduction with parameter λ ∈ F Q p . The parameter λ constructed from c = 0belongs to F Q p . Above, we have remarked that Γ( D λ ) ∼ = A ( V n, F ) Γ n . Moreover, for q ≫ the stabilizer of this parameter in W × Z / Z is trivial. So we get 2 | W | different Frobeniusconstant quantizations with required global sections. Procesi bundles produced by themare different as well, as was checked in [L4, Section 3.3].4.4.3. Canonical Procesi bundle.
By a canonical Procesi bundle we mean P such that ν P is as in (10). According to [L4, Section 4.2], this bundle has the following property: thesubbundle P Γ n − coincides with the rank n | Γ | bundle T on X = M θ ( nδ, ǫ ) inducedby the G -module L i ∈ Q ( C nδ i ) ⊕ δ i . We will write P θ for this bundle. Recall that for w ∈ W × Z / Z we get an isomorphism M θ ( nδ, ǫ ) ∼ = M wθ ( nδ, ǫ ) that yields the map c red = H ( M θ ( nδ, ǫ )) ⊕ C → H ( M wθ ( nδ, ǫ )) ⊕ C = c red equal to w . It follows that ν w ∗ P θ = wν . So every other Procesi bundle on M θ ( nδ, ǫ ) is obtained as a push-forwardof the canonical Procesi bundle P wθ on M wθ ( nδ, ǫ ).Note that when P is a Procesi bundle, then so is P ∗ . Indeed, End O X ( P ∗ , P ∗ ) ∼ =End O X ( P , P ) opp . The algebra C [ V n ] n is identified with its opposite via v v, γ γ − , v ∈ V ∗ n , γ ∈ Γ n and this gives a Procesi bundle structure on P ∗ . We have ν P θ ∗ = w σν P ∗ , where w is the longest element in W and σ is the image of 1 in Z / Z , see [L4,Remark 4.4]. 5. Macdonald positivity and categories O In this section we provide some applications of results of Section 4.In Section 5.1, we will produce equivalences between categories D b ( H ,c ) and D b (Coh( D λ )).Starting from Section 5.2, we will only consider the groups Γ n with cyclic Γ . Here Γ n isa complex reflection group and the corresponding algebra H t,c (called a Rational Chered-nik algebra) in this case admits a triangular decomposition. This decomposition allowsto define Verma modules and, for t = 1, category O for H ,c that has a so called highestweight structure. We can also define the category O for D λ , this will be a subcategoryin Coh( D λ ). We will show that the derived equivalence D b ( H ,c -mod) ∼ = D b (Coh( D λ ))restricts to categories O . This was used in [GL] to establish [R, Conjecture 5.6] for thegroups Γ n .In Section 5.3 we prove Theorem 1.3 and also its generalization to the groups Γ n dueto Bezrukavnikov and Finkelberg. The proof is based on studying the algebras H ,c andtheir Verma modules.Finally, in Section 5.4 we prove an analog of the Beilinson-Bernstein localization the-orem, [BB], for the Rational Cherednik algebras associated to the groups Γ n . More pre-cisely, we answer the question when the derived equivalence D b (Coh( D λ ) → D b ( H ,c -mod)restricts to an equivalence Coh( D λ ) → H ,c -mod.5.1. Derived equivalence.
Deformed derived McKay correspondence.
Similarly to 4.1.2, the functor R Γ( P ⊗ O X • ) defines an equivalence D b (Coh X ) ∼ −→ D b ( C [ V n ] n -mod) with quasi-inverse P ∗ ⊗ L C [ V n ] n • . These equivalence automatically upgrade to the categories of C × -equivariant objects: D b (Coh C × X ) ∼ = D b ( C [ V n ] n -mod C × ) defined in the same way.Now let us consider the deformation ˜ P ~ of P to a right C × -equivariant e D ~ -module. Itgives a functor ˜ F := R Γ( ˜ P ~ ,fin ⊗ e D ~ ,fin • ) : D b (Coh C × ( ˜ D ~ ,fin )) → D b ( H -mod C × ). Thisfunctor has left adjoint and right inverse ˜ G = ˜ P ∗ ~ ,fin ⊗ L H • . So we get the adjunction ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 37 morphism ˜
G ◦ ˜ F → id. One can show (see [GL, Section 5] for details) that since thismorphism is an isomorphism modulo c univ , it is an isomorphism itself.5.1.2. Specialization.
The equivalence ˜ F can be specialized to a numerical parameter.In particular, we get equivalences D b (Coh( D λ )) → D b ( H ,c -mod), where λ is recov-ered from c as in Theorem 3.14. This is done in two steps. First, one gets a derivedequivalence between Coh C × ( R ~ / ( D λ )) and R ~ / ( H ,c ) -mod C × , the corresponding sheafand algebra are obtained from e D ~ ,fin , H by base change (and the equivalence we needcomes from the corresponding base change of ˜ P ~ ,fin ). To do the second step we recallthat H ,c -mod is the quotient R ~ / ( H ,c ) -mod C × by the full subcategory of the C [ ~ ]-torsion modules and the similar claim holds for Coh( D λ ), see Lemma 2.9. It follows that D b ( H ,c -mod) is the quotient of D b ( R ~ / ( H ,c ) -mod C × ) by the category of all complexeswhose homology are C [ ~ ]-torsion and a similar claim holds for D λ . Since the equivalence D b ( R ~ / ( H ,c ) -mod C × ) ∼ = D b (Coh C × ( R ~ / ( D λ ))) is C [ ~ ]-linear by the construction, theyinduce(15) D b ( H ,c -mod) ∼ = D b (Coh( D λ )) . Application: shift equivalences.
The equivalences (15) can be applied to produc-ing a result that only concerns the symplectic reflection algebras. Namely, we say thatparameters c, c ′ for H ? have integral difference if λ − λ ′ ∈ Z Q for the corresponding pa-rameters λ . Recall that we can view χ ∈ Z Q as a character of G . So χ defines a linebundle on X , explicitly, O χ = π ∗ ( O µ − (0) θ − ss ) G,χ . This line bundle can be quantized to a D λ + χ - D λ -bimodule to be denoted by D λ,χ . Explicitly, D λ,χ := π ∗ ( D ss / D ss { Φ( x ) − h λ, x i} ) G,χ . This bundle carries a natural filtration and an isomorphism gr D λ,χ ∼ = O χ follows fromthe flatness of the moment map.Note that there is a natural (multiplication) homomorphism D λ + χ,χ ′ ⊗ D λ + χ D λ,χ →D λ,χ + χ ′ that becomes the isomorphism O χ ′ ⊗ O χ → O χ + χ ′ after passing to the associatedgraded. So the multiplication homomorphism itself is an isomorphism. It follows that afunctor D λ,χ ⊗ D λ • : Coh( D λ ) → Coh( D λ + χ ) is a category equivalence. We conclude thatcategories D b ( H ,c -mod) and D b ( H ,c ′ -mod) are equivalent provided c, c ′ have integraldifference.5.2. Category O . Starting from now on, we assume that Γ is a cyclic group Z /ℓ Z . Re-call that in this case the space V n (equal to C n when ℓ > C n − when ℓ = 1) splitsas h ⊕ h ∗ , where h is a standard reflection representation of the group Γ n . The embeddings h , h ∗ ֒ → H extend to algebra embeddings S ( h ) , S ( h ∗ ) ֒ → H . These embeddings give riseto the triangular decomposition H = S ( h ∗ ) ⊗ S ( c univ )Γ n ⊗ S ( h ). We can also consider thespecialization H ,c = S ( h ∗ ) ⊗ C Γ n ⊗ S ( h ) (here and below c is a numerical parameter) ofthis decomposition.5.2.1. Category O for H ,c . By definition, the category O for H ,c consists of all H ,c -modules M such that(i) h acts locally nilpotently on M .(ii) M is finitely generated over H ,c .Note that, modulo (i), the condition (ii) is equivalent to (ii ′ ) M is finitely generated over S ( h ∗ ).An example of an object in the category O is a Verma module constructed as follows.Pick an irreducible representation τ of Γ n and view it as a S ( h ) n -module by making h act by 0. Then set ∆ ,c ( τ ) := H ,c ⊗ S ( h ) n τ . As a S ( h ∗ ) W -module, ∆ ,c ( τ ) is naturallyidentified with S ( h ∗ ) ⊗ τ (the algebra S ( h ∗ ) acts by multiplications from the left, and W acts diagonally).The algebra H ,c carries an Euler grading given by deg h = − , deg h ∗ = 1 , deg W = 0.This grading is internal: we have an element h ∈ H ,c with [ h, a ] = da for a ∈ H ,c ofdegree d . Explicitly, the element h is given by m X i =1 x i y i + X s ∈ S c ( s )1 − λ s s. Here the notation is as follows. We write y , . . . , y m for a basis in h (of course, m = n for ℓ > m = n − ℓ = 1) and x , . . . , x m for the dual basis in h ∗ . By S we, as usual,denote the set of reflections in Γ n and c ( s ) stands for c i if s ∈ S i (note that the formulafor h is different from the usual formula for the Euler element, see, e.g., [BE, Section 2.1],because our c ( s ) is rescaled). Finally, λ s is the eigenvalue of s in h ∗ different from 1.Using the element h , we can show that every Verma module ∆ ,c ( τ ) has a unique simplequotient. These quotients form a complete collection of the simple objects in O . Also onecan show that every object in O has finite length. These claims are left as exercises tothe reader.5.2.2. Category O for D λ . We have a C × -action on D ( R ) induced by the C × -action on R given by t.r := t − r . This action is Hamiltonian, the corresponding quantum comomentmap Φ : C → D ( R ) sends 1 to the Euler vector field. The action descends to a Hamiltonian C × -action on D λ for any λ .Consider the corresponding Hamiltonian C × -action on X = M θ ( nδ, ǫ ). Recall thatthe resolution of singularities morphism X → ( h ⊕ h ∗ ) / Γ n becomes C × -equivariant if weequip the target variety with the C × -action induced by t. ( a, b ) = ( t − a, tb ) , a ∈ h , b ∈ h ∗ . This action has finitely many fixed points that are in a natural bijection with theirreducible representations of Γ n , see [G3, 5.1]. Namely, X C × is in a natural bijectionwith M p ( nδ, ǫ ) C × , where p ∈ g ∗ G is generic. Indeed, M p ( nδ, ǫ ) = M θp ( nδ, ǫ ) and thesets M θp ( nδ, ǫ ) C × are identified for all p by continuity. Let c be a parameter correspondingto p (meaning that ν (0 , c ) = (0 , p )). Then we can consider the Verma module ∆ ,c ( τ ) := H ,c ⊗ S ( h ) n τ . The subalgebra S ( h ∗ ) Γ n is easily seen to be central. Let us write S ( h ∗ ) Γ n + for the augmentation ideal in S ( h ∗ ) Γ n . Following [G1], consider the baby Verma module ∆ ,c ( τ ) := ∆ ,c ( τ ) /S ( h ∗ ) Γ n + ∆ ,c ( τ ) ∼ = S ( h ∗ ) / ( S ( h ∗ ) Γ n + ) ⊗ τ (the last isomorphism is that of S ( h ∗ ) n -modules). This module is easily seen to be indecomposable so it has a centralcharacter that is a point of Spec( Z ( H ,c )) = M p ( nδ, ǫ ). Clearly, this point is fixed by C × and this defines a map Irr(Γ n ) → M p ( nδ, ǫ ) C × , τ z τ , that was shown to be a bijectionin [G3].Fix some p ∈ g ∗ G . Consider the attracting locus Y p ⊂ M θp ( nδ, ǫ ) for the C × -action.Since this action has finitely many fixed points, we see that Y p is a lagrangian subvarietywith irreducible components indexed by Irr(Γ n ). Namely, to τ ∈ Irr(Γ n ) we assign theattracting locus Y p ( τ ) := { z ∈ M θp ( nδ, ǫ ) | lim t → t.z = z τ } . The irreducible components ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 39 of Y p are the closures Y p ( τ ). When p is Zariski generic, the subvarietes Y p ( τ ) are alreadyclosed.By the category O loc for D λ we mean the full category of coherent D λ -modules thatare supported on Y (see 2.3.4) and admit a C × -equivariant structure compatible with the C × -action on D λ . Such categories were systematically studied in [BLPW]. In particular,it was shown that all modules in O loc have finite length and are indexed by M θ ( nδ, ǫ ) C × ,see [BLPW, Sections 3.3,5.3].5.2.3. Choice of identification X C × ∼ = Irr(Γ n ) . We note that despite our identificationof X C × with Irr(Γ n ) is natural, there are other natural choices as well. The choice wehave made is good for working with the category O . We could also consider the category O ∗ , where the modules are locally nilpotent for h ∗ , not for h (and are still finitely gen-erated over H ,c ). Consequently, we need to use the opposite Hamiltonian C × -action on X, M p ( nδ, ǫ ) and Verma modules ∆ ∗ ,c ( τ ) := H ,c ⊗ S ( h ∗ ) n τ . Let us explain how thebijection X C × ∼ = Irr(Γ n ) changes.All simple constituents of ∆ ,c ( τ ) are isomorphic modules of dimension | Γ n | (indeed, H ,c is the endomorphism algebra of the rank | Γ n | bundle ˜ P p on M p ( nδ, ǫ )). Let us denotethis simple module by L ,c ( τ ). This module is graded, the highest graded component is τ .Let us determine the lowest graded component in L ,c ( τ ). This component coincides withthe lowest graded component in ∆ ,c ( τ ) that is the tensor product of τ with the lowestdegree component in C [ h ] / ( C [ h ] Γ n ) + . It is easy to see that the latter is Λ top h . Abusingthe notation, we will denote τ ⊗ Λ top h by τ t . When Γ = { } we can use the standardidentification of Irr( S n ) with the set of Young diagrams of n boxes. In this case, Λ top h isthe sign representation of S n and τ t indeed corresponds to the transposed Young diagramof τ .The previous paragraph shows that there is an epimorphism ∆ ∗ ,c ( τ t ) ։ L p ( τ ). So ournew bijection sends the point z τ ∈ X C × to τ t .We also note that the identification X C × ∼ = Irr(Γ n ) , τ z τ , depends on the choice of aProcesi bundle P but we are not going to use this.5.2.4. Highest weight structures.
Let us recall the definition of a highest weight category.Let C be an abelian category that is equivalent to the category of modules over a finitedimensional algebra, equivalently, the category C has finitely many simples, enough pro-jectives and finite dimensional Hom’s (and hence every object has finite length). Let T denote an indexing set of the simple objects in C , we write L ( τ ) for the simple objectindexed by τ ∈ T and P ( τ ) for its projective cover. The additional structure of a highestweight category is a partial order on T and a collection of so called standard objects∆( τ ) , τ ∈ T , satisfying the following axioms:(1) Hom C (∆( τ ) , ∆( τ ′ )) = 0 implies τ τ ′ ,(2) End C (∆( τ )) = C .(3) P ( τ ) ։ ∆( τ ) and the kernel admits a filtration with quotients ∆( τ ′ ) for τ ′ > τ . Remark 5.1.
Let us point out that the standard objects are uniquely recovered fromthe partial order. Namely, consider the category C τ that is the Serre span of the simples L ( τ ′ ) with τ ′ τ . Then ∆( τ ) is the projective cover of L ( τ ) in C τ .Both categories O , O loc that were described above are highest weight, see [GGOR,Sections 2.6,3.2] for O and [BLPW, Section 5.3] for O loc . The standard objects ∆( λ ) are the Verma modules. The order can be introduced as follows. Recall the element h ∈ H ,c introduced in 5.2.1. It acts on τ ⊂ ∆( τ ) by X s ∈ S c ( s )1 − λ s s. The latter element in C Γ n iscentral and so acts on τ by a scalar, denote that scalar by c τ . Then we set τ τ ′ if c τ − c τ ′ ∈ Z > .Let us provide a formula for c τ . We start with ℓ = 1. Then a classical computationshows that c τ = c cont( τ ) /
2, where the integer cont( τ ) is defined as follows. For thebox b ∈ τ lying in x th column and y th row, we set cont( b ) := x − y . Then cont( τ ) := P b ∈ τ cont( b ). Now let us proceed to ℓ >
1. In this case, the irreducible representationsof Γ n are parameterized by the ℓ -multipartitions ( τ (1) , . . . , τ ( ℓ ) ) of n . Define elements λ , . . . , λ ℓ by requiring that λ i , i = 1 , . . . , ℓ − , is recovered from c as in Theorem 3.14and P ℓi =1 λ i = 0. For a box b ∈ τ ( j ) set d c ( b ) := c ℓ cont( b ) / ℓλ j . Then, up to asummand independent of τ , we have c τ = P b ∈ τ d c ( b ), see [R, Proposition 6.2] or [GL,2.3.5] (in both papers the notation is different from what we use).In fact, one can take a weaker ordering on Irr(Γ n ) making O into a highest weight cat-egory. Namely, according to [Gr], for two boxes b, b ′ in j th and j ′ th diagrams respectivelywe say that b b ′ if d c ( b ) − d c ( b ′ ) is congruent to j − j ′ modulo ℓ and is in Z > . Then λ λ ′ if one can order boxes b , . . . , b n of λ and b ′ , . . . , b ′ n of λ ′ in such a way that b i b ′ i for all i .Let us proceed to the categories O loc . They are highest weight with respect to the order (we will often write θ to indicate the dependence on θ ) defined as follows. We firstdefine a pre-order ′ by setting τ ′ τ ′ if z τ ∈ Y τ ′ and then define as the transitiveclosure of ′ . Example 5.2.
When ℓ = 1 and θ <
0, the bijection between the C × h -fixed points andpartitions is the standard one. A combinatorial description of θ follows from [Nak2,Section 4]: we have τ θ τ ′ if τ τ ′ as Young diagrams.In the case when ℓ > O loc into a highest weight category) was described by Gordon in [G3, Section 7] in combina-torial terms. The standard modules are recovered from θ as before. Below we will seethat they can be described using the deformations of the Procesi bundle.5.2.5. Derived equivalence.
Here we are going to produce a derived equivalence D b ( O ) ∼ = D b ( O loc ).Inside D b ( H ,c -mod) we can consider the full subcategory D b O ( H ,c -mod) consisting ofall complexes whose homology lie in the category O . We then have a natural functor D b ( O ) → D b O ( H ,c -mod). This functor is an equivalence by [E, Proposition 4.4]. Wecan also consider the category D b O (Coh( D λ )), the functor D b ( O loc ) → D b O (Coh( D λ )) is anequivalence as well, this follows from [BLPW, Corollary 5.13] and [BPW, Corollary 5.12].The equivalence D b ( H ,c -mod) ∼ −→ D b (Coh( D λ )) is compatible with the supports inthe following sense. Recall that we have two commuting C × -actions. The Hamiltoniantorus will be denoted by C × h , while, for the contracting torus (which is present even whenΓ is not cyclic), we will write C × c . Pick a closed subvariety Y ⊂ ( h ⊕ h ∗ ) / Γ n that isstable under the C × c -action. Consider the full subcategory D bY ( H ,c -mod) in D b ( H ,c ) ofall complexes with homology supported on Y . Set Y := ρ − ( Y ), where, recall, ρ standsfor the resolution of singularities morphism ρ : X → V n / Γ n and consider the subcategory ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 41 D bY (Coh( D λ )) ⊂ D b (Coh( D λ )). Then the equivalence D b (Coh( D λ )) ∼ = D b ( H ,c -mod)restricts to D bY (Coh( D λ )) ∼ = D bY ( H ,c -mod).Note that the bundle P on X is ( C × ) -equivariant. Therefore the deformation ˜ P ~ is( C × ) -equivariant as well. It follows that the equivalence D b (Coh( D λ )) ∼ = D b ( H ,c -mod)preserves complexes whose homology admit C × -equivariant liftings. Combined with theprevious paragraph, this means that we get an equivalence D b O ( H ,c -mod) ∼ = D b O (Coh( D λ ))and hence an equivalence D b ( O ) ∼ = D b ( O loc ).This was used in [GL, Section 5] to prove a conjecture of Rouquier, [R, Conjecture 5.6].Namely, suppose that we have parameters c, c ′ such that the corresponding parameters λ, λ ′ have integral difference. Then we have an abelian equivalence Coh( D λ ) ∼ −→ Coh( D λ ′ ),given by tensoring with the bimodule D λ,λ ′ − λ . This bimodule is C × h -equivariant, thisfollows from the construction. Also it is clear that tensoring with D λ,λ ′ − λ preserves thesupports. So we conclude that O locλ ∼ −→ O locλ ′ . It follows that the categories O c and O c ′ arederived equivalent that was conjectured by Rouquier (in the generality of all Cherednikalgebras).5.3. Macdonald positivity.
Consider the H -module ∆ ( λ ) := H ⊗ S ( h ) n λ . Recall thederived equivalence D b (Coh( e D ~ ,fin )) ∼ −→ D b ( H -mod) given by F := Γ( e P ~ ,fin ⊗ e D ~ ,fin • )and its inverse G . It turns out that the study of the objects G ( ∆ ( λ )) leads to the proofof the Macdonald positivity. The proof that we provide below is morally similar to butdifferent from the original proof in [BF].5.3.1. Flatness.
A key step in the proof is to establish the flatness over C [ h ] of an arbitraryProcesi bundle P , where we view P ( C [ h ] acts on P via the inclusion C [ h ] ֒ → S ( h ⊕ h ∗ ) n = End O X ( P )). This will imply that the Koszul complex P ← h ∗ ⊗ P ← Λ h ∗ ⊗ P ← . . . ← Λ n h ∗ ⊗ P is a resolution of P / h ∗ P . The proof of the flatness is taken from the proof of [BF, Lemma3.7].Note that, since Γ n is a complex reflection group, C [ h ] is free over C [ h ] Γ n . So it isenough to show that P is flat over C [ h ] Γ n .Let us recall how P was constructed, see 4.1.4 (construction of one Procesi bundle incharacteristic p ≫ D of X F , where F is analgebraically closed field of characteristic 0.(2) Then we take a splitting bundle B of D| X (1) ∧ F .(3) We form a bundle P ′ on X (1) ∧ F that is the sum of indecomposable summands of S ∗ with suitable multiplicities. Then we extend this bundle to X (1) F and get a Procesibundle P (1) F on X (1) F .(4) Since X (1) F ∼ = X F as F -varieties, we can view P (1) F as a bundle P F on X .(5) Then we lift P F to characteristic 0.The procedure in (5) implies that if P F is flat over F [ h ] Γ n , then P is flat over C [ h ] Γ n (the reader is welcome to verify the technical details). Obviously, P F is flat over F [ h ] Γ n if and only if P (1) F is flat over F [ h (1) ] Γ n . The latter is equivalent to B ∗ being flat over F [[ h (1) ]] Γ n , which, in turn, is equivalent to the claim that D is a flat F [ h (1) ] Γ n -module. Butgr D ∼ = Fr X ∗ O X F . So it is enough to verify that O X F is flat over F [ h (1) ] Γ n . Since F [ h ] Γ n is flatover F [ h (1) ] Γ n , we reduce to proving that X F is flat over h F / Γ n , equivalently, all fibers of X F → h F / Γ n have the same dimension, equivalently, the zero fiber has dimension dim h .But the zero fiber of this map is precisely the contracting variety for the Hamiltonian F × -action and so is lagrangian. This completes the proof.Similarly, P is flat over C [ h ∗ ]. Also let us recall, see 4.4.3, that P ∗ can be equipped witha structure of the Procesi bundle, for which we need to convert the right S ( h ⊕ h ∗ ) n -module into a left S ( h ⊕ h ∗ ) n using a natural anti-automorphism of S ( h ⊕ h ∗ ) n .This shows that P ∗ is a flat right module over both C [ h ] and C [ h ∗ ]. This is what we aregoing to use below.5.3.2. Upper triangularity.
Let θ be a generic stability condition and take X = X θ . Thisgives rise to the partial order θ on the set Irr(Γ n ) described in 5.2.2. Recall that wewrite z τ for the C × h -fixed point in X corresponding to τ as explained in 5.2.2. We write Y τ for the C × h -contracting component of z τ , a lagrangian subvariety in X θ . Further, write e τ for a primitive idempotent in C Γ n corresponding to τ so that τ ∼ = ( C Γ n ) e τ . Proposition 5.3.
Let P be the canonical Procesi bundle on X θ . Then the sheaf ( P ∗ / P ∗ h ) e τ is supported on S τ ′ θ τ Y τ ′ .Proof. Consider the deformation e P ∗ of P ∗ to e X . It is flat over C [ g ∗ G , h ∗ ]. Therefore e P ∗ / e P ∗ h is flat over C [ g ∗ G ]. It follows that Supp(( P ∗ / P ∗ h ) e τ ) ⊂ C × c Supp( P ∗ p / P ∗ p h ) e τ for a generic p ∈ g ∗ G . But ( P ∗ p / P ∗ p h ) e τ is nothing else but e ∆ ,c ( τ ). We claim thatSupp ∆ ,c ( τ ) ⊂ Y p,τ . Indeed, we have shown in 5.2.2 that ∆ ,c ( τ ) /S ( h ∗ ) Γ n + ∆ ,c ( τ ) is sup-ported in z p,τ , the point in M p ( nδ, ǫ ) C × h indexed by τ . If Supp ∆ ,c ( τ ) Y p,τ , then thereis τ ′ = τ with z p,τ ′ ∈ Supp ∆ ,c ( τ ) (because the latter is closed and contained in Y p ). Thesupport of ∆ ,c ( τ ) is disconnected and so the module ∆ ,c ( τ ) is indecomposable. Fromhere one deduces that z p,τ ′ lies in the support of ∆ ,c ( τ ) /S ( h ∗ ) Γ n + ∆ ,c ( τ ), contradiction.Now the inclusion Supp (( P ∗ / P ∗ h ) e τ ) ⊂ [ τ ′ θ τ Y τ ′ follows from C × Y p,τ ∩ X θ ⊂ [ τ ′ θ τ Y τ ′ , see [BF, Lemma 3.8]. (cid:3) In fact, e ∆ ,c ( τ ) = C [ Y p,τ ] but we do not need this fact.5.3.3. Wreath-Macdonald positivity.
Now we are ready to prove the Macdonald positivitytheorem, Theorem 1.3, and its “wreath-generalization” due to Bezrukavnikov and Finkel-berg.First of all, Proposition 5.3 implies that if the fiber of [ P ∗ / P ∗ h ] e τ in z τ ′ is nonzero,then τ ′ θ τ . It follows that if τ ∗ is a constituent of the fiber ( P ∗ / P ∗ h ) z τ ′ , then τ > θ τ ′ .But since P ∗ is a flat right C [ h ]-module, we see that the class of [ P ∗ / P ∗ h ∗ ] z τ ′ in the K of bigraded Γ n -modules coincides with that of the Koszul resolution P ∗ z τ ′ ← P ∗ z τ ′ ⊗ h ← . . . ROCESI BUNDLES AND SYMPLECTIC REFLECTION ALGEBRAS 43
Taking the duals, we see that if τ occurs in the class P z τ ′ ⊗ dim h X i =0 ( − i Λ i h ∗ , then τ ′ θ τ . When Γ = { } , this yields (a) from Definition 1.2.To get (b) in that definition (and its wreath-generalization), we consider [ P ∗ / P ∗ h ∗ ] e τ .This sheaf is supported on the union of repelling components for C × h and can have nonzerofibers only in the fixed points z τ ′ with z τ ′ > θ z τ t meaning τ t θ τ ′ . In other words, if τ appears in P z τ ′ ⊗ dim h X i =0 ( − i Λ i h , then τ t θ τ ′ . When Γ = { } , this yields (b) in Definition 1.2. (c) there follows because P is normalized.5.4. Localization theorem.
Let P ,λ denote the the right D λ -module obtained by spe-cializing e P ~ . One can ask when (i.e., for which λ ) the functor Γ( P ,λ ⊗ D λ • ) : O locλ → O c is a category equivalence. The following result answers this question. Theorem 5.4.
Suppose that there is an order on Irr(Γ n ) refining θ and making both O locλ , O c into highest weight categories. Then Γ : Coh( D λ ) → H ,c -mod , O locλ → O c areequivalences of categories. This theorem can be viewed as an analog of the Beilinson-Bernstein localization theo-rem, [BB], from the Lie representation theory.
Sketch of proof.
It is enough to prove that Γ gives an equivalence between the categories O , see [L5, Section 3.3]. So below in the proof we only deal with the categories O .Set ∆ loc ( λ ) := [ P ∗ ,λ / P ∗ ,λ h ] e λ . Further, let F stand for R Γ( P ,λ ⊗ D λ • ). The flatness of P over S ( h ) from the previous subsection implies that(16) F ∆ locλ ( τ ) = ∆ c ( τ ) . We have ∆ locλ ( τ ) ∈ O loc θ λ . The condition on the orders implies that ∆ locλ ( τ ) is the standardobject in O locλ . Now the claim of Theorem 5.4 follows from the next general claim. (cid:3) Lemma 5.5.
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