Etingof conjecture for quantized quiver varieties II: affine quivers
aa r X i v : . [ m a t h . R T ] N ov ETINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II:AFFINE QUIVERS
IVAN LOSEV
Abstract.
We study the representation theory of quantizations of Gieseker modulispaces. Namely, we prove the localization theorems for these algebras, describe theirfinite dimensional representations and two-sided ideals as well as their categories O insome special cases. We apply this to prove our conjecture with Bezrukavnikov on thenumber of finite dimensional irreducible representations of quantized quiver varieties forquivers of affine type. Contents
1. Introduction 21.1. Classical and quantum quiver varieties 21.2. Results in the Gieseker cases 31.3. Counting result 51.4. Content of the paper 52. Preliminaries 52.1. Symplectic resolutions 62.2. Gieseker moduli spaces 72.3. Symplectic leaves 82.4. Quantizations 92.5. Localization theorems 102.6. Duality and wall-crossing functor 112.7. Categories O n and singular parameters 28 MSC 2010: Primary 16G99; Secondary 16G20,53D20,53D55. O ( A λ ( n, r )) 295.4. Completion of proofs 326. Affine wall-crossing and counting 336.1. Functor • † ,x A λ ( n, r ) 366.3. Computation of Tor’s 386.4. Faithfulness 396.5. Affine wall-crossing functor and counting 40References 401. Introduction
Classical and quantum quiver varieties.
This paper continues the study of therepresentation theory of quantized quiver varieties initiated in [BL]. So we start byrecalling Nakajima quiver varieties and their quantizations.Let Q be a quiver (=oriented graph, we allow loops and multiple edges). We canformally represent Q as a quadruple ( Q , Q , t, h ), where Q is a finite set of vertices, Q is a finite set of arrows, t, h : Q → Q are maps that to an arrow a assign its tail andhead. In this paper we are interested in the case when Q is of affine type, i.e., Q is anextended quiver of type A, D, E .Pick vectors v, w ∈ Z Q > and vector spaces V i , W i with dim V i = v i , dim W i = w i .Consider the (co)framed representation space R = R ( v, w ) := M a ∈ Q Hom( V t ( a ) , V h ( a ) ) ⊕ M i ∈ Q Hom( V i , W i ) . We will also consider its cotangent bundle T ∗ R = R ⊕ R ∗ , this is a symplectic vectorspace that can be identified with M a ∈ Q (cid:0) Hom( V t ( a ) , V h ( a ) ) ⊕ Hom( V h ( a ) , V t ( a ) ) (cid:1) ⊕ M i ∈ Q (Hom( V i , W i ) ⊕ Hom( W i , V i )) . The group G := Q k ∈ Q GL( V k ) naturally acts on T ∗ R and this action is Hamiltonian. Itsmoment map µ : T ∗ R → g ∗ is dual to x x R : g → C [ T ∗ R ], where x R stands for thevector field on R induced by x ∈ g .Fix a stability condition θ ∈ Z Q that is thought as a character of G via θ (( g k ) k ∈ Q ) = Q k ∈ Q det( g k ) θ k . Then, by definition, the quiver variety M θ ( v, w ) is the GIT Hamiltonianreduction µ − (0) θ − ss //G . We are interested in two extreme cases: when θ is generic (andso M θ ( v, w ) is smooth and symplectic) and when θ = 0 (and so M θ ( v, w ) is affine). Wewill write M ( v, w ) for Spec( C [ M θ ( v, w )]), this is an affine variety independent of θ anda natural projective morphism ρ : M θ ( v, w ) → M ( v, w ) is a resolution of singularities.Under an additional restriction on v , we have the equality M ( v, w ) = M ( v, w ).Namely, let g ( Q ) be the affine Kac-Moody algebra associated to Q . Let us set ω := P i ∈ Q w i ω i , ν := ω − P i ∈ Q v i α i , where we write ω i for the fundamental weight and α i for a simple root corresponding to i ∈ Q . Then we have M ( v, w ) = M ( v, w ) provided ν is dominant.Note also that we have compatible C × -actions on M θ ( v, w ) , M ( v, w ) induced from theaction on T ∗ R given by t. ( r, α ) := ( t − r, t − α ) , r ∈ R, α ∈ R ∗ . TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 3
A special case of most interest and importance for us in this paper is the Giesekermoduli spaces M θ ( n, r ) where n, r ∈ Z > . It corresponds to the case when Q is a quiverwith a single vertex and a single arrow (that is a loop), with v = n, w = r . This spaceparameterizes torsion free sheaves of rank r and degree n on P trivialized at the line atinfinity (but we will not need this description). The importance of this case in our workis of the same nature as in the work of Maulik and Okounkov, [MO], on computing thequantum cohomology of quiver varieties.Now let us proceed to the quantum setting. We will work with quantizations of M θ ( v, w ) , M ( v, w ). Consider the algebra D ( R ) of differential operators on R . The group G naturally acts on D ( R ) with a quantum comoment map Φ : g → D ( R ) , x x R . Wecan consider the quantum Hamiltonian reduction A λ ( v, w ) = [ D ( R ) /D ( R ) { x R −h λ, x i} ] G .It is a quantization of M ( v, w ) = M ( v, w ) when ν is dominant. In the general case onecan define a quantization of M ( v, w ) in two equivalent way: as an algebra A λ ′ ( v ′ , w ) forsuitable λ ′ and v ′ (thanks to quantized LMN isomorphisms from [BL, 2.2]) or as the alge-bra of global section of a suitable microlocal sheaf on M θ ( v, w ) (where θ is generic). Letus recall the second approach. We can microlocalize D ( R ) to a sheaf in conical topology(i.e., the topology where “open” means “Zariski open” and C × -stable) so that we canconsider the restriction of D ( R ) to the ( T ∗ R ) θ − ss , let D θ − ss denote the restriction. Let π stand for the quotient morphism µ − (0) θ − ss → µ − (0) θ − ss /G = M θ ( v, w ). Let us noticethat D θ − ss / D θ − ss { x R − h λ, x i} is scheme-theoretically supported on µ − (0) θ − ss and so canbe regarded as a sheaf in conical topology on that variety. Set A θλ ( v, w ) := [ π ∗ ( D θ − ss / D θ − ss { x R − h λ, x i} )] G , this is a sheaf (in conical topology) of filtered algebras on M θ ( v, w ) such that gr A θλ ( v, w ) = O M θ ( v,w ) . By the Grauert-Riemenschneider theorem, H i ( O M θ ( v,w ) ) = 0 for i >
0. Itfollows that A θλ ( v, w ) has no higher cohomology as well, and gr Γ( A θλ ( v, w )) = C [ M ( v, w )].One can show, see [BPW, 3.3] or [BL, 2.2], that Γ( A θλ ( v, w )) is independent of the choiceof θ . We will write A λ ( v, w ) for Γ( A θλ ( v, w )).In this paper we will be interested in the representation theory of the algebras A λ ( v, w )and, especially, of A λ ( n, r ) (quantizations of M ( n, r )). Let us point out that the rep-resentations of A θλ ( v, w ) and of A λ ( v, w ) are closely related. Namely, we can considerthe category of coherent A θλ ( v, w )-modules to be denoted by A θλ ( v, w ) -mod and the cat-egory A λ ( v, w ) -mod of all finitely generated A λ ( v, w )-modules. When the homologicaldimension of A λ ( v, w ) is finite, we get adjoint functors R Γ θλ : D b ( A θλ ( v, w ) -mod) ⇄ D b ( A λ ( v, w ) -mod) : L Loc θλ , where R Γ θλ is the derived global section functor, and L Loc θλ := A θλ ( v, w ) ⊗ L A λ ( v,w ) • . Itturns out that these functors are equivalences, [MN1]. In particular, they restrict tomutually inverse equivalences(1.1) D bρ − (0) ( A θλ ( v, w ) -mod) ⇄ D bfin ( A λ ( v, w ) -mod) , where on the left hand side we have the category of all complexes with homology supportedon ρ − (0), while on the right hand side we have all complexes with finite dimensionalhomology.1.2. Results in the Gieseker cases.
In this paper we are mostly dealing with thealgebras A λ ( n, r ). Note that R = C ⊕ ¯ R , where ¯ R := sl n ( C ) ⊕ Hom( C n , C r ) and the actionof G on C is trivial. So we have M θ ( n, r ) = C × ¯ M θ ( n, r ) and A λ ( n, r ) = D ( C ) ⊗ ¯ A λ ( n, r ), IVAN LOSEV where ¯ M θ ( n, r ) , ¯ A λ ( n, r ) are the reductions associated to the G -action on ¯ R . We willconsider the algebra ¯ A λ ( n, r ) rather than A λ ( n, r ), all interesting representation theoreticquestions about A λ ( n, r ) can be reduced to those about ¯ A λ ( n, r ).There is one case that was studied very explicitly in the last decade: r = 1. Here thevariety M θ ( n,
1) is the Hilbert scheme Hilb n ( C ) of n points on C and M ( r, n ) = C n / S n (the n th symmetric power of C ). The quantization ¯ A λ ( n, r ) is the spherical subalgebrain the Rational Cherednik algebra H λ ( n ) for the pair ( h , S n ), where h is the reflectionrepresentation of S n , see [GG] for details. The representation theory of ¯ A λ ( n,
1) wasstudied, for example, in [BEG, GS1, GS2, R, KR, BE, L3, W]. In particular, it is known(1) when (=for which λ ) this algebra has finite homological dimension, [BE],(2) how to classify its finite dimensional irreducible representations, [BEG],(3) how to compute characters of irreducible modules in the so called category O , [R],(4) how to determine the supports of these modules, [W],(5) how to describe the two-sided ideals of ¯ A λ ( n, A λ ( n, r ) (as well as relatively easy parts of(3) and (4)) in the present paper. We plan to address an analog of (4) in a subsequentpaper, while (3) is a work in progress.Before we state our main results, let us point out that there is yet another case whenthe algebra ¯ A λ ( n, r ) is classical, namely when n = 1. In this case, ¯ A λ (1 , r ) = D λ ( P r − ),the algebra of λ -twisted differential operators on P r − .First, let us give answers to (1) and (6). Theorem 1.1.
The following is true. (1)
The algebra ¯ A λ ( n, r ) has finite global dimension (equivalently, R Γ θλ is an equiva-lence) if and only if λ is not of the form sm , where m n and − rm < s < . (2) For θ > , the abelian localization holds for λ (i.e., Γ θλ is an equivalence) if λ isnot of the form sm , where m n and s < . For θ < , the abelian localizationholds for λ if and only if λ is not of the form sm with m n and s > − rm . In fact part (2) is a straightforward consequence of (1) and results of McGerty andNevins, [MN2].Let us proceed to classification of finite dimensional representations.
Theorem 1.2.
The following holds. (1)
The sheaf ¯ A θλ ( n, r ) has a representation supported on ¯ ρ − (0) if and only if λ = sn with s and n coprime. If that is the case, then the category ¯ A θλ ( n, r ) -mod ρ − (0) isequivalent to Vect . (2) The algebra ¯ A λ ( n, r ) has a finite dimensional representation if and only if λ = sn with s and n coprime and the homological dimension of ¯ A λ ( n, r ) is finite. If thatis the case, then the category ¯ A θλ ( n, r ) -mod ρ − (0) is equivalent to Vect . In fact, (2) is an easy consequence of (1) and Theorem 1.1.Now let us proceed to the description of two-sided ideals (in the finite homologicaldimension case).
Theorem 1.3.
Assume that ¯ A λ ( n, r ) has finite homological dimension and let m standfor the denominator of λ (equal to + ∞ if λ is not rational). Then there are ⌊ n/m ⌋ propertwo-sided ideals in ¯ A λ ( n, r ) , all of them are prime, and they form a chain. TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 5
Finally, let us explain some partial results on a category O for ¯ A θλ ( n, r ), we will recallnecessary definitions below in Subsection 2.7. We use the notation O ( A θλ ( n, r )) for thiscategory. What we need to know now is the following: • The category O ( A θλ ( n, r )) is a highest weight category so it makes sense to speakabout standard objects ∆( p ). • The labeling set for standard objects is naturally identified with the set of r -multipartitions of n . Theorem 1.4.
If the denominator of λ is bigger than n , then the category O ( A θλ ( n, r )) is semisimple. If the denominator of λ equals n , the category O ( A θλ ( n, r )) has only onenontrivial block. That block is equivalent to the nontrivial block of O ( A θ /nr ( nr, . In some cases, we can say which simple objects belong to the nontrivial block, we willdo this below.1.3.
Counting result.
We are going to describe K ( A θλ ( v, w ) -mod ρ − (0) ) (we always con-sider complexified K ) in the case when Q is of affine type, confirming [BL, Conjecture1.1] in this case. The dimension of this K coincides with the number of finite dimen-sional irreducible representations of A λ ( v, w ) provided λ is regular, i.e., the homologicaldimension of A λ ( v, w ) is finite.Recall that, by [Nak1], the homology group H mid ( M θ ( v, w )) (where “mid” stands fordim C M θ ( v, w )) is identified with the weight space L ω [ ν ] of weight ν (see Subsection 1.1)in the irreducible integrable g ( Q )-module L ω with highest weight ω . Further, by [BaGi],we have a natural inclusion K ( A λ ( v, w ) -mod ρ − (0) ) ֒ → H mid ( M θ ( v, w )) given by thecharacteristic cycle map CC λ . We will elaborate on this below in Subsection 2.5. Wewant to describe the image of CC λ .Following [BL], we define a subalgebra a (= a λ ) ⊂ g ( Q ) and an a -submodule L a ω ⊂ L ω .By definition, a is spanned by the Cartan subalgebra t ⊂ g ( Q ) and all root spaces g β ( Q )where β = P i ∈ Q b i α i is a real root with P i ∈ Q b i λ i ∈ Z . For L a ω we take the a -submoduleof L ω generated by the extremal weight spaces (those where the weight is conjugate tothe highest one under the action of the Weyl group). Theorem 1.5.
Let Q be of affine type. The image of K ( A λ ( v, w ) -mod ρ − (0) ) in L ω [ ν ] under CC λ coincides with L a ω ∩ L ω [ ν ] . Content of the paper.
Section 2 contains some known results and construction.In Section 3 we introduce our main tool for inductive study of categories O . In Section 4we will prove Theorems 1.2 (most of it, in fact) 1.4. In Section 5 we prove Theorem 1.1.Finally, in Section 6 we prove Theorem 1.3 and also complete the proof of Theorem 1.5.In the beginning of each section, its content is described in more detail. Acknowledgments . I would like to thank Roman Bezrukavnikov, Dmitry Korb, Dav-esh Maulik, Andrei Okounkov and Nick Proudfoot for stimulating discussions. My workwas supported by the NSF under Grant DMS-1161584.2.
Preliminaries
This section basically contains no new results. We start with discussing conical sym-plectic resolutions. Then, in Subsection 2.2, we list some further properties of Giesekermoduli spaces. Subsection 2.3 describes the symplectic leaves of the varieties M ( v, w ). IVAN LOSEV
After that, we proceed to quantizations. We discuss some further properties, withemphasis on the Gieseker case, in Subsection 2.4. We discuss (derived and abelian)localization theorems for quantized quiver varieties, Subsection 2.5. Then we proceed tothe homological duality and wall-crossing functors, one of our main tools to study therepresentation theory of quantized quiver varieties, Subsection 2.6. In Subsection 2.7, werecall the definition of categories O and list some basic properties. Then, in Subsection2.8, we recall one more important object in this representation theory, Harish-Chandrabimodules. Our main tool to study those is restriction (to so called quantum slices )functors, defined in this context in [BL]. We recall quantum slices in Subsection 2.9 andthe restriction functors in Subsection 2.10.2.1. Symplectic resolutions.
Although in this paper we are primarily interested inthe case of Nakajima quiver varieties for quiver of affine types (and, more specifically,Gieseker moduli spaces) some of our results easily generalize to symplectic resolutions ofsingularities. Here we recall the definition and describe some structural theory of thesevarieties due to Namikawa, [Nam2]. Our exposition follows [BPW, Section 2].Let X be a smooth symplectic algebraic variety. By definition, X is called a symplecticresolution of singularities if C [ X ] is finitely generated and the natural morphism X → X := Spec( C [ X ]) is a resolution of singularities. In this paper we only consider symplecticresolutions X that are projective over X . We also only care about resolutions comingwith additional structure, a C × -action satisfying the following two conditions: • The grading induced by the C × -action on C [ X ] is positive, i.e., C [ X ] = L i > C [ X ] i and C [ X ] = C . • C × rescales the symplectic form ω , more precisely, there is a positive integer d such that t.ω = t d ω for all t ∈ C × .We call X equipped with such a C × -action a conical symplectic resolution .We remark that X admits a universal Poisson deformation, e X , over H ( X ). Thisdeformation comes with a C × -action and the C × -action contracts ˜ X to X , see [L4, 2.2]or [BPW, 2.1] for details. The generic fiber of e X is affine.Namikawa associated a Weyl group W to X that acts on H ( X, R ) as a crystallographicreflection group. We have Pic X = H ( X, Z ). The (closure of the) movable cone of X in H ( X, R ) is a fundamental chamber for W . Furthermore, there are open subset U ⊂ X, U ′ ⊂ X ′ with complements of codimension bigger than 1 that are isomorphic. Sowe get an isomorphism Pic( X ) ∼ = Pic( X ′ ) that preserves the movable cones.Namikawa has shown that there are finitely many isomorphism classes of conical sym-plectic resolutions. Moreover, he proved there is a finite W -invariant union H of hyper-planes in H ( X, R ) with the the following properties: • The union of the complexifications of the hyperplanes in H is precisely the locusin H ( X, C ) over which e X → e X is not an isomorphism. • The closure of the movable cone is the union of some chambers for H . • Each chamber inside a movable cone is the nef cone of exactly one symplecticresolution.For θ ∈ H ( X, R ) \ H , let X θ be the resolution corresponding to the element of W θ lying in the movable cone.
TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 7
We will not need to compute the Namikawa Weyl group. Let us point out that it istrivial provided X has no leaves of codimension 2 (we remark that it is known that thenumber of leaves is always finite).An example of a conical symplectic resolution is provided by M θ ( v, w ) → M ( v, w ) inthe case when Q is an affine quiver ( d = 2 in this case). We always have a natural map P := C Q → H ( M θ ( v, w )) that is always injective. In the case of an affine quiver, thismap can be actually shown to be an isomorphism, but we will not need that: in the quivervariety setting one can retell the constructions above using P instead of H ( M θ ( v, w )).2.2. Gieseker moduli spaces.
We will need some additional facts about varieties M θ ( n, r ).First of all, let us point out that dim M θ ( n, r ) = 2 nr .Let us note that we have an isomorphism M θ ( n, r ) ∼ = M − θ ( n, r ) (of symplectic varietieswith C × -actions). Define R ∨ := End( V ∗ ) ⊕ ⊕ Hom( V ∗ , W ∗ ) ⊕ Hom( W ∗ , V ∗ ). We have anisomorphism R → R ∨ given by ι : ( A, B, i, j ) ( − B ∗ , A ∗ , − j ∗ , i ∗ ), here we write i for anelement in Hom( W, V ) and j for an element of Hom( V, W ). This is a symplectomorphism.Choosing bases in V and W , we identify R with R ∨ . Note that, under this identification, ι is not G -equivariant, we have ι ( g.r ) = ( g t ) − ι ( r ), where the superscript “t” stands forthe matrix transposition. Also note that ( A, B, i, j ) is det-stable (equivalently, there isno nonzero
A, B -stable subspace in ker j ) if and only if ( − B ∗ , A ∗ , − j ∗ , i ∗ ) is det − -stable(i.e., C h B ∗ , A ∗ i im j ∗ = V ∗ ). It follows that M θ ( n, r ) ∼ = M − θ ( n, r ).We will also need some information on the cohomology of M θ ( n, r ). Lemma 2.1.
We have H i ( M θ ( n, r )) = 0 for odd i or for i > nr and dim H ( M θ ( n, r )) =dim H nr − ( M θ ( n, r )) = 1 . In particular, dim H nr − ( M θ ( n, r )) = 1 .Proof. That the odd cohomology groups vanish is [NY, Theorem 3.7,(4)] (or a general factabout symplectic resolutions, see [BPW, Proposition 2.5]). According to [NY, Theorem3.8], we have X i dim H i ( M θ ( n, r )) t i = X λ t P ri =1 ( r | λ ( i ) |− i ( λ ( i ) t ) ) , where the summation is over the set of the r -multipartitions λ = ( λ (1) , . . . , λ ( r ) ). Thehighest power of t in the right hand side is rn −
1, it occurs for a single λ , namely, for λ =(( n ) , ∅ , . . . , ∅ ). This shows dim H nr − ( M θ ( n, r )) = 1. The equality dim H nr − ( M θ ( n, r )) =1 follows. Also there is a single r -multipartition of n with P ri =1 ( r | λ ( i ) | − i ( λ ( i ) t ) ) = 1, itis ( ∅ , . . . , , n − H ( M θ ( n, r )) = 1. (cid:3) It follows, in particular, that the universal deformation of M θ ( n, r ) coincides with the“universal quiver variety” M θ P ( n, r ) := µ − ( g ∗ G ) θ − ss /G . The isomorphism M θ ( n, r ) ∼ = M − θ ( n, r ) extends to M − θ P ( n, r ) ∼ = M θ P ( n, r ) that is, however, not an isomorphism ofschemes over P , but rather induces the multiplication by − M θ ( n, r ) we have an action of GL( r ) × C × induced from the following action on T ∗ R : ( X, t ) . ( A, B, i, j ) = ( tA, t − B, Xi, jX − ). We will need a description of certaintorus fixed points. First, let T denote the maximal torus in GL( r ). Then, see [Nak2,Lemma 3.2, Section 7], we see that(2.1) M θ ( n, r ) T = G n + ... + n r = n r Y i =1 M θ ( n i , , The embedding Q ri =1 M θ ( n i , ֒ → M θ ( n, r ) is induced from L ri =1 T ∗ R ( n i , ֒ → T ∗ R ( n, r ). IVAN LOSEV
Now set ˜ T := T × C × . Then M θ ( n, r ) ˜ T is a finite set that is in a natural bijection withthe set of the r -multipartitions of n , this follows from (2.1) and the classical fact that M θ ( n i , C × is identified with the set of the partitions of n i . More precisely, M θ ( n i , C × = M θ ( n i , C × × C × , where the second copy of C × is contracting. When θ >
0, we label thefixed point corresponding to (
A, B, i, j ) (we automatically have j = 0) by the partitionsof n i into sizes of Jordan blocks of B .2.3. Symplectic leaves.
Here we want to describe the symplectic leaves of M λ ( v, w ) := µ − ( λ ) //G and study the structure of the variety near a symplectic leaf.Let us, first, study the leaf containing 0 ∈ M ( v, w ). Similarly to Subsection 1.2,consider the space ¯ R that is obtained similarly to R but with assigning sl ( V i ) instead of gl ( V i ) to any loop a with t ( a ) = h ( a ) = i so that R = R ⊕ C k , where k is the total numberof loops. Let ¯ M ( v, w ) be the reduction of T ∗ ¯ R so that M ( v, w ) = ¯ M ( v, w ) × C k . Lemma 2.2.
The point is a single leaf of ¯ M ( v, w ) .Proof. It is enough to show that the maximal ideal of 0 in C [ T ∗ ¯ R ] G is Poisson. Since ¯ R does not include the trivial G -module as a direct summand, we see that all homogeneouselements in C [ T ∗ ¯ R ] G have degree 2 or higher. It follows that the bracket of any twohomogeneous elements also has degree 2 or higher and our claim is proved. (cid:3) Now let us describe the slices to symplectic leaves in M λ ( v, w ), see, for example, [BL,2.1.6]. Pick x ∈ M λ ( v, w ). We can view T ∗ R as the representation space of dimension( v,
1) for the double DQ w of the quiver Q w obtained from Q by adjoining the additionalvertex ∞ with w i arrows from i to ∞ . Pick a semisimple representation of DQ w lying over x . This representation decomposes as r ⊕ r ⊗ U ⊕ . . . ⊕ r k ⊗ U k , where r is an irreduciblerepresentation of DQ w with dimension ( v ,
1) and r , . . . , r k are pairwise nonisomorphicirreducible representations of DQ with dimension v , . . . , v k . All representations r , . . . , r k are mapped to λ under the moment map. Consider the quiver Q := Q x with vertices1 , . . . , k and − ( v i , v j ) arrows between vertices i, j with i = j and 1 − ( v i , v i ) loops at thevertex i . We consider the dimension vector v := (dim U i ) ki =1 and the framing w = ( w i ) ki =1 with w i = w · v i − ( v , v i ). Proposition 2.3.
The following is true: (1)
The symplectic leaves of M λ ( v, w ) are parameterized by the decompositions v = v + v v ⊕ . . . ⊕ v k v k (we can permute summands with v i = v j and v i = v j ) subjectto the following conditions: there is an irreducible representation r of DQ w ofdimension ( v , and pairwise different irreducible representations r , . . . , r k of DQ of dimensions v , . . . , v k , all of them mapping to λ under the moment map. (2) The leaf corresponding to the decomposition as above consists of the isomorphismclasses of the representations r ⊕ r ⊗ U ⊕ . . . ⊕ r k ⊗ U k , where r , . . . , r k are asabove. (3) There is a transversal slice to the leaf as above that is isomorphic to the formalneighborhood of 0 in the quiver variety ¯ M ( v, w ) for the quiver Q .Proof. We have a decomposition M λ ( v, w ) ∧ x ∼ = D × ¯ M ( v, w ) ∧ of Poisson formal schemes,where D stands for the symplectic formal disk and • ∧ x indicates the formal neighborhoodof x . From Lemma 2.2 it now follows that the locus described in (2) is a union of leaves.So in order to prove the entire proposition, it remains to show that the locus in (2) isirreducible. This follows from [CB, Theorem 1.2]. (cid:3) TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 9
Now assume that X → X is a symplectic resolution (not necessarily conical). Then X has finitely many symplectic leaves. Pick a point x ∈ X and consider its formalneighborhood X ∧ x . Then, according to Kaledin, [K], the Poisson formal scheme X ∧ x decomposes into the product of two formal schemes: the symplectic formal disk D , and a“slice” X ′ that is a Poisson formal scheme, where x is a single symplectic leaf.2.4. Quantizations.
Let X = X θ be a conical symplectic resolution corresponding toa parameter θ . Now let us consider quantizations of X . We will work with microlocalquantizations. Those are sheaves A θ of algebras in conical topology equipped with thefollowing additional structures: • a complete and separated ascending Z -filtration, A θ = S i ∈ Z A θ i , • an action of Z /d Z (where d has the same meaning as above) on A θ by filteredalgebra automorphisms such that 1 ∈ Z /d Z acts on A θ i / A θ i − by exp(2 πi √− /d ). • an isomorphism gr A θ ∼ = O X of sheaves of graded algebras.Consider the subsheaf R ~ ( A θ ) of Z /d Z -invariants in the Rees sheaf R ~ /d ( A ). Complet-ing R ~ ( A ) with respect to the ~ -adic topology, we get a homogeneous quantization of X in the sense of [L4, 2.3]. To get back we take C × -finite sections and mod out ~ −
1. It fol-lows that the microlocal quantizations of X are canonically parameterized by H ( X, C ).We write A θ ˆ λ for the quantization corresponding to ˆ λ ∈ H ( X ) (and we call ˆ λ the period of the quantization A θ ˆ λ ). In fact, we can also quantize the universal deformation ˜ X by amicrolocal sheaf ˜ A θ of C [ H ( X )]-algebras (the canonical quantization from [BK, L4]). Weremark that A θ,opp ˆ λ ∼ = A θ − ˆ λ , as sheaves of algebras on X , this follows from the definition ofa canonical quantization.We write A ˆ λ , ˜ A for the global sections of A θ ˆ λ , ˜ A θ , these algebras are independent of θ by [BPW, 3.3].When X is a quiver variety M θ ( v, w ), the quantization A θλ ( v, w ) satisfies the assump-tions above. As we have mentioned in Subsection 2.1, we can embed P = C Q into H ( X ).We remark however, that A θ ˆ λ = A θ ˆ λ − ̺ ( v, w ), where ̺ is half the character of the action of G on V top R ∗ , see, e.g., [BL, 2.2].Let us now consider the Gieseker case. Here ̺ = r/
2. So we have(2.2) A λ ( n, r ) opp ∼ = A − λ − r ( n, r ) . Lemma 2.4.
We have A λ ( n, r ) ∼ = A − λ − r ( n, r ) .Proof. Recall that A λ ( n, r ) ∼ −→ Γ( A ± θλ ( n, r )). Also recall the identification M θ P ( n, r ) →M − θ P ( n, r ) that induces − P . It follows that Γ( A θ ˆ λ ) ∼ = Γ( A − θ − ˆ λ ). Since A θ ˆ λ ∼ = A θ ˆ λ − r/ ( n, r ), our claim follows. (cid:3) We conclude that A λ ( n, r ) opp ∼ = A λ ( n, r ).We remark that the isomorphism of Lemma 2.4 is similar in spirit to isomorphismsfrom [BPW, Section 3] provided by the Namikawa Weyl group action. We would like topoint out however that our isomorphism does not reduce to that. Indeed, when r > M ( n, r ). Localization theorems.
We assume that X is a conical symplectic resolution of X := Spec( C [ X ]). Recall that we write ρ : X → X for the canonical morphism. Let A θ be a quantization of X and A be its algebra of global sections.Consider the categories of modules A -Mod ⊃ A -mod consisting of all and of finitelygenerated A -modules. Also consider the category A θ -Mod of all quasi-coherent A θ -modules and A θ -mod of all coherent A θ -modules, i.e., modules that have a global goodfiltration (a filtration is called good if the associated graded object is a coherent sheaf of O X -modules).We have the global section, Γ θ , and localization, Loc θ := A θ ⊗ A • , functorsΓ θ : A θ -Mod ⇄ A -Mod : Loc θ , Γ θ : A θ -mod ⇄ A -mod : Loc θ . For objects in A θ -mod , A -mod we can define supports, those are closed C × -stable sub-varieties in X, X , respectively. For a subvariety Y ⊂ X , we write A -mod Y for thefull subcategory of A -mod consisting of all modules supported inside Y . Similarly, for Y ⊂ X , we consider the subcategory A θ -mod Y . The functors Γ θ , Loc θ restrict to functorsbetween the subcategories A θ -mod ρ − ( Y ) , A -mod Y .The functors Γ θ , Loc θ admit derived functors R Γ θ : D b ( A θ -Mod) → D b ( A -Mod)(given by taking the ˇCech complex for a cover by affine open subsets) and L Loc θ : D − ( A -Mod) → D − ( A θ -Mod). If the homological dimension of A is finite, we also have L Loc θ : D b ( A -Mod) → D b ( A θ -Mod). Clearly, the functors R Γ θ , L Loc θ preserve thesubcategories D ? ( A θ -mod) , D ? ( A -mod) (as in the case of usual coherent sheaves, the for-mer is identified with the full subcategory in D ? ( A -Mod) with coherent homology). Alsothe functors R Γ θ , L Loc θ restrict to functors between the subcategories D ? Y ( A -mod) and D ? ρ − ( Y ) ( A θ -mod), consisting of all complexes with homology supported on Y , ρ − ( Y ).Now let us suppose that X := M θ ( v, w ) and so X = M ( v, w ). We will write Γ θλ , R Γ θλ to indicate the dependence on the quantization parameter λ .Let us recall some results on when (i.e., for which λ ) the functors Γ θλ , Loc θλ are derivedor abelian equivalences. Proposition 2.5 ([MN1]) . The functor R Γ θλ : D b ( A θλ ( v, w ) -mod) → D b ( A λ ( v, w ) -mod) is an equivalence of triangulated categories if and only if A λ ( v, w ) has finite homologicaldimension. In this case, the inverse equivalence is given by L Loc θλ . In the situation of the previous proposition, we say that the derived localization holds(for λ ), such parameters are called regular . [BL, Conjecture 9.1] describes a precise locusof the singular (=non-regular) parameters λ and we prove this conjecture, Theorem 1.1.We say that the abelian localization holds for ( λ, θ ) if Γ θλ is an equivalence of abeliancategories. The following result was proved in [BPW, Corollary 5.12] for arbitrary sym-plectic resolutions. Proposition 2.6.
For any λ there is k ∈ Z > such that the abelian localization holds for ( λ + kθ, θ ) whenever k > k . There are also results of McGerty and Nevins, [MN2], that provide a sufficient conditionfor the functor Γ θλ to be exact. We will elaborate on these results applied to the specialcase of A λ ( n, r ) below. We will see that for A λ ( n, r ) this sufficient condition is alsonecessary.To conclude this section let us mention the characteristic cycle map. Suppose we arein the general case of a projective symplectic resolution X → X . Then to a module TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 11 M ′ ∈ A θ -mod ρ − (0) we can assign its characteristic cycle. By definition, it coincides withthe sum of irreducible components of ρ − (0) with multiplicities, where the multiplicityof the component equals to the generic rank of gr M ′ on the component. Of course, thecharacteristic cycle defines a group homomorphism CC θ : K ( A θ -mod ρ − (0) ) → H mid ( X ).Now to a module M ∈ A -mod fin we can assign CC θ (Loc θλ M ). It was shown in [BL, 3.2]that, in the quiver variety case, this map is actually independent of θ . Here is anotherresult that will be of crucial importance for us. This was proved in an unpublished workof Baranovsky and Ginzburg. Proposition 2.7.
The map CC θ is injective. Duality and wall-crossing functor.
We still consider a conical projective resolu-tion X of X . Let us take a quantization A θ of X that satisfies the abelian localizationtheorem. In this case the homological dimension of A (equivalently, of A θ ) does notexceed the homological dimension of O X equal to dim X .It turns out that dimension of support for certain modules can be computed via a suit-able functor: the homological duality. Namely, recall the functor D := RHom A ( • , A )[ N ]where N = dim X , defines an equivalence D b ( A -mod) → D b ( A opp -mod) opp . Moreover,for a simple object L in A -mod we have H i ( DL ) = 0 if i > N or i < N − dim Supp L ,see [BL, 4.2]. In particular, L is finite dimensional if and only if H i ( DL ) = 0 for i < N .In the case when A = A ˆ λ , we can view D as a functor D b ( A ˆ λ -mod) → D b ( A − ˆ λ -mod) opp .We will need a technical property of D . Lemma 2.8.
Let L be a simple A ˆ λ -module with dim Supp L = dim X . Then H i ( DL ) is supported on the complement of the open symplectic leaf of X for i > .Proof. As we know from Commutative Algebra, the irreducible components of the sup-port of Ext i (gr L, C [ X ]) intersecting the open leaf have dimension smaller than dim X provided i > dim X . Thanks to a standard spectral sequence for the homology of afiltered quotient, we see that the irreducible components of Supp H i ( DL ) for i > dim X . However, no nonzero A − ˆ λ -modulecan have this property as the support of any A − ˆ λ -module is a coisotropic subvariety byGabber’s theorem. (cid:3) Now let us consider a different family of functors: wall-crossing functors. Below wewill recall a connection of some of those functors with the duality introduce above thatwas discovered in [BL, Section 4]. We will make some additional assumptions. Let usassume that the all conical projective symplectic resolutions of X are strictly semismall.Recall that ρ : X → X is called strictly semismall, if all components of X i := { x ∈ X | dim ρ − ( x ) = i } have codimension 2 i and all components of ρ − ( x ) have the samedimension.Pick χ ∈ Pic( X ). We can uniquely quantize the corresponding line bundle O ( χ ) on X to a A θ ˆ λ + χ - A θ ˆ λ -bimodule to be denoted by A θ ˆ λ,χ . Let A ( θ )ˆ λ,χ denote the global sections.We remark that taking the tensor product with A θ ˆ λ,χ gives rise to an equivalence T θ ˆ λ,χ : A θ ˆ λ -Mod → A θ ˆ λ + χ -Mod.Pick a different stability condition θ ′ and ˆ λ ′ ∈ ˆ λ + Z Q such that the abelian localizationholds for (ˆ λ ′ , θ ′ ). We consider the wall-crossing functor WC ˆ λ → ˆ λ ′ := Γ θ ′ ˆ λ ′ ◦ T θ ′ ˆ λ, ˆ λ ′ − ˆ λ ◦ L Loc θ ′ ˆ λ : D b ( A ˆ λ -mod) → D b ( A ˆ λ ′ -mod) . Here we write Γ θ ′ ˆ λ ′ for the global section functor A θ ′ ˆ λ ′ -Mod → A ˆ λ ′ -Mod. According to[BPW, Proposition 6.29], we have WC ˆ λ → ˆ λ ′ = A ( θ ′ )ˆ λ, ˆ λ ′ − ˆ λ ⊗ L A ˆ λ • . In the case of quiver varieties,we will write WC λ → λ ′ for a functor D b ( A λ ( v, w ) -Mod) → D b ( A λ ′ ( v, w ) -Mod).By a long wall-crossing functor we mean WC ˆ λ → ˆ λ ′ , where (ˆ λ, θ ) , (ˆ λ ′ , − θ ) satisfy theabelian localization. A connection between the contravariant duality and the long wall-crossing functor is as follows. We say that an A ˆ λ -module M is strongly holonomic if everynonempty intersection of Supp M with a symplectic leaf in X is lagrangian in that leaf.Thanks to the assumption that X → X is strictly semismall, this is equivalent to thecondition that ρ − (Supp M ) ⊂ X is lagrangian. Set ˆ λ − := ˆ λ − nθ for sufficiently large n .The choice of n guarantees that the abelian localization holds for (ˆ λ − , − θ ).Consider the subcategory D bshol ( A ˆ λ -mod) ⊂ D b ( A ˆ λ -mod) of all complexes with stronglyequivariant homology. It is easy to see that D restricts to an equivalence D bshol ( A ˆ λ -mod) ∼ −→ D bshol ( A − ˆ λ -mod) opp . On the other hand, WC ˆ λ → ˆ λ − restricts to an equivalence D bshol ( A ˆ λ -mod) ∼ −→ D bshol ( A ˆ λ − -mod). The following result was proved in [BL, Section 4]. Proposition 2.9.
There is an equivalence ι : D bshol ( A ˆ λ − -mod) ∼ −→ D bshol ( A − ˆ λ -mod) opp preserving the natural t -structures such that ι ◦ WC ˆ λ → ˆ λ − = D . We will need a corollary of this proposition.
Corollary 2.10.
Let M be a simple strongly holonomic A -module. The following areequivalent: (1) dim Supp M < dim X . (2) M is annihilated by a proper ideal of A .Proof. Let us note that the associated graded of a proper ideal in A is a Poisson ideal in C [ X ]. So its associated variety does not intersect the open leaf in X . Obviously, thesupport of M is contained in that associated variety. Since M is strongly holonomic, theimplication (2) ⇒ (1) follows.Let us prove (1) ⇒ (2). Consider the A = A ˆ λ -bimodule D := A ( θ )ˆ λ − ,Nθ ⊗ A ˆ λ − A ( − θ )ˆ λ, − Nθ .Obviously, H ( WC λ − → λ ◦ WC λ → λ − M ) = D ⊗ A ˆ λ M . There is a natural homomorphism D → A ˆ λ (compare to [BL, (5.15)]) that becomes an isomorphism after microlocalizationto the open leaf of X because ρ is an isomorphism over the open leaf. So the imageis a nonzero ideal in A ˆ λ , say J . If M is not annihilated by that ideal, we see that H ( WC λ → λ − M ) = 0. It follows that dim Supp M = dim X . (1) ⇒ (2) is proved. (cid:3) We will also need a straightforward corollary of Lemma 2.8 for strongly holonomicmodules.
Corollary 2.11.
Let L be a simple strongly holonomic A ˆ λ -module. Then dim Supp H i ( DL ) < dim X provided i > . Categories O . Basically, all results of this section can be found in [BLPW].First of all, let A be an associative algebra equipped with a rational action α of C × byalgebra automorphism. Then we can consider the eigendecomposition A = L i ∈ Z A i andset A > := L i > A i , A > := L i> A i and C α ( A ) := A > / ( A > ∩ AA > ). We remark that C α ( A ) is an algebra because AA > ∩ A > is a two-sided ideal in A > . We remark that C α is a functor from the category of algebras equipped with a C × -action to the categoryof algebras, we call it the Cartan functor . This name is justified by the observation that
TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 13 if A = U ( g ) for a semisimple Lie algebra g and α comes from a regular one-parametricsubgroup of Ad( g ), then C α ( A ) is the universal enveloping algebra of the correspondingCartan subalgebra.Let X be a symplectic resolution of X and A θ be a quantization of X . We assumethat that X is equipped with compatible Hamiltonian C × -action α . We require that theaction on X commutes with the contracting C × -action. It is not difficult to see that α lifts to a Hamiltonian C × -action on A θ again denoted by α . This action preserves thefiltration and the Z /d Z -grading. Let h α ∈ A denote the image of 1 under the quantumcomoment map for α .Consider the category O ( A ) consisting of all modules with locally finite action of h α , A > . The action of A > is automatically locally nilpotent. If α ( C × ) has finitely manyfixed points, this definition coincides with the definition of the category O a from [BLPW,3.2] (this because the algebra C α ( A ) is finite dimensional, which is proved analogously to[GL, 3.1.4]).The category O ( A ) has analogs of Verma modules. More precisely, there is an inductionfunctor C α ( A ) -mod → O ( A ) , M ∆( M ) := A ⊗ A > M . By a Verma module, wemean ∆( M ) with simple M .Now let us consider the case when α ( C × ) has finitely many fixed points. For ˆ λ ∈ H ( X )lying in a Zariski open subset, the algebra C α ( A ˆ λ ) is naturally isomorphic to C [ X α ( C × ) ],see [BLPW, 5.1]. In this case, for p ∈ X α ( C × ) , we will write ∆ ˆ λ ( p ) for the correspondingVerma module.Now let us define the category O for A θ following [BLPW, 3.3]. Let Y stand for thecontracting locus for α , i.e, the subvariety of all points x ∈ X such that lim t → α ( t ) x exists (and automatically lies in X α ( C × ) ). We remark that Y is a lagrangian subvariety in X stable with respect to the contracting C × -action. We also remark that Y = ρ − ( Y ),where Y stands for the contracting locus of the C × -action on X induced by α . This isbecause, under our assumptions on X α ( C × ) , the fixed point set X α ( C × )0 is a single pointand because ρ is proper. We remark that if X is strictly semismall, then every module in O ( A ) is strongly holonomic. This is because Y = ρ − ( Y ) is lagrangian.By definition, the category O ( A θ ) consists of all modules M ∈ A θ -mod Y that admit aglobal good h α -stable filtration.We write D b O ( A ˆ λ ) , D b O ( A θ ˆ λ ) for the categories of all complexes (in the correspondingderived categories) with homology in the categories O .Let us summarize some properties of categories O ( A θ ) , O ( A ). Proposition 2.12.
Assume that the action α has finitely many fixed points. (1) We have Γ θ ( O ( A θ )) ⊂ O ( A ) , Loc θ ( O ( A )) ⊂ O ( A θ ) , R Γ θ ( D b O ( A θ -mod)) ⊂ D b O ( A -mod) ,L Loc θ ( D b O ( A -mod)) ⊂ D b O ( A θ -mod) . (2) The functor A θ ˆ λ,χ ⊗ A θ ˆ λ • maps O ( A θ ˆ λ ) to O ( A θ ˆ λ + χ ) . (3) The categories O ( A ) , O ( A θ ) are length categories, i.e., all objects have finite length. (4) O ( A ) ⊂ A -mod , O ( A θ ) ⊂ A θ -mod are Serre subcategories. (5) All modules in O ( A ) , O ( A θ ) can be made weakly α ( C × ) -equivariant. (6) Conversely, all weakly α ( C × ) -equivariant modules in A -mod Y (resp., A θ -mod Y )are in O ( A ) (resp., in O ( A θ ) ).Proof. (1)-(4) were established in [BLPW, Section 3]. The proof of (5) for O ( A ) is stan-dard: we decompose a module in O ( A ) into the direct sum of submodules according to the class of eigenvalues of h α modulo Z . It is easy to introduce a weakly equivariantstructure on each summand.Since the localization functor is α ( C × )-equivariant, we see that Loc θ ( M ) can be madeweakly α ( C × )-equivariant. So (5) for O ( A θ ˆ λ ) is true provided the abelian localizationholds for ˆ λ . So it also holds for ˆ λ + χ for any integral χ . This is because of (2) and theobservation that A θ ˆ λ,χ is α ( C × )-equivariant. Now our claim follows from Proposition 2.6.Let us prove (6). Again, thanks to the equivariance of the localization functor, it isenough to prove the claim for A . Pick a weakly α ( C × )-equivariant module M ∈ A -mod Y and let M = L i ∈ Z M i be the eigen-decomposition for the α ( C × )-action. Since M issupported on Y , we see that M i = 0 for i ≫
0. Since M is finitely generated, it is easyto see that all weight spaces are finite dimensional. Our claim follows. (cid:3) Now let us discuss highest weight structure on O ( A θ ). Consider the so called geometricorder on X α ( C × ) defined as follows. For p ∈ X α ( C × ) set Y p := { x ∈ X | lim t → α ( t ) x = p } ,the contracting locus of p so that we have Y = F p ∈ X α ( C × ) Y p . We consider the relation θ on X α ( C × ) that is the transitive order of the pre-order p θ p ′ if p ∈ Y p ′ . We write Y p ′ for F p θ p ′ Y p and Y
The natural functor D b ( O ( A θ ˆ λ )) → D b O ( A θ ˆ λ -mod) is an equivalence. Fur-thermore, if ˆ λ ∈ H ( X ) lies in a suitable Zariski open subset and the abelian localizationholds for (ˆ λ, θ ) , then D b ( O ( A ˆ λ )) → D b O ( A θ ˆ λ ) is an equivalence. We are going to identify D b ( O ( A θ ˆ λ )) with D b O ( A θ ˆ λ -mod) and D b ( O ( A ˆ λ )) with D b O ( A ˆ λ ).In the case of A λ ( n, r ) we consider the categories O defined for the action of a genericone-dimensional subtorus in GL( r ) × C × . Then the fixed point are in one-to-one corre-spondence with the r -multipartitions of n . Different choices of a generic torus may giverise to different categories O .2.8. Harish-Chandra bimodules.
Let us recall some basics on Harish-Chandra (shortly,HC) bimodules. Let A θ , A ′ θ be two quantizations of X in the sense of Subsection 2.4 and A , A ′ be their global sections. For technical reasons, we make a restriction on the con-tracting C × -action β on X . Namely, we require that there are commuting actions β and γ of C × on X with the following properties: • β = β ′ d γ , • and γ is Hamiltonian.Then γ lifts to A θ , A ′ θ and these sheaves acquire new filtrations, coming from β . In theremainder of the section we consider A , A ′ with these new filtrations. Let us point outthat X = M θ ( v, w ) does satisfy our additional condition: we can take β induced by theaction t. ( r, α ) = ( r, t − α ). TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 15
Let us recall the definition of a HC A - A ′ -bimodule. By definition, this is a finitelygenerated A - A ′ -bimodule B with a filtration that is compatible with those on A , A ′ suchthat gr B is a C [ X ]-module. By a homomorphism of Harish-Chandra bimodules wemean a bimodule homomorphism. The category of HC A ′ - A -bimodules is denoted byHC( A ′ - A ). We also consider the full subcategory D bHC ( A ′ − A ) of the derived categoryof A ′ - A -bimodules with Harish-Chandra homology.By the associated variety of a HC bimodule B (denoted by V( B )) we mean the supportin X of the coherent sheaf gr B , where the associated graded is taken with respect to afiltration as in the previous paragraph (below we call such filtrations good ). It is easy tosee that gr B is a Poisson C [ X ]-module so V( B ) is the union of symplectic leaves.Using associated varieties and the finiteness of the number of the leaves it is easy toprove the following standard result. Lemma 2.14.
Any HC bimodule has finite length.
For B ∈ HC( A ′ - A ) and B ∈ HC( A ′′ - A ′ ) we can take their tensor product B ⊗ A ′ B .This is easily seen to be a HC A ′′ - A -bimodule. Also the derived tensor product of theobjects from D bHC ( A ′′ - A ′ ) , D bHC ( A ′ - A ) lies in D − HC ( A ′′ - A ) (and in D bHC ( A ′′ - A ) provided A ′ has finite homological dimension).2.9. Quantum slices.
Let X → X , where the contracting C × -action satisfies the addi-tional assumptions imposed in the previous subsection. Pick a quantization A θ of X . Wewrite A θ ~ for the ~ -adic completion of the Rees sheaf R ~ ( A θ ) of A θ (with respect to theaction β so that t. ~ = t ~ ). We write A ~ for the algebra of the global sections of A θ ~ , thisis the ~ -adic completion of the Rees algebra of A .Pick a point x ∈ X . Then we can form the completions A ∧ x ~ of A ~ at x and A θ ∧ x ~ of A ∧ x ~ at ρ − ( x ). Consider the homogenized Weyl algebra A ~ for the tangent space tothe symplectic leaf in x . Then we have an embedding A ∧ ~ ֒ → A ∧ x ~ , see [L5, 2.1] fora proof. It was checked in [L5, 2.1] that we have the tensor product decomposition A ∧ x ~ = A ∧ ~ b ⊗ C [[ ~ ]] A ′ ~ that lifts the decomposition X ∧ x ∼ = D × X ′ mentioned in Subsection2.3. The algebra A ′ ~ is independent of the choices up to an isomorphism, as was explainedin [L5, 2.1]. For a similar reason, we have a decomposition A θ ∧ x ~ ∼ = A ∧ ~ b ⊗ C [[ ~ ]] A θ ~ ′ , where A ′ θ ~ is a formal quantization of the slice X ′ . By the construction, A ′ ~ = Γ( A ′ θ ~ ).Now suppose that A θ has period ˆ λ . Then the period of A θ ∧ x ~ coincides with the imageˆ λ ′ of ˆ λ under the natural map between the ˇCech-De Rham cohomology groups H ( X ) → H ( X ∧ x ) = H ( X ′ ). It follows that A ′ θ ~ also has period ˆ λ ′ .Assume that X ′ is again equipped with a contracting C × action with the same integer d satisfying the additional restriction in Subsection 2.8, this holds in the quiver varietysetting, for example. So X ′ is the formal neighborhood at 0 of a conical symplecticresolution X . The formal quantization A ′ θ ~ is homogeneous (by the results of [L4, 2.3]).It follows that it is obtained by completion at 0 of A θ ~ for some quantization A θ of X .So the product A θ ∧ x ~ = A ∧ ~ b ⊗ C [[ ~ ]] A ′ θ ~ comes equipped with a C × -action by algebraautomorphisms satisfying t. ~ = t ~ . On the other hand, the C × -action on A θ ~ produces aderivation of A θ ∧ x ~ . The difference between this derivation and the one produced by the C × -action on A θ ∧ x ~ has the form ~ [ a, · ] for some a ∈ A ∧ x ~ , see [BL, Lemma 5.7].Let us now elaborate on the Gieseker case, which was already considered (in a moregeneral quiver variety case) in [BL, 5.4]. Recall that the symplectic leaves in ¯ M ( n, r ) areparameterized by partitions ( n , . . . , n k ) with n + . . . + n k n . For x in the corresponding leaf, we have ¯ A λ ( n, r ) = ¯ A λ ( n , r ) ⊗ ¯ A λ ( n , r ) ⊗ . . . ⊗ ¯ A λ ( n k , r ) . Also we have a similar decomposition for ¯ A θλ ( n, r ).2.10. Restriction functors for HC bimodules.
In this subsection, we will recall re-striction functors • † ,x : HC( A ′ - A ) → HC( A ′ - A ), where A ′ , A are slice algebras for A ′ , A ,respectively (it is in order for these functors to behave nicely that we have introducedour additional technical assumption on the contracting C × -action). These functors weredefined in [BL, Section 5] in the case when A ′ , A are of the form A λ ( v, w ), but the gen-eral case (under the assumption that the slice to x is conical and the contracting actionsatisfies the additional assumption) is absolutely analogous. We will need several factsabout the restriction functors established in [BL, Section 5]. Proposition 2.15.
The following is true. (1)
The functor • † ,x is exact and H ( X ) -linear. (2) The associated variety V( B † ,x ) is uniquely characterized by the property D × V( B † ,x ) ∧ =V( B ) ∧ x . (3) The functor • † ,x intertwines the Tor’s: for B ∈ HC( A ′′ - A ′ ) and B ∈ HC( A ′ - A ) we have a natural isomorphism Tor A ′ i ( B , B ) † ,x = Tor A ′ i ( B † ,x , B † ,x ) . Parabolic induction
The first goal of this section is to elaborate on the Cartan functor that appeared inSubsection 2.7. There we were basically dealing with the case when the Hamiltonianaction only has finitely many fixed points. Here we consider a more general case andour goal is to better understand the structure of C α ( A ). The second goal is to introduceparabolic induction for categories O .Not surprisingly, the case of actions on smooth symplectic (even non-affine) varieties iseasier to understand. We extend the definition of C α to sheaves in Subsection 3.1. Therewe show that if A θ is a quantization of X , then C α ( A θ ) is a quantization of X α ( C × ) . InSubsection 3.2 we compare C α ( A ) (an algebra which is hard to understand directly) withΓ( C α ( A θ )) in the case of symplectic resolutions. We will see that, for a Zariski genericquantization parameter, the two algebras coincide. Next, in Subsection 3.3, we determinethe quantization parameter (=period) of C α ( A θ ) from that of A θ . Subsection 3.4 appliesthis result to some particular action α in the Gieseker case.Finally, in Subsection 3.5 we introduce parabolic induction.3.1. Cartan functor for sheaves.
We start with a symplectic variety X equipped witha C × -action that rescales the symplectic form and also with a commuting Hamiltonianaction α . Of course, it still makes sense to speak about quantizations of X that areHamiltonian for α . We want to construct a quantization C α ( A θ ) of X α ( C × ) starting froma Hamiltonian quantization A θ of X .The variety X can be covered by ( C × ) -stable open affine subvarieties. Pick such asubvariety X ′ with ( X ′ ) α ( C × ) = ∅ . Define C α ( A θ )( X ′ ) as C α ( A θ ( X ′ )). We remark thatthe open subsets of the form ( X ′ ) α ( C × ) form a base of the Zariski topology on X α ( C × ) .The following proposition defines the sheaf C α ( A θ ). Proposition 3.1.
The following holds.
TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 17 (1)
Suppose that the contracting α -locus in X ′ is a complete intersection defined by ho-mogeneous (for α ( C × ) ) equations of positive weight. Then the algebra C α ( A θ ( X ′ )) is a quantization of C [ X ′ α ( C × ) ] . (2) There is a unique sheaf C α ( A θ ) of X α ( C × ) whose sections on X ′ α ( C × ) with X ′ asabove coincide with C α ( A θ ( X ′ )) . This sheaf is a quantization of X α ( C × ) . (3) If X ′ is a ( C × ) -stable affine subvariety, then C α ( A θ )( X ′ ) = C α ( A θ ( X ′ )) .Proof. Let us prove (1). To simplify the notation, we write A for A θ ( X ′ ). The algebra A is Noetherian because it is complete and separated with respect to a filtration whoseassociated graded is Noetherian. Let us show that gr AA > = C [ X ′ ] C [ X ′ ] > , this willcomplete the proof of (1).In the proof it is more convenient to deal with ~ -adically completed homogenized quan-tizations. Namely, let A ~ stand for the ~ -adic completion of R ~ ( A ). The claim thatgr AA > = C [ X ′ ] C [ X ′ ] > is equivalent to the condition that A ~ A ~ ,> is ~ -saturated mean-ing that ~ a ∈ A ~ A ~ ,> implies that a ∈ A ~ A ~ ,> .Recall that we assume that there are α -homogeneous elements f , . . . , f k ∈ C [ X ′ ] > thatform a regular sequence generating the ideal C [ X ′ ] C [ X ′ ] > . We can lift those elementsto homogeneous ˜ f , . . . , ˜ f k ∈ A ~ ,> . We claim that these elements still generate A ~ A ~ ,> .Indeed, it is enough to check that A ~ ,> ⊂ Span A ~ ( ˜ f , . . . , ˜ f k ). For a homogeneous element f ∈ A ~ ,> \ ~ A ~ we can find homogeneous elements g , . . . , g k such that f − P ki =1 g i ˜ f i stillhas the same α ( C × )-weight and is divisible by ~ . Divide by ~ and repeat the argument.Since the ~ -adic topology is complete and separated, we see that f ∈ Span A ~ ( ˜ f , . . . , ˜ f k ).So it is enough to check that Span A ~ ( ˜ f , . . . , ˜ f k ) is ~ -saturated.Pick elements ˜ h , . . . , ˜ h k such that P kj =1 ˜ h j ˜ f j is divisible by ~ . Let h j ∈ C [ X ′ ] becongruent to ˜ h j modulo ~ so that P kj =1 h j f j = 0. Since f , . . . , f k form a regular sequence,we see that there are elements h ij ∈ C [ X ′ ] such that h j ′ j = − h jj ′ and h j = P kℓ =1 h jℓ f ℓ . Liftthe elements h jj ′ to ˜ h jj ′ ∈ A ~ with ˜ h jj ′ = − ˜ h j ′ j . So we have ˜ h j = P kℓ =1 ˜ h jℓ ˜ f ℓ + ~ ˜ h ′ j for some˜ h ′ j ∈ A ~ . It follows that P kj =1 ˜ h j ˜ f j = ~ P kj =1 ˜ h ′ j ˜ f j + P kj,ℓ =1 ˜ h jℓ ˜ f ℓ ˜ f j . But P kj,ℓ =1 ˜ h jℓ ˜ f ℓ ˜ f j = P j<ℓ ˜ h jℓ [ ˜ f ℓ , ˜ f j ]. The bracket is divisible by ~ . But ~ [ ˜ f ℓ , ˜ f j ] is still in A ~ ,> and so inSpan A ~ ( ˜ f , . . . , ˜ f k ). This finishes the proof of (1).Let us proceed to the proof of (2). Let us show that we can choose a covering of X α ( C × ) by X ′ α ( C × ) , where X ′ is as in (1). This is easily reduced to the affine case. Herethe existence of such a covering is deduced from the Luna slice theorem applied to afixed point for α . In more detail, for a fixed point x , we can choose an open affineneighborhood U of x in X//α ( C × ) with an ´etale morphism U → T x X//α ( C × ) such that π − ( U ) ∼ = U × T x X//α ( C × ) T x X , where π stands for the quotient morphism for the action α .The subset π − ( U ) then obviously satisfies the requirements in (1).It is easy to see that the algebras C α ( A θ ( X ′ )) form a presheaf with respect to thecovering X ′ α ( C × ) (obviously, if X ′ , X ′′ satisfy our assumptions, then their intersectiondoes). Since the subsets X ′ α ( C × ) form a base of topology on X α ( C × ) , it is enough to showthat they form a sheaf with respect to the covering. This is easily deduced from the twostraightforward claims: • C α ( A θ ( X ′ )) is complete and separated with respect to the filtration (here we usean easy claim that, being finitely generated, the ideal A θ ( X ′ ) A θ ( X ′ ) > is closed). • The algebras gr C α ( A θ ( X ′ )) = C [ X ′ α ( C × ) ] do form a sheaf – the structure sheaf O X α ( C × ) .The proof of (2) is now complete.To prove (3) it is enough to assume that X is affine. Let π denote the categoricalquotient map X → X//α ( C × ). It is easy to see that, for every open ( C × ) -stable affinesubvariety X ′ that intersects X α ( C × ) non-trivially, and any point x ∈ X ′ α ( C × ) , there issome C × -stable open affine subvariety Z ⊂ X//α ( C × ) with x ∈ π − ( Z ) ⊂ X ′ . So we canassume, in addition, that all covering affine subsets X i are of the form π − (?). Moreover,we can assume that they are all principal (and so are given by non-vanishing of α ( C × )-invariant and C × -semiinvariant elements of A θ ( X )). Then all algebras C α ( A θ ( X ′ )) areobtained from C α ( A θ ( X )) by microlocalization. Our claim follows from standard proper-ties of microlocalization. (cid:3) Comparison between algebra and sheaf levels.
Now let us suppose that X is aconical symplectic resolution of X . We write A θλ for the quantization of X correspondingto λ and A λ for its algebra of global sections. By the construction, for any λ ∈ H ( X ),there is a natural homomorphism C α ( A λ ) → Γ( C α ( A θλ )). Our goal in this section is toprove the following result. Proposition 3.2.
Suppose that H i ( X α ( C × ) , O ) = 0 for i > . There is a Zariski opensubset subset Z ⊂ H ( X ) such that the homomorphism C α ( A λ ) → Γ( C α ( A θλ )) is anisomorphism provided λ ∈ Z .Proof. Let ˜ X be the universal deformation of X over H ( X ) and ˜ X be its affinization.Consider the natural homomorphism C α ( C [ ˜ X ]) → C [ ˜ X α ( C × ) ]. It is an isomorphismoutside of H C (the union of singular hyperplanes) since ˜ X → ˜ X is an isomorphismprecisely outside that locus. Now consider the canonical quantization ˜ A θ of ˜ X . Similarlyto the previous section, C α ( ˜ A θ ) is a quantization of ˜ X α ( C × ) . The cohomology vanishingfor X α ( C × ) implies that for ˜ X α ( C × ) . It follows that gr Γ( C α ( ˜ A θ )) = C [ ˜ X α ( C × ) ]. Alsothere is a natural epimorphism C α ( C [ ˜ X ]) → gr C α ( ˜ A ) and a natural homomorphismgr C α ( ˜ A ) → gr Γ( C α ( ˜ A θ )). The resulting homomorphism gr C α ( ˜ A ) → gr Γ( C α ( ˜ A θ )) is, onone hand, the associated graded of the homomorphism C α ( ˜ A ) → Γ( C α ( ˜ A θ )) and on theother hand, an isomorphism over the complement of H C . We deduce that the supportsof the associated graded modules of the kernel and the cokernel of C α ( ˜ A ) → Γ( C α ( ˜ A θ ))are supported on H C as C [ H ( X )]-modules. It follows that the support of the kernel andof the cokernel of C α ( ˜ A ) → Γ( C α ( ˜ A θ )) are Zariski closed subvarieties of H ( X ). We notethat Γ( C α ( ˜ A θ )) is flat over H ( X ) and the specialization at λ coincides with Γ( C α ( A θλ )),this is because of the vanishing assumption on the structure sheaf. So C α ( ˜ A ) is genericallyflat over H ( X ), while the specialization at λ always coincides with C α ( A λ ). This impliesthe claim of the proposition. (cid:3) Correspondence between parameters.
Our next goal is to understand how torecover the periods of the direct summands C α ( A θ ) from that of A θ . We will assumethat X α ( C × ) satisfies the cohomology vanishing conditions on the structure sheaf, butwe will not require that of X , the period map still makes sense, see [BK]. Consider thedecomposition X α ( C × ) = F i X i into connected components. Let Y i denote the contractinglocus of X i and let A θ i be the restriction of C α ( A θ ) to X i . To determine the period of TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 19 A θ i , we will quantize Y i and then use results from [BGKP] on quantizations of line bundleson lagrangian subvarieties.First of all, let us consider the case when X is affine and so is quantized by a singlealgebra, A . We will quantize the contracting locus Y by a single A - A -bimodule (where A stands for C α ( A )), this bimodule is A / AA > . Lemma 3.3.
Under the above assumptions, the associated graded of A / AA > is C [ Y ] .Proof. This was established in the proof of Proposition 3.1. More precisely, the case when Y is a complete intersection given by α ( C × )-semiinvariant elements of positive weightfollows from the proof of assertion (1), while the general case follows similarly to theproof of (3). (cid:3) Now let us consider the non-affine case. Let us cover X \ S k = i X k with ( C × ) -stableopen affine subsets X j . We may assume that X j either does not intersect Y i or itsintersection with Y i is of the form π − i ( X j ∩ X i ), where π i : Y i → X i is the projection.For this we first choose some covering by ( C × ) -stable open affine subsets. Then we delete Y i \ π − i ( X j ∩ X i ) from each X j , we still have a covering. We cover the remainder of each X j by subsets that are preimages of open affine subsets on X j //α ( C × ), it is easy to seethat this covering has required properties. Let us replace X with the union of X j thatintersect Y i .After this replacement, we can quantize Y i by a A θ - C α ( A θ )-bimodule. We have natural A θ ( X j )- C α ( A θ )( X j ∩ X i )-bimodule structures on A θ ( X j ) / A θ ( X j ) A θ ( X j ) > and glue thebimodules corresponding to different j together along the intersections X i ∩ X j (we havehomomorphisms A θ ( X i ) → A θ ( X i ∩ X j ) that give rise to A θ ( X i ) / A θ ( X i ) A θ ( X i ) > →A θ ( X i ∩ X j ) / A θ ( X i ∩ X j ) A θ ( X i ∩ X j ) > and to C α ( A θ ( X i )) → C α ( A θ ( X i ∩ X j ))). Sim-ilarly to the proof of (2) in Proposition 3.1, we get a sheaf of A θ - C α ( A θ )-bimodules on Y i that we denote by A θ / A θ A θ> . The following is a direct consequence of the construction. Lemma 3.4.
The associated graded of A θ / A θ A θ> coincides with the O X - O X i -bimodule O Y i . Now we want to realize Y i a bit differently (we still use X as in the paragraph precedingLemma 3.4, and so can write Y instead of Y i and X instead of X i ). Namely, let ι denotethe inclusion Y ֒ → X and π be the projection Y → X . We embed Y into X × X via( ι, π ). We equip X × X with the symplectic form ( ω, − ω ), where ω is the restrictionof ω to X . With respect to this symplectic form Y is a lagrangian subvariety. Further, A θ b ⊗ C α ( A θ ) opp is a quantization of X × X with period ( λ, − λ ), where λ, λ are periodsof A θ , C α ( A θ ). Proposition 3.5.
The period λ coincides with ι ∗ ( λ + c ( K Y ) / ∈ H ( X ) = H ( Y ) ,where K Y denotes the canonical class of Y and ι is the inclusion X ֒ → X .Proof. The period of A θ b ⊗ C α ( A θ ) opp coincides with p ( λ ) − p ( λ ), where p : X × X → X, p : X × X → X are the projections. So the pull-back of the period to Y is ι ∗ ( λ ) − π ∗ ( λ ). The structure sheaf of Y admits a quantization to a A θ b ⊗ C α ( A θ ) opp -bimodule,By [BGKP, (1.1.3),Theorem 1.1.4], we have ι ∗ ( λ ) − π ∗ ( λ ) = − c ( K Y ). Restricting thisequality to X , we get the equality required in the proposition. (cid:3) Gieseker case.
Now we want to apply Proposition 3.5 to the case when X = M θ ( n, r ) and α comes from a generic one-dimensional torus in GL( w ) given by t ( t d , . . . , t d r ) with d ≫ d ≫ . . . ≫ d r . Recall that the fixed point components are pa-rameterized by partitions of n . Let X µ denote the component corresponding to a partition µ of n and let Y µ be its contracting locus. So Y µ → X µ is a vector bundle. We will need todescribe this vector bundle. The description is a slight ramification of [Nak2, Proposition3.13].First, consider the following situation. Set V := C n , W = C r . Choose a decomposition W = W ⊕ W with dim W i = r i and consider the one-dimensional torus in GL( w ) actingtrivially on W and by t t on W . The components of the fixed points in M θ ( n, r ) arein one-to-one correspondence with decompositions on n into the sum of two parts. Picksuch a decomposition n = n + n and consider the splitting V = V ⊕ V into the sumof two spaces of the corresponding dimensions and let X = M θ ( n , r ) × M θ ( n , r ) ⊂M θ ( n, r ) α ( C × ) be the corresponding component. We assume that θ > Y → X . This is the bundle on X = M θ ( n , r ) ×M θ ( n , r ) that is induced from the GL( n ) × GL( n )-module ker β / im α ,where α , β are certain GL( n ) × GL( n )-equivariant linear mapsHom( V , V ) α −−→ Hom( V , V ) ⊕ ⊕ Hom( W , V ) ⊕ Hom( V , W ) β −−→ Hom( V , V )We do not need to know the precise form of the maps α , β , what we need is that α is injective while β is surjective. So ker β / im α ∼ = Hom( W , V ) ⊕ Hom( V , W ), anisomorphism of GL( n ) × GL( n )-modules.It is easy to see that if α ′ : C × → GL( r ) is a homomorphism of the form t diag( t d , . . . , t d k ) with d , . . . , d k ≫
0, then the contracting bundle for the one-parametricsubgroup ( α ′ ,
1) : C × → GL( W ) × GL( W ) coincides with the sum of the contractingbundles for α ′ and for ( t, Lemma 3.6.
Consider α : C × → GL( r ) of the form t diag( t d , . . . , t d r ) with d ≫ d ≫ . . . ≫ d r . Consider the irreducible component of M θ ( n, r ) α ( C × ) corresponding to thedecomposition n = n + . . . + n r . Then its contracting bundle is induced from the following Q ri =1 GL( n i ) -module: P ri =1 (( C n i ) ⊕ r − i ⊕ ( C n i ∗ ) ⊕ i − ) . For A θλ ( n , . . . , n r ) denote the summand of C α ( A θλ ( n, r )) corresponding to the decom-position n = n + . . . + n r . Let us recall that the value of the period for A θλ ( n, r ) is λ + r .Using Lemma 3.6 and Proposition 3.5, we deduce the following claim. Corollary 3.7.
We have A θλ ( n , . . . , n r ) = N ri =1 A θλ +( i − ( n i , . Parabolic induction.
Let X be a conical symplectic resolution of X . We assumethat X comes with a Hamiltonian action of a torus T such that X T is finite. Let C standfor Hom( C × , T ). We introduce a pre-order ≺ λ on C as follows: α ≺ λ α ′ if A λ A λ,> ,α ⊂A λ A λ,> ,α ′ .This gives an equivalence relation ∼ λ on C . Both extend naturally to C Q := Q ⊗ Z C .The following lemma explains why this ordering is important. Lemma 3.8.
Suppose α ≺ α ′ . Then C α ′ ( C α ( A λ )) = C α ′ ( A λ ) . Further, let ∆ α ′ : C α ′ ( A λ ) -mod → A λ -mod , ∆ α : C α ( A λ ) -mod → A λ -mod , ∆ : C α ′ ( A ) -mod → C α ( A λ ) -mod be the Verma module functors. We have ∆ α ′ = ∆ α ◦ ∆ . The proof is straightforward.The lemma shows that the Verma module functor can be studied in stages. This iswhat we mean by the parabolic induction.
TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 21
Our goal now is to describe the pre-order ≺ λ for λ Zariski generic. We say that α ≺ α ′ if, for each x ∈ X T , we have T x X > ,α ⊂ T x X > ,α ′ . This automatically implies T x X > ,α ⊃ T x X > ,α ′ (via taking the skew-orthogonal complement) and T x X < ,α ⊂ T x X < ,α ′ . Proposition 3.9.
Fix α, α ′ . For λ in a Zariski open subset, α ≺ λ α ′ is equivalent α ≺ α ′ .Proof. The proof is in several steps. Suppose α ≺ α ′ and let us check that α ≺ λ α ′ . Step 1 . We need to check that, for a Zariski generic λ , we have A λ,> ,α ⊂ A λ A λ,> ,α ′ or, equivalently, α has no positive weights on A λ / A λ A λ,> ,α ′ . This will follow if we checkthat the ˜ A -submodule in ˜ A / ˜ A ˜ A > ,α ′ generated by the elements of positive weight for α is torsion over C [ H ( X )] (here, as usual, ˜ A stands for the algebra of global sections ofthe canonical quantization ˜ A θ of ˜ X ). This, in turn, will follow if we prove an analogousstatement for gr ˜ A / ˜ A ˜ A > ,α ′ . Step 2 . We have an epimorphism C [ ˜ X ] / C [ ˜ X ] C [ ˜ X ] > ,α ′ ։ gr ˜ A / ˜ A ˜ A > ,α ′ . We claim thatits kernel is again torsion over C [ H ( X )], in fact, it is supported on H C . Consider the ~ -adic completion ˜ A ~ of R ~ ( A ). Let ˜ A reg ~ denote the (completed) localization of ˜ A ~ to H ( X ) \ H C . Then ˜ A reg ~ / ˜ A reg ~ ˜ A reg ~ ,> ,α ′ coincides with the localization of ˜ A ~ / ˜ A ~ ˜ A ~ ,> ,α ′ . Onthe other hand, over H ( X ) \H C , the ideal C [ ˜ X ] C [ ˜ X ] > ,α ′ is a locally complete intersection(given by elements of positive α ′ -weight), compare to the proof of (2) in Proposition 3.1.As in the proof of (1) of Proposition 3.1, this implies that ˜ A reg ~ / ˜ A reg ~ ˜ A reg ~ ,> ,α ′ is flat over C [ ~ ]. So the ~ -torsion in ˜ A ~ / ˜ A ~ ˜ A ~ ,> ,α ′ is supported on H C . This implies the claim inthe beginning of this step. Step 3 . So we need to check, that under the assumption α ≺ α ′ , the submodule in C [ ˜ X ] / C [ ˜ X ] C [ ˜ X ] > ,α ′ generated by the elements of positive α -weight is supported on H C .This is equivalent to the claim that C [ X z ] > ,α ⊂ C [ X z ] C [ X z ] > ,α ′ for z
6∈ H C . Here wewrite X z for the fiber of ˜ X → H ( X ) over z . Note that we still have ( T x X z ) > ,α ⊂ ( T x X z ) > ,α ′ for all x ∈ X α ′ ( C × ) z . The inclusion C [ X z ] > ,α ⊂ C [ X z ] C [ X z ] > ,α ′ now followsfrom the Luna slice theorem (for α ( C × ) α ′ ( C × ) applied to α ′ ( C × )-fixed points, we wouldlike to point out that such points are automatically α ( C × )-fixed).The proof of α ≺ α ′ ⇒ α ≺ λ α ′ is now complete. We can reverse the argument to seethat if α ≺ λ α ′ for Zariski generic λ , then α ≺ α ′ . (cid:3) The equivalence classes for ≺ are cones in C Q and the pre-order is by inclusion of theclosures. In particular, there are finitely many equivalence classes. So there is a Zariskiopen subset where ≺ λ refines ≺ .Sometimes we will need to determine when α ≺ λ α ′ for a fixed (non Zariski generic) λ .Pick one-parameter subgroups α, β : C × → T . Lemma 3.10.
For m ≫ , we have α ≺ λ mα + β for all λ .Proof. Clearly, α ∼ λ mα for all m . The algebra gr A > ,α = C [ X ] > ,α is finitely generated,as in the proof of [GL, Lemma 3.1.2]. So we can choose finitely many T -semiinvariantgenerators of the ideal C [ X ] > ,α in C [ X ] > ,α , say f , . . . , f k . Let ˜ f , . . . , ˜ f k denote theirlifts to T -semiinvariant elements in A := A λ , these lifts are generators of the ideal A > ,α in A > ,α . Let a , . . . , a k > α and b , . . . , b k be their weights for β . Take m ∈ Z > such that ma i + b i > i . The elements ˜ f , . . . , ˜ f k then lie in A > ,mα + β and so AA > ,α ⊂ AA > ,mα + β . (cid:3) Finite dimensional representations in the Gieseker case
In this section we will prove (1) of Theorem 1.2 and Theorem 1.4. First, we prove thatthe homological duality realizes the Ringel duality of highest weight categories, Subsection4.1.Then, in Subsection 4.2, we prove part (1) of Theorem 1.2. The ideas of the proof areas follows: we use the Cartan construction to show that we cannot have finite dimensionalrepresentations when the denominator is different from n and also that, in the denominator n case, the category O is not semisimple. Thanks to Subsection 4.1, this means thatthere is a module with support of dimension < dim X in O (for the algebra ¯ A λ ( n, r )with Zariski generic λ ). Using the restriction functors, we see that this module is finitedimensional. Proposition 2.7 then implies that there is a unique finite dimensional module.Finally, in Subsection 4.3 we prove Theorem 1.4. The main idea is to recover thecategory from the homological shifts produced by the Ringel duality.4.1. Homological duality vs Ringel duality.
We start by proving that the homolog-ical duality functor D realizes the contravariant Ringel duality on categories O .Here we deal with the case when X → X is a conical symplectic resolution (satisfyingthe additional assumption from Subsection 2.8). We assume that X comes equipped witha Hamiltonian C × -action α that has finitely many fixed points. We choose a period ˆ λ such that(i) C α ( ± ˆ λ ) ∼ = C [ X α ( C × ) ] the categories O ( A ˆ λ ) , O ( A − ˆ λ ) are highest weight with stan-dard objects being Verma modules.(ii) D b ( O ( A ˆ λ )) ∼ −→ D b O ( A ˆ λ ) , D b ( O ( A − ˆ λ )) ∼ −→ D b O ( A − ˆ λ ).We recall that these two conditions hold for a Zariski generic ˆ λ .Let us recall the definition of the (contravariant) Ringel duality. Let C , C be twohighest weight categories. Suppose we have a contravariant equivalence R : C ∆1 ∼ −→ C ∆2 (the superscript ∆ means the full subcategories of standardly filtered objects). Then itrestricts to a contravariant duality between C -proj and C -tilt. The former denotes thecategory of the projective objects in C , while the latter is the category of tilting objectsin C , i.e., objects that are both standardly and costandardly filtered. The equivalence R extends to an equivalence D b ( C ) ∼ −→ D b ( C ) opp . Moreover, the category C gets iden-tified with End( T ) -mod and, under this identification, the derived equivalence above isRHom C ( • , T ). Here T is the tilting generator of C , i.e., the direct sum of all indecompos-able tiltings. For the proofs of the claims above in this paragraph see [GGOR, Proposition4.2].We say that C is a Ringel dual of C and write C ∨ for C . Proposition 4.1.
Take ˆ λ in a Zariski open set and such that the abelian localization holdsfor (ˆ λ, θ ) , ( − ˆ λ, − θ ) . Then there is an equivalence O ( A − ˆ λ ) ∼ −→ O ( A ˆ λ ) ∨ that intertwinesthe homological duality functor D : D b ( O ( A ˆ λ )) → D b ( O ( A − ˆ λ )) opp and the contravariantRingel duality functor RHom O ( A ˆ λ ) ( • , T ) : D b ( O ( A ˆ λ )) → D b ( O ( A ˆ λ ) ∨ ) opp . Let ∆ ˆ λ denote the sum of all standard objects in O ( A ˆ λ ). Of course, ∆ ˆ λ = A ˆ λ / A ˆ λ A ˆ λ,> .We write θ for an element of the ample cone of X . Lemma 4.2.
For a parameter ˆ λ in a Zariski open subset, the object D (∆ ˆ λ ( p )) is con-centrated in homological degree and, moreover, its characteristic cycle (an element of TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 23 the vector space with basis formed by the irreducible components of the contracting variety Y ) coincides with the class of (the degeneration of ) the contracting component of p at ageneric fiber of ˜ X → H ( X ) .Proof. Let us prove the first claim. What we need to prove is that Ext i (∆ ˆ λ , A ˆ λ ) = 0provided i = dim X for ˆ λ in a Zariski open space. Our claim will follow follow ifwe show that the support of Ext i ( ˜∆ , e A ) in H ( X ) is not dense in H ( X ) and that the C [ H ( X )]-module Ext i ( ˜∆ , e A ) is generically flat. Here we write e ∆ = e A / e A e A > .We can take a graded free resolution of gr e ∆ and lift it to a free resolution of e ∆. Itfollows that the right e A -modules Ext i ( e ∆ , e A ) are naturally filtered and that the associatedgraded modules are subquotients of Ext i (gr e ∆ , C [ e X ]). The claim about generic flatnessfollows (compare with [BL, Lemma 5.5, Corollary 5.6]). Also to prove that claim in theprevious paragraph that the support is not dense it is enough to prove a similar claim forExt i (gr e ∆ , C [ e X ]).Set e ∆ cl := C [ e X ] / C [ e X ] C [ e X ] > . We have e ∆ cl ։ gr e ∆. Moreover, the support of thekernel in H ( X ) is contained in H C , see Step 2 of the proof of Proposition 3.9. So itis enough to show that the support of Ext i ( e ∆ cl , C [ e X ]) is not dense when i = dim X .This follows from the observation that, generically over H ( X ), the ideal C [ e X ] C [ e X ] > isa locally complete intersection in a smooth variety.The argument above also implies that the associated graded of D (∆ ˆ λ ( p )) coincides withthat of Ext dim X (∆ cl,λ ( p ) , C [ X λ ]) for a Zariski generic element λ ∈ H ( X ). The latteris just the class of the contracting component Y λ,p (defined as the sum of components of X ∩ C × Y λ,p with obvious multiplicities). (cid:3) Proof of Proposition 4.1.
We write ∆ ˆ λ ( p ) ∨ for D (∆ ˆ λ ( p )), thanks to Lemma 4.2, thisis an object in O ( A − ˆ λ ) (and not just a complex in its derived category). We haveEnd(∆ ˆ λ ( p ) ∨ ) = C and Ext i (∆ ˆ λ ( p ) ∨ , ∆ ˆ λ ( p ′ ) ∨ ) = 0 if i > p θ p ′ . We remark thatthe orders θ and − θ can be refined to opposite partial orders (first we refine them tothe orders coming by the values of the real moment maps for the actions of S ⊂ α ( C × ),and then refine those), compare with [G, 5.4]. So it only remains to prove that thecharacteristic cycle of ∆ ˆ λ ( p ) ∨ consists of the contracting components Y p ′ with p ′ − θ p .The characteristic cycle of ∆ ˆ λ ( p ) ∨ coincides with C × Y λ,p ∩ X , by Lemma 4.2. But thecharacteristic cycle of ∆ − ˆ λ ( p ) is the same. Our claim follows. (cid:3) Remark 4.3.
We also have covariant Ringel duality given by RHom( T, • ), it maps co-standard objects to standard ones. Under the assumption that the conical symplecticresolutions of X are strictly semismall, Propositions 4.1 and 2.9 imply that the longwall-crossing functor is inverse of the covariant Ringel duality. This proves a part of[BLPW, Conjecture 8.27].4.2. Proof of Theorem 1.2.
Here we prove (1) of Theorem 1.2. The proof is in severalsteps.
Step 1 . Let us establish a criterium for the semisimplicity of a highest weight categoryvia the Ringel duality.
Lemma 4.4.
Let C be a highest weight category and R : D b ( C ) → D b ( C ∨ ) opp denote thecontravariant Ringel duality. The following conditions are equivalent: (1) C is semisimple. (2) We have H ( R ( L )) = 0 for every simple object L . (3) every simple lies in the socle of a standard object.Proof. The implication (1) ⇒ (2) is clear. The implication (2) ⇒ (3) follows from the factthat every standard object in a highest weight category is included into an indecomposabletilting.Let us prove (3) ⇒ (1). Let λ be a maximal (with respect to the coarsest highest weightordering) label. Then the simple L ( λ ) lies in the socle of some standard, say ∆( µ ). Butall simple constituents of ∆( µ ) are L ( ν ) with ν µ . It follows that µ = λ . Since L ( λ )lies in the socle of ∆( λ ) and also coincides with the head, we see that ∆( λ ) = L ( λ ). So L ( λ ) is projective and therefore spans a block in the category. Since this holds for anymaximal λ , we deduce that the category C is semisimple. (cid:3) Let us remark that for the category O ( ¯ A λ ( n, r )) condition (2) is equivalent to every sim-ple having support of dimension rn −
1. This follows from Subsection 2.6 and Proposition4.1.Below in this proof we assume that λ is chosen as in Proposition 4.1, in particular, thecategories O ( ¯ A λ ( n, r )) and O ( ¯ A θλ ( n, r )) are equivalent. In the definition of categories O we choose the torus of the form ( α, α : C × → GL( r ) is given by t ( t d , . . . , t d r )with d ≫ d ≫ . . . ≫ d r . Step 2 . Let us prove that the category O ( ¯ A θλ ( n, r )) is semisimple, when λ Q or thedenominator of λ is bigger than n . The proof is by induction on n (for n = 0 the claimis vacuous).By Corollary 2.10, we see that a simple ¯ A λ ( n, r )-module whose support has dimension < rn − A λ ( n, r ). We claim that any such idealhas finite codimension under our assumption on λ . Indeed, otherwise some proper slicealgebra has an ideal of finite codimension, see Proposition 2.15, which contradicts ourinductive assumption. So the support of a simple has dimension either rn − rn −
1, we are done. Butthanks to Corollary 3.7, Proposition 3.2 and known results on finite dimensional ¯ A λ ( n, C α ( ¯ A λ ( n, r )) has no finite dimensional modules (we obviouslyhave C α ( A λ ( n, r )) = D ( C ) ⊗ C α ( ¯ A λ ( n, r )) and none of the summands of A λ ( n, r ) hassimple of GK dimension 1 in category O ). Step 3 . The description of C α ( ¯ A λ ( n, r )) shows that there are no finite dimensional¯ A λ ( n, r )-modules in the case when the denominator of λ is less than n . Step 4 . Now consider the case of denominator n . Similarly to Step 2, all simplesare either finite dimensional or have support of dimension rn −
1. By Lemma 2.1, thedimension of the middle homology of ¯ M θ ( n, r ) is 1. Thanks to Proposition 2.7, the numberof finite dimensional irreducibles is 0 or 1. If there is one such module, then the category offinite dimensional modules is semisimple because O ( ¯ A λ ( n, r )) is a highest weight category.Thanks to Step 1, we only need to show that O ( ¯ A λ ( n, r )) is not semisimple.One-parameter subgroups α : t diag( t d , . . . , t d r ) with d ≫ . . . ≫ d r form oneequivalence class for the pre-order ≺ . This cone is a face of the equivalence class containing( α, α ≺ λ ( α, , ∆ for the Verma module functors ∆ : C [ P r ( n )] → O ( C α ( A λ ( n, r )))and ∆ : O ( C α ( A λ ( n, r ))) → O ( A λ ( n, r )), here we write P r ( n ) for the set of the r -multipartitions of n . By Lemma 3.8, we have ∆ = ∆ ◦ ∆ . The category O ( C α ( A λ ( n, r )))is not semisimple: there is a nonzero homomorphism ϕ : ∆ ( p ) → ∆ ( p ), where TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 25 p = ( ∅ r − , ( n )) , p = ( ∅ r − , ( n − , ϕ ) : ∆( p ) =∆(∆ ( p )) → ∆(∆ ( p )) = ∆( p ). The highest α -weight components of ∆( p ) , ∆( p )coincide with ∆ ( p ) , ∆ ( p ), respectively, by the construction. The homomorphism∆ ( p ) → ∆ ( p ) induced by ∆( ϕ ) coincides with ϕ . It follows that ∆( ϕ ) = 0. Weconclude that O ( A λ ( n, r )) is not semisimple.This completes the proof of all claims of the theorem but the claim that the categoryof modules supported on ρ − (0) is semisimple. The latter is an easy consequence of theobservation that, in a highest weight category, we have Ext ( L, L ) = 0. We would like topoint out that the argument of the previous paragraph generalizes to the denominatorsless than n . So in those cases there are also simple ¯ A λ ( n, r )-modules of support withdimension < rn − Proof of Theorem 1.4.
In this subsection we will prove Theorem 1.4. We havealready seen in the previous subsection that if the denominator is bigger than n , thenthe category O is semisimple. The case of denominator n will follow from a more precisestatement, Theorem 4.5.Let us introduce a certain model category. Let C n denote the nontrivial block for thecategory O for the Rational Cherednik algebra H /n ( n ) for the symmetric group S n . Letus summarize some properties of this category.(i) Its coarsest highest weight poset is linearly ordered: p n < p n − < . . . < p .(ii) The objects I ( p i ) for i > → ∇ ( p i ) → I ( p i ) →∇ ( p i − ) →
0. Here we write ∇ ( p i ) , I ( p i ) for the costandard and the indecom-posable injective objects of C n labeled by p i .(iii) The indecomposable tilting objects T ( p i − ) for i > I ( p i ).(iv) The simple objects L ( p i ) with i > C n ( L ( p ) , T )is concentrated in homological degree n .(v) There is a unique simple in C ∨ n that appears in the higher cohomology of RHom C n ( • , T ). Theorem 4.5.
Consider a parameter of the form λ = qn with coprime q, n . Then thefollowing is true. (1) The category O ( ¯ A θλ ( n, r )) has only one nontrivial block that is equivalent to C rn .This block contains an irreducible representation supported on ¯ ρ − (0) . (2) Suppose the one parameter torus used to define the category O is of the form t ( α ( t ) , t ) , where α ( t ) = diag( t d , . . . , t d r ) with d i − d i +1 > n for all i . Thenthe labels in the non-trivial block of O ( ¯ A θλ ( n, r )) are hooks h i,d = ( ∅ , . . . , ( n + 1 − d, d − ) , . . . , ∅ ) (where i is the number of the diagram where the hook appears)ordered by h ,n > h ,n − > . . . > h , > h ,n > . . . > h , > . . . > h r, .Proof. The proof is in several steps. We again deal with the realization of our cate-gory as O ( ¯ A λ ( n, r )), where λ is Zariski generic and such that ( λ, θ ) satisfies the abelianlocalization. Step 1 . As we have seen in Step 4 of the proof of Theorem 1.2, all simples have maximaldimension of support, except one, let us denote it by L , which is finite dimensional. Soall blocks but one consist of modules with support of maximal dimension. Now arguingas in the first two steps of the proof of Theorem 1.2, we see that the blocks that do notcontain L are simple. Let C denote the nontrivial block. The label of L , denote it by p max , is the largest in any highest weight ordering. For all other labels p the simple L ( p ) lies in the socle of the tilting generator T . In other words an analog of (iv) above holdsfor C with rn instead of n . In the subsequent steps we will show that C ∼ = C rn . Step 2 . Let us show that an analog of (v) holds for C . By Corollary 2.11, the highercohomology of D ( L ) cannot have support of maximal dimension. It follows that thehigher cohomology is finite dimensional and so are direct sums of a single simple in O ( ¯ A − r − λ ( n, r )). Since the Ringel duality is the same as the homological duality (up toan equivalence of abelian categories, see Proposition 4.1), we are done. Step 3 . Let us show that there is a unique minimal label for C , say p min . This isequivalent to C ∨ having a unique maximal label because the orders on C and C ∨ areopposite. But C ∨ is equivalent to the nontrivial block in O ( ¯ A − r − λ ( n, r )). So we are doneby Step 1 (applied to − r − λ instead of λ ) of this proof. Step 4 . Let us show that (v) implies that any tilting in C but one is injective. Let R ∨ denote the Ringel duality equivalence D b ( C ∨ ) → D b ( C ) opp . Let us label the tiltings by thelabel of the top costandard in a filtration with costandard subsequent quotients. We haveExt i ( L ( p ′ ) , T ( p )) = Hom( L ( p ′ )[ i ] , T ( p )) = Hom(( R ∨ ) − T ( p )[ i ] , ( R ∨ ) − L ( p ′ )). The objects( R ∨ ) − T ( p ) are projective so Ext i ( L ( p ′ ) , T ( p )) = Hom(( R ∨ ) − T ( p ) , H i (( R ∨ ) − L ( p ′ ))).Similarly to the previous step (applied to C ∨ instead of C and ( R ∨ ) − instead of R ),there is a unique indecomposable projective P ∨ ( p ∨ ) in C ∨ that can map nontrivially to ahigher homology of ( R ∨ ) − L ( p ). So if ( R ∨ ) − T ( p ) = P ∨ ( p ∨ ), then T ( p ) is injective. Step 5 . We remark that ∇ ( p max ) is injective but not tilting, while ∇ ( p min ) is tiltingbut not injective. So the injectives in C are ∇ ( p max ) and T ( p ) for p = p min . Similarly, thetiltings are I ( p ) , p = p max , and ∇ ( p min ). Step 6 . Let Λ denote the highest weight poset for C . Let us define a map ν : Λ \{ p min } → Λ \{ p max } . It follows from Step 5 that the socle of any tilting in C is simple. By definition, ν ( p ) is such that L ( ν ( p )) is the socle of T ( p ). We remark that ν ( p ) p for any highestweight order. Step 7 . Let us show that any element p ∈ Λ has the form ν i ( p max ). Assume the converseand let us pick the maximal element not of this form, say p ′ . Since p ′ = p max , we seethat L ( p ′ ) lies in the socle of some tilting. But the socle of any indecomposable tilting issimple. So ∇ ( p ′ ) is a bottom term of a filtration with constandard subsequent quotients.By the definition of ν and the choice of p ′ , ∇ ( p ′ ) is tilting itself. Any indecomposabletilting but ∇ ( p min ) is injective and we cannot have a costandard that is injective andtilting simultaneously. So p ′ = p min . But let us pick a minimal element p ′′ in Λ \ { p min } .By above in this step, ν ( p ′′ ) < p ′′ . So ν ( p ′′ ) = p min . The claim in the beginning of thestep is established. This proves (i) for C . Step 8 . (ii) for C follows from Step 7 and (iii) follows from (ii) and Step 5. Step 9 . Let us show that rn . The minimal injective resolution for ∇ ( p min )has length ∇ ( p max ). It followsthat RHom( L ( p max ) , ∇ ( p min )) is concentrated in homological degree −
1. The othertiltings are injectives and RHom’s with them amount to Hom’s. Since RHom( L ( p max ) , T )is concentrated in homological degree rn − Step 10 . Let us complete the proof of (1). Let us order the labels in Λ decreasingly, p > . . . > p rn . Using (ii) we get the following claims. • End( I ( p i )) = C [ x ] / ( x ) for i > I ( p )) = C . • Hom( I ( p i ) , I ( p j )) is 1-dimensional if | i − j | = 1 and is 0 if | i − j | > TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 27
Choose some basis elements a i,i +1 , i = 1 , . . . , rn − I ( p i ) , I ( p i +1 )) and also basiselements a i +1 ,i ∈ Hom( I ( p i +1 ) , I ( p i )). We remark that the image of the compositionmap Hom( I ( p i ) , I ( p i +1 )) × Hom( I ( p i +1 ) , I ( p i )) → End( I ( p i )) spans the maximal ideal.Choose generators a ii in the maximal ideals of End( I ( p i )) , i = 2 , . . . , rn . Normalize a by requiring that a a = a , automatically, a a = 0. Normalize a by a a = a and then normalize a by a = a a . We continue normalizing a i +1 ,i and a i +1 ,i +1 inthis way. We then recover the multiplication table in End( L I ( λ i )) in a unique way. Thiscompletes the proof of (1). Step 11 . Now let us prove (2). Let us check that the labeling set Λ for the nontrivialblock of O ( ¯ A θλ ( n, r )) consists of hooks. For this, it is enough to check that ∆( h i,d ) does notform a block. This in turn, will follow if we check that there is a nontrivial homomorphismbetween ∆( h i,d ) and some other ∆( h i,d ′ ). This is done similarly to the second paragraphof Step 4 in the proof of Theorem 1.2. Now, according to [Ko], the hooks are ordered asspecified in (2) with respect to the geometric order on the torus fixed points in M θ ( n, r )(note that the sign conventions here and in [Ko] are different). (cid:3) Remark 4.6.
We can determine the label of the simple supported on ¯ ρ − (0) in the cat-egory O corresponding to an arbitrary generic torus. Namely, note that ¯ ρ − (0) coincideswith the closure of a single contracting component and that contracting component corre-sponds to the maximal point. Now we can use results of [Ko] to find a label of the point:it always has only one nontrivial partition and this partition is either ( n ) or (1 n ).5. Localization theorems in the Gieseker case
In this section we prove Theorem 1.1. The proof is in the following steps. • We apply results of McGerty and Nevins, [MN2], to show that, first, if the abelianlocalization fails for ( λ, θ ), then λ is a rational number with denominator notexceeding n , and, second, the parameters λ = qm with m n and − r < λ < θλ is exact when λ > − r, θ > λ < , θ < A θλ ( n, r ) ∼ = A − θ − λ − r ( n, r ), this reduces the conjecture tochecking that the abelian localization holds for λ = qm with q > , m n . • Then we reduce the proof to the case when the denominator is precisely n and λ, θ > • Then we will study a connection between the algebras C α ( ¯ A λ ( n, r )) , Γ( C α ( ¯ A θλ ( n, r ))).We will show that the numbers of simples in the categories O for these algebrascoincide. We deduce the localization theorem from there.The last step is a crucial one and it does not generalize to other quiver varieties.5.1. Results of McGerty and Nevins and consequences.
In [MN2], McGerty andNevins found a sufficient condition for the functor Γ θλ : A θλ ( n, r ) -mod → A λ ( n, r ) -mod tobe exact (they were dealing with more general Hamiltonian reductions but we will onlyneed the Gieseker case). Let us explain what their result give in the case of interest for us.Consider the quotient functors π λ : D R -mod G,λ ։ A λ ( n, r ) -mod and π θλ : D R -mod G,λ ։ A θλ ( n, r ) -mod. Proposition 5.1.
The inclusion ker π det λ ⊂ ker π λ holds provided λ > − r . Similarly, ker π det − λ ⊂ π λ provided λ < . I would like to thank Dmitry Korb for explaining me the required modifications to[MN2, Section 8].
Proof.
We will consider the case θ = det, the opposite case follows from A − θλ ( n, r ) ∼ = A θ − r − λ ( n, r ). The proof closely follows [MN2, Section 8], where the case of r = 1 isconsidered. Instead of R = End( V ) ⊕ Hom(
V, W ) they use R ′ = End( V ) ⊕ Hom(
W, V ),then, thanks to the partial Fourier transform, we have D ( R ) -mod G,λ ∼ = D ( R ′ ) -mod G,λ + r .The set of weights in R ′ for a maximal torus T ⊂ GL( V ) is independent of r so wehave the same Kempf-Ness subgroups as in the case r = 1: it is enough to consider thesubgroups β with tangent vectors (in the notation of [MN2, Section 8]) e + . . . + e k . Theshift in loc.cit. becomes rk (in the computation of loc.cit. we need to take the secondsummand r times, that is all that changes). So we get that ker π det λ ⊂ ker π λ provided k ( − r − λ ) rk + Z > for all possible k meaning 1 k n (the number − r − λ is c ′ in loc.cit. ). The condition simplifies to λ
6∈ − r − k Z > . This implies the claim of theproposition. (cid:3) Reduction to denominator n and singular parameters. Proposition 5.1 allowsus to show that certain parameters are singular.
Corollary 5.2.
The parameters λ with denominator n and − r < λ < are singular.Proof. Assume the converse. Since R Γ ± θλ are equivalences and Γ ± θλ are exact, we see thatΓ ± θλ are equivalences of abelian categories. From the inclusions ker π ± θλ ⊂ ker π λ , wededuce that the functors π ± θλ are isomorphic. So the wall-crossing functor WC λ → λ − = π − θλ − ◦ ( C λ − − λ ⊗ • ) ◦ Lπ θ ∗ λ (see [BL, (2.8)] for the equality) is an equivalence of abeliancategories (where we modify λ by adding a sufficiently large integer). However, we havealready seen that it does shift some modules, since not all modules in O ( ¯ A λ ( n, r )) havesupport of maximal dimension (see the end of the proof of Theorem 1.2). (cid:3) Now let us observe that it is enough to check that the abelian localization holds for λ > θ >
0. This follows from an isomorphism A θλ ( n, r ) ∼ = A − θ − λ − r ( n, r ). This anisomorphism of sheaves on M θ ( n, r ) ∼ = M − θ ( n, r ) (see the proof of Lemma 2.4).Now let us reduce the proof of Theorem 1.1 to the case when λ has denominator n . Let the denominator n ′ be less then n . As we have seen in [BL, Section 5], theabelian localization holds for ( λ, θ >
0) if and only if the bimodules A λ,χ ( n, r ) :=[ D ( R ) /D ( R ) { x R − h λ, x i} ] G,χ , A λ + χ, − χ ( n, r ) with χ ≫ A λ,χ ( n, r ) ⊗ A λ ( n,r ) A λ + χ, − χ ( n, r ) → A λ + χ ( n, r ) , A λ + χ, − χ ( n, r ) ⊗ A λ + χ ( n,r ) A λ,χ ( n, r ) → A λ ( n, r )(5.1)are isomorphisms.Assume the converse. Let K , C , K , C denote the kernel and the cokernel of the firstand of the second homomorphism, respectively. If one of these bimodules is nontrivial,then we can find x ∈ M ( n, r ) such that K i † ,x , C i † ,x are finite dimensional, and, at least oneof these bimodules is nonzero. From the classification of finite dimensional irreducibles,we see that the slice algebras must be of the form ¯ A ? ( n ′ , r ) ⊗ k . But then A λ + χ, − χ ( n, r ) † ,x =¯ A λ + χ, − χ ( n ′ , r ) ⊗ k , A λ,χ ( n, r ) † ,x = ¯ A λ,χ ( n ′ , r ) ⊗ k . Further, applying • † ,x to (5.1) we again getnatural homomorphisms. But the localization theorem holds for the algebra ¯ A λ ( n ′ , r )thanks to our inductive assumption, so the homomorphisms of the ¯ A λ ( n ′ , r ) ⊗ k -bimodulesare isomorphisms. This contradiction justifies the reduction to denominator n . TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 29
Number of simples in O ( A λ ( n, r )) . So we need to prove that the localizationtheorem holds for positive parameters λ with denominator n (the case λ = 0 occurs onlyif n = 1 and in that case this is a classical localization theorem for differential operatorson projective spaces). We will derive the proof from the claim that the number of simpleobjects in the categories O ( ¯ A λ ( n, r )) and O ( ¯ A θλ ( n, r )) is the same. For this we will needto study the natural homomorphism ϕ : C α ( ¯ A λ ( n, r )) → Γ( C α ( ¯ A θλ ( n, r ))). Here, as before, α : C × → GL( r ) is of the form t ( t d , . . . , t d r ), where d ≫ d ≫ . . . ≫ d r .Recall that Γ( C α ( ¯ A θλ ( n, r ))) = L ¯ A λ ( n , . . . , n r ; r ), where the summation is taken overall compositions n = n + . . . + n r and ¯ A λ ( n , . . . , n r ; r ) ⊗ D ( C ) = N ri =1 A λ + i − ( n i , D ( C ) is embedded into the right hand side “diagonally”). Let B denote themaximal finite dimensional quotient of Γ( C α ( ¯ A θλ ( n, r ))). Proposition 5.3.
The composition of ϕ with the projection Γ( C α ( ¯ A θλ ( n, r ))) ։ B issurjective.Proof. The proof is in several steps.
Step 1 . We claim that it is sufficient to prove that the composition ϕ i of ϕ withthe projection Γ( C α ( ¯ A θλ ( n, r ))) → ¯ A λ + i ( n,
1) is surjective. Indeed, each ¯ A λ + i ( n, , i =0 , . . . , r − λ = qn , then the dimensionis ( q + n − q ! n ! . So B is the sum of r pairwise non-isomorphic matrix algebras. Therefore thesurjectivity of the homomorphism C α ( ¯ A λ ( n, r )) → B follows from the surjectivity of allits r components. We remark that the other summands of C α ( ¯ A λ ( n, r )) have no finitedimensional representations. Step 2 . Generators of ¯ A λ + i ( n,
1) are known. Namely, recall that ¯ A λ + i ( n,
1) is thespherical subalgebra in the Cherednik algebra H c ( n ) for the reflection representation h of S n with c = λ + i . The latter is generated by h , h ∗ . Then algebra eH c ( n ) e is generatedby S ( h ) W , S ( h ∗ ) W , see [EG]. On the level of quantum Hamiltonian reduction, S ( h ) W coincides with the image of S ( g ) G , while S ( h ∗ ) W coincides with the image of S ( g ∗ ) G . Herewe write g for sl n . We will show that these images lie in the image of ϕ i : C α ( ¯ A λ ( n, r )) → ¯ A λ + i ( n, Step 3 . Let us produce a natural homomorphism S ( g ∗ ) G → C α ( ¯ A λ ( n, r )). First of all,recall that ¯ A λ ( n, r ) is a quotient of D ( g ⊕ ( C ∗ n ) r ) G . The algebra S ( g ∗ ) G is included into D ( g ⊕ ( C ∗ n ) ⊕ r ) G as the algebra of invariant functions on g . So we get a homomorphism S ( g ) G → ¯ A λ ( n, r ). Since the C × -action α used to form C α ( ¯ A λ ( n, r )) is nontrivial only on( C ∗ n ) ⊕ r , we see that the image of S ( g ∗ ) G lies in ¯ A λ ( n, r ) α ( C × ) . So we get a homomorphism ι : S ( g ∗ ) G → C α ( ¯ A λ ( n, r )). Step 4 . We claim that ϕ i ◦ ι coincides with the inclusion S ( g ∗ ) G → ¯ A λ + i ( n, D ( g ⊕ ( C ∗ n ) ⊕ r ) by the order of a differential operator. This inducesfiltrations on ¯ A λ ( n, r ) , ¯ A θλ ( n, r ). We have similar filtrations on the algebras ¯ A λ + i ( n, A λ ( n, r ) , ¯ A θλ ( n, r ) are preserved by α and hence we have filtrationson C α ( ¯ A λ ( n, r )) , Γ( C α ( ¯ A θλ ( n, r ))). It is clear from the construction of the projectionΓ( C α ( ¯ A θλ ( n, r ))) → ¯ A λ + i ( n,
1) that it is compatible with the filtration. On the otherhand, the images of S ( g ∗ ) G in both C α ( ¯ A λ ( n, r )) , ¯ A λ + i ( n,
1) lies in the filtration degree 0.So it is enough to prove the coincidence of the homomorphisms in the beginning of thestep after passing to associate graded algebras.
Step 5 . The associated graded homomorphisms coincide with analogous homomor-phisms defined on the classical level. Recall that the components of M θ ( n, r ) α ( C × ) that are Hilbert schemes are realized as follows. Pick an eigenbasis w , . . . , w r for the fixed r -dimensional torus in GL r . Then the i th component that is the Hilbert schemes con-sists of G -orbits of ( A, B, , j ), where j : C n → C r is a map with image in C w j . Inparticular, the homomorphism S ( g ∗ ) G → gr A λ + i ( n,
1) is dual to the morphism given by(
A, B, , j ) → A .On the other hand, the component of M θ ( n, r ) α ( C × ) in consideration maps onto M ( r, n ) //α ( C × )(via sending the orbit of ( A, B, , j ) to the orbit of the same element). The correspondinghomomorphism of algebras is the associated graded of ¯ A λ ( n, r ) α ( C × ) → ¯ A λ + i ( n, M ( r, n ) //α ( C × ) → g //G given by ( A, B, , j ) A . The corre-sponding homomorphism of algebras is the associated graded of S ( g ∗ ) G → ¯ A λ ( n, r ) α ( C × ) .We have checked that the associated graded homomorphism of ϕ i ◦ ι : S ( g ∗ ) G → ¯ A λ + i ( n, S ( g ∗ ) G → ¯ A λ + i ( n, Step 6 . The coincidence of similar homomorphisms S ( g ) G → ¯ A λ + i ( n,
1) is establishedanalogously. The proof of the surjectivity of C α ( ¯ A λ ( n, r )) → ¯ A λ + i ( n,
1) is now complete. (cid:3)
We still have a Hamiltonian action of C × on C α ( ¯ A λ ( n, r )) that makes the homo-morphism C α ( ¯ A λ ( n, r )) → Γ( C α ( ¯ A θλ ( n, r ))) equivariant. So we can form the category O ( C α ( ¯ A λ ( n, r ))) for this action. By Lemma 3.10, we have α ≺ λ ( mα,
1) for m ≫
0. Werescale α and assume that m = 1. Recall, Lemma 3.8, that we have an isomorphism C ( C α ( ¯ A θλ ( n, r ))) ∼ = C ( α, ( ¯ A θλ ( n, r )). So there is a natural bijection between the sets ofsimples in O ( C α ( ¯ A λ ( n, r ))) and in O ( ¯ A λ ( n, r )). Proposition 5.4.
The number of simples in O ( C α ( ¯ A λ ( n, r ))) is bigger than or equal tothat in O (Γ( C α ( ¯ A θλ ( n, r )))) .Proof. The proof is again in several steps.
Step 1 . We have a natural homomorphism C [ g ] G → L ¯ A λ ( n , . . . , n r ; r ). It can bedescribed as follows. We have an identification C [ g ] G ∼ = C [ h ] S n . This algebra embedsinto ¯ A λ ( n , . . . , n r ; r ) (that is a spherical Cherednik algebra for the group Q ri =1 S n i act-ing on h ) via the inclusion C [ h ] S n ⊂ C [ h ] S n × ... × S nr . For the homomorphism C [ g ] G → L ¯ A λ ( n , . . . , n r ; r ) we take the direct sum of these embeddings. Similarly to Steps 4,5of the proof of Proposition 5.3, the maps C [ g ] G → C α ( ¯ A λ ( n, r )) , Γ( C α ( ¯ A θλ ( n, r ))) are in-tertwined by the homomorphism C α ( ¯ A λ ( n, r )) → Γ( C α ( ¯ A θλ ( n, r ))). Step 2 . Let δ ∈ C [ g ] G be the discriminant. We claim that C α ( ¯ A θλ ( n, r ))[ δ − ] ∼ −→ Γ( C α ( ¯ A θλ ( n, r )))[ δ − ]. Since δ is α ( C × )-stable, we have C α ( ¯ A λ ( n, r ))[ δ − ] = C α ( ¯ A λ ( n, r )[ δ − ]).We will describe the algebra C α ( ¯ A λ ( n, r )[ δ − ]) explicitly and see that C α ( ¯ A λ ( n, r )[ δ − ]) ∼ −→ Γ( C α ( ¯ A θλ ( n, r )))[ δ − ]. Step 3.
We start with the description of ¯ A λ ( n, r )[ δ − ]. Let g reg denote the locus ofthe regular semisimple elements in g . Then ¯ A λ ( n, r )[ δ − ] = D ( g reg × Hom( C n , C r )) /// λ G .Here /// λ denotes the quantum Hamiltonian reduction with parameter λ .Recall that g reg = G × N G ( h ) h reg and so g reg × Hom( C n , C r ) = G × N G ( h ) ( h reg × Hom( C n , C r )). It follows that D ( g reg × Hom( C n , C r )) /// λ G = D ( h reg × Hom( C n , C r )) /// λ N G ( h ) =( D ( h reg ) ⊗ D (Hom( C n , C r )) /// λ H ) S n = (cid:0) D ( h reg ) ⊗ D λ ( P r − ) ⊗ n (cid:1) S n . TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 31
Here, in the second line, we write H for the Cartan subgroup of G and take the diagonalaction of S n . In the last expression, it permutes the tensor factors. A similar argumentshows that ¯ M θ ( n, r ) δ = ( T ∗ ( h reg ) × T ∗ ( P r − ) n ) / S n and the restriction of ¯ A θλ ( n, r ) to thisopen subset is (cid:0) D h reg ⊗ ( D λ P r − ) ⊗ n (cid:1) S n . Step 4 . Now we are going to describe the algebra C α ( (cid:0) D ( h reg ) ⊗ D λ ( P r − ) ⊗ n (cid:1) S n ). Firstof all, we claim that(5.2) C α ( (cid:0) D ( h reg ) ⊗ D λ ( P r − ) ⊗ n (cid:1) S n ) = ( C α (cid:0) D ( h reg ) ⊗ D λ ( P r − ) ⊗ n (cid:1) ) S n There is a natural homomorphism from the left hand side to the right hand side. To provethat it is an isomorphism one can argue as follows. First, note, that since the S n -actionon h reg is free, we have D ( h reg ) ⊗ D λ ( P r − ) ⊗ n = D ( h reg ) ⊗ D ( h reg ) S n (cid:0) D ( h reg ) ⊗ D λ ( P r − ) ⊗ n (cid:1) S n Since D ( h reg ) is α ( C × )-invariant, the previous equality implies (5.2). Step 5 . Now let us describe C α ( (cid:0) D ( h reg ) ⊗ D λ ( P r − ) ⊗ n (cid:1) = D ( h reg ) ⊗ C α (cid:0) ( D λ ( P r − )) ⊗ n (cid:1) .The C × -action on the tensor product ( D λ ( P r − )) ⊗ n is diagonal and it is easy to see that C α (cid:0) ( D λ ( P r − )) ⊗ n (cid:1) = (cid:0) C α ( D λ ( P r − )) (cid:1) ⊗ n . So we need to compute C α ( D λ ( P r − )). Weclaim that this algebra is isomorphic to C ⊕ r . Indeed, D λ ( P r − ) is a quotient of the centralreduction U ˜ λ ( sl r ) of U ( sl r ) at the central character ˜ λ := λω r . We remark that λω r + ρ is regular because λ >
0. We have C α ( U ˜ λ ( sl r )) = C ⊕ r ! and C α ( D λ ( P r − )) is a quotient ofthat. The number of irreducible representations of C α ( D λ ( P r − )) equals to the number ofsimples in the category O for D λ ( P r − ) that coincides with r since the localization holds.An isomorphism C α ( D λ ( P r − )) = C ⊕ r follows. Step 6 . So we see that C α ( ¯ A λ ( n, r )[ δ − ]) = ( D ( h reg ) ⊗ ( C ⊕ r ) ⊗ n ) S n . By similar rea-sons, we have Γ([ ¯ M θ ( n, r ) δ ] α ( C × ) , C α ( ¯ A θλ ( n, r ))) = ( D ( h reg ) ⊗ ( C ⊕ r ) ⊗ n ) S n . The naturalhomomorphism(5.3) C α ( ¯ A λ ( n, r )[ δ − ]) → Γ(( ¯ M θ ( n, r ) δ ) α ( C × ) , C α ( ¯ A θλ ( n, r )))is an isomorphism by the previous two steps. Also we have a natural homomorphism(5.4) Γ( C α ( ¯ A θλ ( n, r )))[ δ − ] → Γ([ M θ ( n, r ) δ ] α ( C × ) , C α ( ¯ A θλ ( n, r ))) . The latter homomorphism is an isomorphism from the explicit description of C α ( ¯ A θλ ( n, r )).Indeed, C α ( ¯ A θλ ( n, r )) is the direct sum of quantizations of products of Hilbert schemes.The morphism Q Hilb n i ( C ) → Q C n i / S n is an isomorphism over the non-vanishinglocus of δ . This implies that (5.4) is an isomorphism.By the construction, (5.3) is the composition of C α ( ¯ A λ ( n, r )[ δ − ]) → Γ( C α ( ¯ A θλ ( n, r )))[ δ − ]and (5.4). So we have proved that C α ( ¯ A λ ( n, r ))[ δ − ] → Γ( C α ( ¯ A θλ ( n, r )))[ δ − ] is an iso-morphism. Step 7 . For p ∈ ¯ M θ ( n, r ) T × C × let L ( p ) be the corresponding irreducible Γ( C α ( ¯ A θλ ( n, r )))-module from category O . These modules are either finite dimensional (those are param-eterized by the multi-partitions with one part equal to ( n ) and others empty) or hassupport of maximal dimension. It follows from Proposition 5.3 that all finite dimensional L ( p ) restrict to pairwise non-isomorphic C α ( ¯ A λ ( n, r ))-modules. Now consider L ( p ) withsupport of maximal dimension. We claim that the localizations L ( p )[ δ − ] are pairwisenon-isomorphic simple Γ( C α ( ¯ A θλ ( n, r )))[ δ − ]-modules. Let us consider p = ( p , . . . , p r ) and p ′ = ( p ′ , . . . , p ′ r ) with | p i | = | p ′ i | for all i and show that the corresponding localizationsare simple and, moreover, are isomorphic only if p = p ′ . This claim holds if we localize to the regular locus for Q ri =1 S | p i | . Indeed, this localization realizes the KZ functor thatis a quotient onto its image. So the images of L ( p ) , L ( p ′ ) under this localization aresimple and non-isomorphic. Then we further restrict the localizations of L ( p ) , L ( p ′ )to the locus where x i = x j for all i, j . But there is no monodromy of the D-modules L ( p )[ δ − ] , L ( p ′ )[ δ − ] along those additional hyperplanes and these D-modules have reg-ular singularities everywhere. It follows that they remain simple and nonisomorphic (if p = p ′ ). Step 8 . So we see that the C α ( ¯ A λ ( n, r ))[ δ − ]-modules L ( p )[ δ − ] are simple and pair-wise non-isomorphic. The C α ( ¯ A λ ( n, r ))-module L ( p ) is not finitely generated a prioribut always lies in the ind-completion of the category O (thanks to the weight decompo-sition). Pick a finitely generated C α ( A λ ( n, r ))-lattice L ( p ) for L ( p )[ δ − ] inside L ( p ).This now an object in the category O . There is a simple constituent L ( p ) of L ( p ) with L ( p )[ δ − ] = L ( p )[ δ − ] because the right hand side is simple. The finite dimensionalmodules L ( p ) together with the modules of the form L ( p ) give a required number ofpairwise nonisomorphic simple A λ ( n, r ) -modules. (cid:3) Completion of proofs.
The following proposition completes the proof of Theorem1.1.
Proposition 5.5.
Let λ be a positive parameter with denominator n . Then the abelianlocalization holds for ( λ, det) .Proof. Let α be the one-parameter subgroup t ( t d , . . . , t d r ) with d ≫ . . . ≫ d r . Let β : C × → T × C × have the form t (1 , t ). Set α ′ = mα + β for m ≫
0. So we have α ≺ λ α ′ for all λ thanks to Lemma 3.10.Since Γ θλ : O α ′ ( ¯ A θλ ( n, r )) → O α ′ ( ¯ A λ ( n, r )) is a quotient functor, to prove that it is anequivalence it is enough to verify that the number of simples in these two categories is thesame. The number of simples in O α ′ ( ¯ A λ ( n, r )) coincides with that for O ( C α ( ¯ A λ ( n, r )))thanks to Lemma 3.8. The latter is bigger than or equal to the number of simples for O ( L ¯ A λ ( n , . . . , n r ; r )) that, in its turn coincides with the number of the r -multipartitionsof n because the abelian localization holds for all summands ¯ A λ ( n , . . . , n r ; r ). We deducethat the number of simples in O α ′ ( ¯ A θλ ( n, r )) and in O α ′ ( ¯ A λ ( n, r )) coincide. So we see thatΓ θλ : O α ′ ( ¯ A θλ ( n, r )) ։ O α ′ ( ¯ A λ ( n, r )) is an equivalence. Now we are going to show thatthis implies that Γ θλ : ¯ A θλ ( n, r ) -mod → ¯ A λ ( n, r ) -mod is an equivalence. Below we write O instead of O α ′ .Since Γ θλ is an equivalence between the categories O , we see that ¯ A (det) λ,χ ( n, r ) ⊗ ¯ A λ ( n,r ) • and ¯ A (det) λ + χ, − χ ( n, r ) ⊗ ¯ A λ + χ ( n,r ) • are mutually inverse equivalences between O ( ¯ A λ ( n, r ))and O ( ¯ A λ + χ ( n, r )) for χ ≫
0. Set B := ¯ A (det) λ + χ, − χ ( n, r ) ⊗ ¯ A λ + χ ( n,r ) ¯ A (det) λ,χ ( n, r ). This is aHC ¯ A λ ( n, r )-bimodule with a natural homomorphism to ¯ A λ ( n, r ) such that the inducedhomomorphism B ⊗ ¯ A λ ( n,r ) M → M is an isomorphism for any M ∈ O ( ¯ A λ ( n, r )). Itfollows from [BL, Proposition 5.15] that the kernel and the cokernel of B → ¯ A λ ( n, r ) haveproper associated varieties and hence are finite dimensional. Let L denote an irreduciblefinite dimensional ¯ A λ ( n, r )-module, it is unique because of the equivalence O ( ¯ A λ ( n, r )) ∼ = O ( ¯ A λ + χ ( n, r )). Since the homomorphism B ⊗ ¯ A λ ( n,r ) L → L is an isomorphism, we seethat B ։ ¯ A λ ( n, r ). Let K denote the kernel. We have an exact sequenceTor A λ ( n,r ) ( ¯ A λ ( n, r ) , L ) → K ⊗ ¯ A λ ( n,r ) L → B ⊗ ¯ A λ ( n,r ) L → L → TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 33
Clearly, the first term is zero, while the last homomorphism is an isomorphism. Wededuce that K ⊗ ¯ A λ ( n,r ) L = 0. But K is a finite dimensional ¯ A λ ( n, r )-bimodule and hencea ¯ A λ ( n, r ) / Ann L -bimodule and so its tensor product with L can only be zero if K = 0.So we see that ¯ A (det) λ + χ, − χ ( n, r ) ⊗ ¯ A λ + χ ( n,r ) ¯ A (det) λ,χ ( n, r ) ∼ = ¯ A λ ( n, r ). Similarly, ¯ A (det) λ,χ ( n, r ) ⊗ ¯ A λ ( n,r ) ¯ A (det) λ + χ, − χ ( n, r ) ∼ = ¯ A λ + χ ( n, r ). It follows that Γ θλ is an equivalence ¯ A θλ ( n, r ) -mod ∼ = ¯ A λ ( n, r ) -mod. (cid:3) Now we can complete the proof of (2) of Theorem 1.2. It remains to show that¯ A λ ( n, r ) with − r < λ < L denote a finite dimensional irreducible representation. Since L Loc θλ ( ¯ A λ ( n, r )) = ¯ A θλ ( n, r ) and R Γ θλ ( ¯ A θλ ( n, r )) = ¯ A λ ( n, r ), we see that R Γ θλ ◦ L Loc θλ isthe identity functor of D − ( ¯ A λ ( n, r ) -mod). The homology of L Loc θλ ( L ) are supported on¯ ρ − (0). It follows that the denominator of λ is n .Recall that Γ θλ is an exact functor. Since R Γ θλ ◦ L Loc θλ is the identity, the functor Γ θλ doesnot kill the simple ¯ A θλ ( n, r )-module ˜ L supported on ¯ ρ − (0). On the other hand, Γ θλ doesnot kill modules whose support intersects ¯ M θ ( n, r ) reg , the open subvariety in ¯ M θ ( n, r ),where ¯ ρ is an isomorphism. In fact, every simple in O ( ¯ A θλ ( n, r )) is either supported on¯ ρ − (0) or its support intersects ¯ M θ ( n, r ) reg . This is true when λ ′ has denominator n andsatisfies the abelian localization theorem. Indeed, every module in O ( ¯ A λ ′ ( n, r )) is strictlyholonomic. So if it has support of dimension rn −
1, then this support intersects regularlocus, if not, the module is finite dimensional. Our claim about ¯ A θλ ( n, r )-modules follows.So we see that Γ θλ does not kill any irreducible module in O ( ¯ A θλ ( n, r )). So it is anequivalence. However, the proof of Proposition 5.5 shows that this is impossible. Thiscompletes the proof of (2) of Theorem 1.2.6. Affine wall-crossing and counting
The main goal of this section is to prove Theorem 1.5. As in [BL, Section 8], theproof follows from the claim that the wall-crossing functor through the wall δ = 0 (theaffine wall-crossing functor) is a perverse equivalence with homological shifts less thandim M θ ( v, w ). As was pointed out in [BL, 9.2], this follows from results that we havealready proved and the following claims yet to be proved.(i) Let L be a symplectic leaf in M ( v, w ). Consider the categories HC fin ( ¯ A ˆ P ( v, w )) ofHC bimodules over the corresponding slice algebra ¯ A ˆ P ( v, w ) that are finitely gen-erated (left and right) modules over C [ ˆ P ] and HC L ( A ˆ P ( v, w )) ⊂ HC L∪ Y ( A ˆ P ( v, w ))of Harish-Chandra bimodules supported on L and on L ∪ Y , where we write Y for the union of all leaves that do not contain L in their closure. Then, for x ∈ L ,there is a functor • † ,x : HC fin ( ¯ A ˆ P ( v, w )) → HC L ( A ˆ P ( v, w )) that is right adjointto • † ,x : HC L∪ Y ( A ˆ P ( v, w )) → HC fin ( ¯ A ˆ P ( v, w )).(ii) Theorem 1.3 holds together with a direct analog of [BL, Lemma 5.21].(iii) For a unique proper ideal J ⊂ ¯ A λ ( n, r ), where λ has denominator n and liesoutside ( − r, i ¯ A λ ( n,r ) ( ¯ A λ ( n, r ) / J , ¯ A λ ( n, r ) / J ) = ¯ A λ ( n, r ) / J if i iseven, between 0 and 2 nr −
2, and 0 otherwise.(iv) The functor • † ,x is faithful for x and λ specified below.We take a Weil generic λ on the hyperplane of the form h δ, ·i = κ , where κ is a fixed rationalnumber with denominator n ′ . The choice of x is as follows. Recall the description of the symplectic leaves of M ( v, w ) in Subsection 2.3. We want x to lie in the leaf correspondingto the decomposition r ⊕ ( r ) ⊕ n ′ ⊕ ( r ) ⊕ n ′ ⊕ . . . ⊕ ( r q ) n ′ ⊕ r q +1 ⊕ . . . ⊕ r n − q ( n ′ − , wheredim r i = δ, i = 1 , . . . , q, q = ⌊ n/n ′ ⌋ and n is given by n := ⌊ w · v − ( v, v ) / w · δ ⌋ so that n is maximal with the property that v = v − nδ is a root of Q w . We remark thatthe slice to x is M ( n ′ , r ) q , where r is given by r := w · δ .In the proof, it is enough to assume that ν is dominant. If this is not true, we canfind an element σ of W ( Q ) such that σν is dominant, then we have a quantum LMNisomorphism A λ ( v, w ) ∼ = A σ · λ ( σ · v, w ) and h λ, δ i = h σ · λ, δ i .6.1. Functor • † ,x . In this subsection we establish (i) above in greater generality.Here we assume that X → X is a conical symplectic resolution whose all slices areconical and satisfy the additional assumption on contracting C × -actions from Subsection2.8. Recall that under these assumptions, for a point x ∈ X , we can define the exactfunctor • † ,x : HC( A P ) → HC( A P ), where we write P for H ( X, C ). In this subsectionwe are going to study its adjoint • † ,x : HC( A P ) → f HC( A P ). Here f HC( A P ) denotes thecategory of A P -bimodules that are sums of their HC subbimodules. The construction ofthe functor is similar to [L2, L3, L6]. Namely, we pick a HC A P -bimodule N , form theRees bimodule N ~ and its completion N ∧ ~ at 0. Then form the A P , ~ -bimodule M ~ thatis the sum of all HC subbimodules in A ∧ ~ b ⊗ C [[ ~ ]] N ∧ ~ (this includes the condition that theEuler derivation acts locally finitely). It is easy to see that if ~ m ∈ M ~ , then m ∈ M ~ .Then we set N † ,x := M ~ / ( ~ − M ~ . Similarly to [L6, 4.1.4], we see that N 7→ N † ,x is afunctor and that Hom( M , N † ,x ) = Hom( M † ,x , N ).Our main result about the functor • † ,x is the following claim. Proposition 6.1. If N is finitely generated over C [ P ] , then N † ,x ∈ HC( A P ) and V( N † ,x ) = L . Here L stands for the symplectic leaf of X containing x .Proof. The proof is similar to analogous proofs in [L2, 3.3,3.4],[L3, 3.7]. As in those proofs,it is enough to show that if M is a Poisson C [ L ] ∧ x -module of finite rank equipped withan Euler derivation, then the maximal Poisson C [ L ]-submodule of M that is the sumof its finitely generated Poisson C [ L ]-submodules (with locally finite action of the Eulerderivation) is finitely generated. By an Euler derivation, we mean an endomorphism eu of M such that • eu ( am ) = ( eu a ) m + a ( eu m ), • eu { a, m } = { eu a, m } + { a, eu m } − d { a, m } .Here by eu on C [ L ] ∧ x we mean the derivation induced by the contracting C × -action. Step 1 . First, according to Namikawa, [Nam1], the algebraic fundamental group π alg ( L )is finite. Let e L be the corresponding Galois covering of L . Being the integral closure of C [ L ] in a finite extension of C ( L ), the algebra C [ e L ] is finite over C [ L ]. The group π ( e L )has no homomorphisms to GL m by the choice of e L . Also let us note that the C × -actionon L lifts to a C × -action on e L possibly after replacing C × with some covering torus. Weremark that the action produces a positive grading on C [ e L ]. Step 2 . Let V be a weakly C × -equivariant D e L -module. We claim that V is the sum ofseveral copies of O e L . Indeed, this is so in the analytic category: V an := O an e L ⊗ O e L V ∼ = TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 35 O an e L ⊗ V fl (where the superscript “fl” means flat sections) because of the assumptionon π ( e L ). But then the space V fl carries a holomorphic C × -action that has to be di-agonalizable and by characters. So we have an embedding Γ( V ) ֒ → Γ( V an ) C × − fin . SinceSpec( C [ e L ]) is normal, any analytic function on e L extends to Spec( C [ e L ]). Since the gradingon C [ e L ] is positive, any holomorphic C × -semiinvariant function must be polynomial. Sothe embedding above reduces to Γ( V ) ֒ → C [ e L ] ⊗ V fl . The generic rank of Γ( V ) coincideswith dim V fl . Since the module C [ e L ] ⊗ V fl has no torsion, we see that Γ( V ) = C [ e L ] ⊗ V fl .It follows that V = O e L ⊗ V fl as a D-module and the claim of this step follows. Step 3 . Let Y be a symplectic variety. We claim that a Poisson O Y -module carries acanonical structure of a D Y -module and vice versa. If N is a D Y -module, then we equipit with the structure of a Poisson module via { f, n } := v ( f ) n . Here f, n are local sectionsof O Y , N , respectively, and v ( f ) is the skew-gradient of f , a vector field on Y . Let usnow equip a Poisson module with a canonical D-module structure. It is enough to dothis locally, so we may assume that there is an etale map Y → C k . Let f , . . . , f k be thecorresponding etale coordinates. Then we set v ( f ) n := { f, n } . This defines a D-modulestructure on N that is easily seen to be independent of the choice of an ´etale chart.Let us remark that a weakly C × -equivariant Poisson module gives rise to a weakly C × -equivariant D-module and vice versa.So the conclusion of the previous 3 steps is that every weakly C × -equivariant Poisson O e L -module is the direct sum of several copies of O e L . Step 4 . Pick a point ˜ x ∈ e L lying over x so that e L ∧ ˜ x is naturally identified with L ∧ x .Of course, any Poisson module over e L ∧ ˜ x is the direct sum of several copies of C [ e L ] ∧ ˜ x . Sothe claim in the beginning of the proof will follow if we check that any finitely generatedPoisson C [ e L ]-bimodule in C [ e L ] ∧ ˜ x with locally finite eu -action coincides with C [ e L ]. Forthis, let us note that the Poisson center of C [ e L ] ∧ ˜ x coincides with C . On the other hand, anyfinitely generated Poisson submodule with locally finite action of eu is the sum of weakly C × -equivariant Poisson submodules. The latter have to be trivial and so are generatedby the Poisson central elements. This implies our claim and completes the proof. (cid:3) We will also need to consider a map between the sets of two-sided ideals Id ( A ) → Id ( A )induced by the functor • † ,x , compare to [L1, L2, L3]. Namely, for I ∈ Id ( A ) we write I † A ,x for the kernel of the natural map A → ( A / I ) † ,x . Alternatively, the ideal I † A ,x can beobtained as follows. Consider the ideal A ∧ ~ b ⊗ C [[ ~ ]] I ∧ ~ ⊂ A ∧ x ~ . Set J ~ := A ∧ ~ b ⊗ C [[ ~ ]] I ∧ ~ ∩ A ~ .This is a C × -stable ~ -saturated ideal in A ~ and we set I † A ,x := J ~ / ( ~ − J ~ .We will need some properties of the map Id ( A ) → Id ( A ) analogous to those establishedin [L1, Theorem 1.2.2]. Proposition 6.2.
The following is true. (1)
J ⊂ ( J † ,x ) † A ,x for all J ∈ Id ( A ) and ( I † A ,x ) † ,x ⊂ I for all I ∈ Id ( A ) . (2) We have I † A ,x ∩ I † A ,x = ( I ∩ I ) † A ,x . (3) If I is prime, then so is I † A ,x .Proof. (1) and (2) follow from the alternative definition of I † A ,x given above. The proof of(3) closely follows that of an analogous statement, [L1, Theorem 1.2.2,(iv)], let us providea proof for readers convenience. It is easy to see that the ideals I ~ , I ∧ ~ , A ∧ b ⊗ C [[ ~ ]] I ∧ ~ areprime because of the bijections between the sets of two-sided ideals in A , A ~ , A ∧ ~ , A ∧ ~ b ⊗ C [[ ~ ]] A ∧ ~ (we only consider the C × -stable ~ -saturated ideals in the last three algebras). So we need to show that the intersection J ~ of a C × -stable ~ -saturated prime ideal I ′ ~ ⊂ A ∧ x ~ with A ~ is prime. Assume the converse, let there exist ideals J ~ , J ~ ) J ~ such that J ~ J ~ ⊂ J ~ . We may assume that both J i ~ are C × -stable and ~ -saturated.Indeed, if they are not ~ -saturated, then we can saturate them. To see that they canbe taken C × -stable one can argue as follows. The radical of J ~ is C × -stable and sowe can take appropriate powers of the radical for J ~ , J ~ if J ~ is not semiprime. If J ~ is semiprime, then its associated prime ideals are C × -stable and we can take theirappropriate intersections for J ~ , J ~ .So let us assume that J ~ , J ~ are ~ -saturated and C × -stable. Then so are ( J ~ ) ∧ x , ( J ~ ) ∧ x .Also let us remark that ( J ~ ) ∧ x ( J ~ ) ∧ x = ( J ~ J ~ ) ∧ x ⊂ J ∧ x ~ ⊂ I ′ ~ . Without loss of gener-ality, we may assume that ( J ~ ) ∧ x ⊂ I ′ ~ . It follows that J ~ ⊂ J ~ = A ~ ∩ I ′ ~ , and we aredone. (cid:3) Two-sided ideals in ¯ A λ ( n, r ) . The goal of this subsection is to prove Theorem 1.3and more technical statements in (ii) above. We use the following notation. We write A for ¯ A λ ( n, r ) and write A for ¯ A λ ( n ′ , r ), where n ′ is the denominator of λ .Let us start with the description of the two-sided ideals in A . Lemma 6.3.
There is a unique proper ideal in A .Proof. We have seen in the proof of Theorem 1.2 that the proper slice algebras for A haveno finite dimensional representations. So every ideal J ⊂ A is either of finite codimensionor V( A / J ) = ¯ M ( n ′ , r ). The algebra A has no zero divisors so the second option is onlypossible when J = { } . Now suppose that J is of finite codimension. Then A / J (viewedas a left A -module) is the sum of several copies of the finite dimensional irreducible A -module. So J coincides with the annihilator of the finite dimensional irreducible module,and we are done. (cid:3) Let J denote the unique two-sided ideal.Now we are going to describe the two-sided ideals in A ⊗ k . For this we need somenotation. Set I i := A ⊗ i − ⊗ J ⊗ A ⊗ k − i − . For a subset Λ ⊂ { , . . . , k } define the ideals I Λ := P i ∈ Λ I i , I Λ := Q i ∈ Λ I i .Recall that a collection of subsets in { , . . . , k } is called an anti-chain if none of thesesubsets is contained in another. Also recall that an ideal I in an associative algebra A iscalled semi-prime if it is the intersection of prime ideals. Lemma 6.4.
The following is true. (1)
The prime ideals in A ⊗ k are precisely the ideals I Λ . (2) For every ideal
I ⊂ A ⊗ k , there is a unique anti-chain Λ , . . . , Λ q of subsets in { , . . . , k } such that I = T pi =1 I Λ i . In particular, every ideal is semi-prime. (3) For every ideal
I ⊂ A ⊗ k , there is a unique anti-chain Λ ′ , . . . , Λ ′ q of subsets of { , . . . , k } such that I = P qi =1 I Λ ′ i . (4) The anti-chains in (2) and (3) are related as follows: from an antichain in (2), weform all possible subsets containing an element from each of Λ , . . . , Λ p . Minimalsuch subsets form an anti-chain in (3). The proof essentially appeared in [L3, 5.8].
Proof.
Let us prove (1). Let I be a prime ideal. Let x be a generic point in an open leaf L ⊂ V( A ⊗ k / I ) of maximal dimension. The corresponding slice algebra A ′ has a finite TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 37 dimensional irreducible and so is again the product of several copies of A . The leaf L istherefore the product of one-point leaves and full leaves in ¯ M ( n ′ , r ) k . An irreducible finitedimensional representation of A ′ is unique, let I ′ be its annihilator. Then I ⊂ I ′† A⊗ k ,x .By Proposition 6.1, V( A ⊗ k / I ′† A⊗ k ,x ) = L . It follows from [BoKr, Corollar 3.6] that I = I ′† A⊗ k ,x . So the number of the prime ideals coincides with that of the non-emptysubsets { , . . . , k } . On the other hand, the ideals I Λ are all different (they have differentassociated varieties) and all prime (the quotient A ⊗ k / I Λ is the product of a matrix algebraand the algebra A ⊗ k −| Λ | that has no zero divisors).Let us prove (2) (and simultaneously (3)). Let us write I Λ ,..., Λ p for T sj =1 I Λ j . Forideals in A ⊗ k − we use notation like I Λ ′ ,..., Λ ′ q . Reordering the indexes, we may assumethat k ∈ Λ , . . . , Λ s and k Λ s +1 , . . . , Λ p . Set Λ ′ j := Λ j \ { k } for j s . Then(6.1) I Λ ,..., Λ p = ( A ⊗ k − ⊗ J + I Λ ′ ,..., Λ ′ s ⊗ A ) ∩ ( I Λ s +1 ,..., Λ p ⊗ A ) . We claim that the right hand side of (6.1) coincides with(6.2) I Λ s +1 ,..., Λ p ⊗ J + I Λ ′ ,..., Λ ′ s , Λ s +1 ,..., Λ p ⊗ A . First of all, we notice that (6.2) is contained in (6.1). So we only need to prove the oppositeinclusion. The projection of (6.1) to A ⊗ k − ⊗ A / J is contained in I Λ ′ ,..., Λ ′ s , Λ s +1 ,..., Λ p andhence also in the projection of (6.2). Also the intersection of (6.1) with A ⊗ k − ⊗ J iscontained in I Λ s +1 ,..., Λ p ⊗ J . So (6.1) is included into (6.2).Repeating this argument with the two summands in (6.2) and other factors of A ⊗ k we conclude that I Λ ,..., Λ p = P j I Λ ′ j , where the subsets Λ ′ j ⊂ { , . . . , k } are formed asdescribed in (4). So we see that the ideals (2) are the same as the ideals in (3) and that(4) holds. What remains to do is to prove that every ideal has the form described in (2).To start with, we notice that every semi-prime ideal has the form as in (2) because of (1).In particular, the radical of any ideal has such form.Clearly, I Λ ′ I Λ ′ = I Λ ′ ∪ Λ ′ . So it follows any sum of the ideals I Λ ′ j coincides with itssquare. So if I is an ideal whose radical is I Λ ,..., Λ p , then I coincides with its radical. Thiscompletes the proof. (cid:3) Now we are ready to establish a result that will imply Theorem 1.3 together withtechnical results required in (ii). Let x i ∈ ¯ M ( n, r ) be a point corresponding to the leafwith slice ¯ M ( n ′ , r ) i (i.e. to the semisimple representations of the form r ⊕ ( r ) n ′ ⊕ . . . ⊕ ( r i ) n ′ ). We set J i := I † A ,x i , where I is the maximal ideal in A ⊗ i , equivalently, theannihilator of the finite dimensional irreducible representation. Proposition 6.5.
The ideals J i , i = 1 , . . . , q , have the following properties. (1) The ideal J i is prime for any i . (2) V( A / J i ) = L i , where L i is the symplectic leaf containing x i . (3) J ( J ( . . . ( J q . (4) Any proper two-sided ideal in A is one of J i . (5) We have ( J i ) † ,x j = A ⊗ j if j < i and ( J i ) † ,x j = P | Λ | = j − i +1 I Λ else.Proof. (1) is a special case of (3) of Proposition 6.2. (2) follows from Proposition 6.1,compare with the proof of (1) in Lemma 6.4. Let us prove (3). Since ( J i ) † ,x i has finite codimension, we see that it coincides with themaximal ideal in A ⊗ i . So ( J j ) † ,x i ⊂ ( J i ) † ,x i for j < i . It follows that J j ⊂ [( J j ) † ,x i ] † A ,x i ⊂ [( J i ) † ,x i ] † A ,x i = J i .Let us prove (4). The functor • † ,x q is faithful. Indeed, otherwise we have a HC bimodule M with V( M ) ∩ L q = ∅ . But M † ,x has to be nonzero finite dimensional for some x andthis is only possible when x ∈ L i for some i . But L q ⊂ L i for all i that shows faithfulness.Since • † ,x q is faithful and exact, it follows that it embeds the lattice of the ideals in A intothat in A ⊗ q . We claim that this implies that every ideal in A is semiprime. Indeed, thefunctor • † ,x q is, in addition, tensor and so preserves products of ideals. Our claim followsfrom (2) of Lemma 6.4. But every prime ideal in A is some J i , this is proved analogouslyto (1) of Lemma 6.4. Since the ideals J i form a chain, any semiprime ideal is prime andso coincides with some J i .Let us prove (5). We will deduce that from the behavior of • † ,x on the associated vari-eties. We have an action of S j on M ( n ′ , r ) j by permuting factors. The action is inducedfrom N G ( G ˜ x ), where ˜ x is a point from the closed G -orbit lying over x . It follows that theintersection of any leaf with the slice is S j -stable. The associated variety V( A ⊗ j / ( J i ) † ,x j )is the union of some products with factors { pt } and ¯ M ( n ′ , r ), where, for the dimensionreasons, ¯ M ( n ′ , r ) occurs j − i times. Because of the S j -symmetry, all products occur.Now we deduce the required formula for ( J i ) † ,x j from the description of the two-sidedideals in ¯ A ⊗ j . This description shows that for each associated variety there is at most onetwo-sided ideal. (cid:3) Computation of Tor’s.
Here we consider the case when the denominator of λ is n and λ is regular. Set A := ¯ A λ ( n, r ) and let J denote a unique proper ideal in thisalgebra. We want to establish (iii). Proposition 6.6.
We have
Tor A i ( A / J , A / J ) = A / J if i is even and i rn − ,and Tor A i ( A / J , A / J ) = 0 else. The proof closely follows that of [BL, Lemma 7.4] but we need to modify some partsof that argument.
Proof.
Thanks to the translation equivalences it is enough to prove the claim when λ isZariski generic.Let L denote a unique finite dimensional irreducible A -module. What we need to showis that Tor A i ( L ∗ , L ) = C if i is even and 0 i nr − A i ( L ∗ , L ) = Ext i A ( L, L ) ∗ . Knowing that, one can argue asfollows. By Lemma 2.13, Ext i A ( L, L ) = Ext i O ( A ) ( L, L ). The block in O ( A ) containing L was described in Theorem 4.5. In this block, we have Ext i ( L, L ) = C when i is even,0 i nr −
2, and Ext i ( L, L ) = 0 otherwise. To see this one considers the so calledBGG resolution, see [BEG], for the first copy of L and its analog with costandard objectsfor the second copy.So we need to show that Tor A i ( L ∗ , L ) ∗ = Ext i A ( L, L ). The proof is similar to that of[BLPW, Theorem A.1]. Namely, we consider the objects ∆ := A / AA > , ∇ ∨ := A / A < A and let ∇ be the restricted dual of ∇ ∨ . Then, since λ is Zariski generic, we see that ∆is the sum of the standard objects in O ( A ), while ∇ is the sum of all costandard objectsin O ( A ). Then, as we have checked in the appendix to [BLPW], we have Tor A i ( ∇ ∨ , ∆) =Ext i A (∆ , ∇ ) = 0 for i > ∇ ∨ ⊗ A ∆) ∗ = Hom A (∆ , ∇ ) (an equality of TINGOF CONJECTURE FOR QUANTIZED QUIVER VARIETIES II: AFFINE QUIVERS 39 C ¯ M θ ( n, r ) T × C × -bimodules). Taking a resolution P of the first copy of L in Ext i A ( L, L )by direct summands of ∆ (which exists because of the structure of O ( A )) and of thesecond copy by direct summands of ∇ , denote this resolution by Q , we get Ext i A ( L, L ) = H i (Hom A ( P, Q )) = H i ( Q ∨ ⊗ A P ) ∗ = Tor A i ( Q ∨ , P ) ∗ = Tor i A ( L ∗ , L ) ∗ . (cid:3) Faithfulness.
Now we are going to establish (iv) for x and λ specified above. Let L be the leaf containing x and L ′ be the leaf corresponding to the decomposition r ⊕ ( r ) ⊕ n ,where r is an irreducible representation of dimension δ . Clearly, L ′ ⊂ L .We are going to prove that L is contained in the associated variety of any HC A λ ( v, w )-bimodule M (or A λ ′ ( v, w )- A λ ( v, w )-bimodule or A λ ( v, w )- A λ ′ ( v, w )-bimodule; thanksto Proposition 6.1, a direct analog of [BL, Corollary 5.19] holds so that the associ-ated variety of any HC bimodule A λ ′ ( v, w )- A λ ( v, w )-bimodule coincides with those of A λ ′ ( v, w ) / J ℓ , A λ ( v, w ) / J r , where J ℓ , J r are the left and right annihilators). This is equiv-alent to saying that • † ,x is faithful. The scheme of the proof is as follows:(a) We first show that L ′ ⊂ V( M ). We do this by showing that M † ,y cannot be finitedimensional nonzero for y from a leaf L such that L ′
6⊂ L .(b) From L ′ ⊂ V( M ) we deduce that L ⊂ V( M ).Let us deal with (a). As in the proof of [BL, Lemma 7.10], it is enough to showthat the slice algebra A corresponding to y has no finite dimensional representations.So let us analyze the structure of the leaves that contain L ′ in their closure. For apartition µ = ( µ , . . . , µ k ) with | µ | < n , we write L ( µ ) for the leaf corresponding to thedecomposition of the form r ⊕ ( r ) ⊕ µ ⊕ . . . ⊕ ( r k ) ⊕ µ k . From [CB, Theorem 1.2] it followsthat L ′ = L ( n ) ⊂ L ( µ ). There is a natural surjection from the set of leaves in ¯ M ( n, r )(this is precisely the slice to L ′ ) to the set of leaves in M ( v, w ) whose closure contains L ′ .As in the proof of [BL, Lemma 7.10,(1)], we need to prove that, for a generic p ∈ ker δ ,the variety ¯ M p ( v, w ) has no single point symplectic leaves as long as the correspondingsymplectic leaf L is different from L ( µ ). The latter is equivalent to the condition thatthe decomposition of v defining L contains real roots. So let this decomposition be v = v + ν δ + . . . + ν ℓ δ + P i ∈ Q m i α i . The slice variety ¯ M p ( v, w ) is the product Q ℓ ¯ M ( ν i , r ) ×M ( v ′ , w ′ ), where v ′ = ( m i ) i ∈ Q and w ′ is given by w ′ i = w i − ( v , α i ). We remark thatdim M ( v ′ , w ′ ) >
0. Indeed, we havedim M ( v ′ , w ′ ) = 2 X i ∈ Q ( w i − ( v , α i )) m i − ( X i m i α i , X i m i α i ) == 2 w · ( v − v ) − v , v − v ) − ( v − v , v − v ) == 2 w · ( v − v ) − v, v − v ) + ( v − v , v − v ) . But w · ( v − v ) > ( v, v − v ) because ν is dominant. We also have ( v − v , v − v ) > v − v = kδ . But if the last equality holds, then we have( v, v − v ) = 0, while w · δ is always positive. The quiver defining M p ( v ′ , w ′ ) has no loopsso that variety cannot have single point leaves. So we have proved that A cannot have afinite dimensional representation. This implies that L ′ ⊂ V( M ).Now let us show that L ⊂ V( M ). The slice algebra corresponding to L ′ is A ′ =¯ A h λ,δ i ( n, r ). It follows that V( M † ,x ′ ) contains the leaf corresponding to the partition µ = ( n ′ q ). It follows that V( M ) contains L ( µ ). Affine wall-crossing functor and counting.
Using (i)-(iv) proved above one getsa direct analog of [BL, Theorem 7.2]. Let us introduce some notation. Let θ, θ ′ be twostability conditions separated by ker δ . Let λ, λ ′ be two parameters with λ ′ − λ ∈ Z Q , h λ, δ i has denominator n ′ n , and ( λ, θ ), ( λ ′ , θ ′ ) satisfy the abelian localization. Set q = ⌊ n/n ′ ⌋ . Theorem 6.7.
There are chains of two-sided ideals { } ( J q ( J q − ( . . . ( J ( A λ ( v, w ) and { } ( J ′ q ( J ′ q − ( . . . ( J ′ ( A λ ′ ( v, w ) with the following properties.Let C i be the subcategory of all modules in A λ ( v, w ) -mod annihilated by J q +1 −⌊ i/ ( rm − ⌋ (this is a Serre subcategory by a direct analog of (1) of [BL, Theorem 7.1] ) and let C ′ i ⊂A λ ′ ( v, w ) -mod be defined analogously. Then the following is true: (1) WC θ → θ ′ , WC θ ′ → θ are perverse equivalences with respect to these filtrations inducingmutually inverse bijections between simples. (2) For a simple S ∈ C j ( rm − \C j ( rm − , the simple S ′ is a quotient of H j ( rm − ( WC θ → θ ′ S ) . (3) The bijection S S ′ preserves the associated varieties of the annihilators. Similarly to [BL, Section 8], this theorem implies Theorem 1.5.
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I.L.: Department of Mathematics, Northeastern University, Boston MA 02115 USA
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