QUIVER VARIETIES AND TENSOR PRODUCTS, II

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Nov 25, 2012 - also by a recent work of Maulik and Okounkov [12] on a geometric con- struction of a tensor product structure on equivariant homology groups.
arXiv:1207.0529v2 [math.QA] 25 Nov 2012

QUIVER VARIETIES AND TENSOR PRODUCTS, II HIRAKU NAKAJIMA Abstract. We define a family of homomorphisms on a collection of convolution algebras associated with quiver varieties, which gives a kind of coproduct on the Yangian associated with a symmetric Kac-Moody Lie algebra. We study its property using perverse sheaves.

Contents Introduction 1. Quiver varieties 2. Tensor product varieties 3. Coproduct 4. Tensor product multiplicities References

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Introduction In the conference the author explained his joint work with Guay on a construction of a coproduct on the Yangian Y (g) associated with an affine Kac-Moody Lie algebra g. It is a natural generalization of the coproduct on the usual Yangian Y (g) for a finite dimensional complex simple Lie algebra g given by Drinfeld [7]. Its definition is motivated also by a recent work of Maulik and Okounkov [12] on a geometric construction of a tensor product structure on equivariant homology groups of holomorphic symplectic varieties, in particular of quiver varieties. The purpose of this paper is to explain this geometric background. For quiver varieties of finite type, the geometric coproduct corresponding to the Drinfeld coproduct on Yangian Y (g), or more precisely the quantum affine algebra Uq (b g), was studied in [21, 17, 22]. (And one corresponding to the coproduct on g was studied also in [11].) But the 2000 Mathematics Subject Classification. Primary 17B37, Secondar 14D21, 55N33. Supported by the Grant-in-aid for Scientific Research (No.23340005), JSPS, Japan. 1

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results depend on the algebraic definition of the coproduct. As it is not known how to define a coproduct on Y (g) for an arbitrary Kac-Moody Lie algebra g, the results cannot be generalized to other types. In this paper, we take a geometric approach and define a kind of a coproduct on convolution algebras associated with quiver varieties together with a C∗ -action preserving the holomorphic symplectic form, and study its properties using perverse sheaves. In fact, we have an ambiguity in the definition of the coproduct, and we have a family of coproducts ∆c , parametrized by c in a certain affine space. This ambiguity of the coproduct was already noticed in [22, Remark in §5.2]. Maulik-Okounkov theory gives a canonical choice of c for a quiver variety of an arbitrary type, and gives the formula of ∆c on standard generators of Y (g). Therefore we can take the formula as a definition of the coproduct and check its compatibility with the defining relations of Y (g). This will be done for an affine Kac-Moody Lie algebra g as we explained in the conference. (The formula is a consequence of results in [12], and hence is not explained here.) Although there is a natural choice, the author hopes that our framework, considering also other possibilities for ∆, is suitable for a modification to other examples of convolution algebras when geometry does not give us such a canonical choice. (For example, the AGT conjecture for a general group. See [20].) Remark also that our construction is specific for Y (g), and is not clear how to apply for a quantum loop algebra Uq (Lg). We need to replace cohomology groups by K groups to deal with the latter, but many of our arguments work only for cohomology groups. Finally let us comment on a difference on the coproduct for quiver varieties of finite type and other types. A coproduct on an algebra A usually means an algebra homomorphism ∆ : A → A ⊗ A satisfying the coassociativity. In our setting the algebra A depends on the dimension vector, or equivalently dominant weight w. Hence ∆ is supposed to be a homomorphism from the algebra A(w) for w to the tensor product A(w1 ) ⊗ A(w2 ) with w = w1 + w2 . For a quiver of type ADE, this is true, but not in general. See Remark 2.4 for the crucial point. The target of ∆ is, in general, larger than A(w1 ) ⊗ A(w2 ). Fortunately this difference is not essential, for example, study of tensor product structures of representations of Yangians. Notations. The definition and notation of quiver varieties related to a coproduct are as in [17], except the followings: • Linear maps i, j are denoted by a, b here.

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• A quiver possibly contains edge loops. Roots are defined as in [6, §2]. They are obtained from coordinate vectors at loop free vertices or ± elements in the fundamental region by applying some sequences of reflections at loop free vertices. e are denote by T, T e here. • Varieties Z, Z We say a quiver is of finite type, if its underlying graph is of type ADE. We way it is of affine type, if it is Jordan quiver or its underlying graph is an extended Dynkin diagram of type ADE. For v = (vi ), v′ = (vi′ ) ∈ ZI , we say v ≤ v′ if vi ≤ vi′ for any i ∈ I. For a variety X, H∗ (X) denote its Borel-Moore homology group. It is the dual to Hc∗ (X) the cohomology group with compact support. We will use the homology group H∗ (L) of a closed variety L in a smooth variety M in several contexts. There is often a preferred degree in the context, which is written as ‘top’ below. For example, if L is lagrangian, it is dimC M. If M has several components Mα of various dimensions, we mean Htop (L) to be the direct sum of Htop (L ∩ Mα ), though the degree ‘top’ changes for each L ∩ Mα . Let D(X) denote the bounded derived category of complexes of constructible C-sheaves on X. When X is smooth, CX ∈ D(X) denote the constant sheaf on X shifted by dim X. If X is a disjoint union of smooth varieties Xα with various dimensions, we understand CX as the direct sum of CXα . The intersection cohomology (IC for short) complex associated with a smooth locally closed subvariety Y ⊂ X and a local system ρ on Y is denoted by IC(Y, ρ) or IC(Y , ρ). If ρ is the trivial rank 1 local system, we simply denote it by IC(Y ) or IC(Y ). 1. Quiver varieties In this section we fix the notation for quiver varieties. See [13, 14] for detail. Suppose that a finite graph is given. Let I be the set of vertices and E the set of edges. In [13, 14] the author assumed that the graph does not contain edge loops (i.e., no edges joining a vertex with itself), but most of results (in particular definitions, natural morphisms, etc) hold without this assumption. Let H be the set of pairs consisting of an edge together with its orientation. So we have #H = 2#E. For h ∈ H, we denote by i(h) (resp. o(h)) the incoming (resp. outgoing) vertex of h. For h ∈ H we denote by h the same edge as h with the reverse orientation. Choose and fix an orientation Ω of the graph, i.e., a subset Ω ⊂ H such that Ω ∪ Ω = H, Ω ∩ Ω = ∅. The pair (I, Ω) is called a quiver.

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Let V = (Vi )i∈I be a finite dimensional I-graded vector space over C. The dimension of V is a vector dim V = (dim Vi )i∈I ∈ ZI≥0 . If V 1 and V 2 are I-graded vector spaces, we define vector spaces by M M def. def. 1 2 Hom(Vo(h) , Vi(h) ). L(V 1 , V 2 ) = Hom(Vi1 , Vi2 ), E(V 1 , V 2 ) = i∈I

h∈H

1

2

For B = (Bh ) ∈ E(V , V ) and C = (Ch ) ∈ E(V 2 , V 3 ), let us define a multiplication of B and C by   X def. CB =  Ch Bh  ∈ L(V 1 , V 3 ). i(h)=i

i

1

2

Multiplications ba, Ba of a ∈ L(V , V ), b ∈ L(V 2 , V 3 ), B ∈ E(V 2 , V 3 ) 1 1 are defined in the P obvious manner. If a ∈ L(V , V ), its trace tr(a) is understood as i tr(ai ). For two I-graded vector spaces V , W with v = dim V , w = dim W , we consider the vector space given by def.

M ≡ M(v, w) = E(V, V ) ⊕ L(W, V ) ⊕ L(V, W ), where we use the notation M when v, w are clear in the context. The above threeL components L L for an element of M will be denoted by B= Bh , a = ai , b = bi respectively. The orientation Ω defines a function ε : H → {±1} by ε(h) = 1 if h ∈ Ω, ε(h) = −1 if h ∈ Ω. We consider ε as an element of L(V, V ). Let us define a symplectic form ω on M by def.

ω((B, a, b), (B ′, a′ , b′ )) = tr(εB B ′ ) + tr(ab′ − a′ b). Let G ≡ Gv be an algebraic group defined by Y def. G ≡ Gv = GL(Vi ).

Its Lie algebra is the direct sum

L

i

i

gl(Vi ). The group G acts on M by def.

(B, a, b) 7→ g · (B, a, b) = (gBg −1, ga, bg −1) preserving the symplectic structure. The moment map vanishing at the origin is given by µ(B, a, b) = εB B + ab ∈ L(V, V ), where the dual of the Lie algebra of G is identified with L(V, V ) via the trace.

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We would like to consider a ‘symplectic quotient’ of µ−1 (0) divided by G. However we cannot expect the set-theoretical quotient to have a good property. Therefore we consider the quotient using the geometric invariant theory. Then the quotient depends on an additional parameter ζ = (ζi )i∈I ∈ ZI as follows: Let us define a character of G by Y def. (det gi )−ζi . χζ (g) = i∈I

Let A(µ−1 (0)) be the coodinate ring of the affine variety µ−1 (0). Set n

def.

A(µ−1 (0))G,χζ = {f ∈ A(µ−1 (0)) | f (g·(B, a, b)) = χζ (g)n f ((B, a, b))}. The direct sum with respect to n ∈ Z≥0 is a graded algebra, hence we can define M n def. A(µ−1 (0))G,χζ ). Mζ ≡ Mζ (v, w) ≡ Mζ (V, W ) = Proj( n≥0

This is the quiver variety introduced in [13]. Since this space is unchanged when we replace χ by a positive power χN (N > 0), this space is well-defined for ζ ∈ QI . We call ζ a stability parameter. We use two special stability parameters in this paper. When ζ = 0, the corresponding M0 is an affine algebraic variety whose coordinate ring consists of the G-invariant functions on µ−1 (0). Another choice is ζi = 1 for all i. In this case, we denote the corresponding variety simply by M. The corresponding stability condition is that an I-graded subspace V ′ of V invariant under B and contained in Ker b is 0 [14, Lemma 3.8]. The stability and semistability are equivalent in this case, and the action of G on the set µ−1 (0)s of stable points is free, and M is the quotient µ−1 (0)s /G. In particular M is nonsingular. 2. Tensor product varieties Let W 2 ⊂ W be an I-graded subspace and W 1 = W/W 2 be the quotient. We fix an isomorphism W ∼ = W 1 ⊕ W 2 . We define a one ∗ parameter subgroup λ : C → GW by λ(t) = idW 1 ⊕ t idW 2 . Then C∗ acts on M, M0 through λ. We fix v, w and w1 = dim W 1 , w2 = dim W 2 throughout this paper. Since we use several quiver varieties with different dimension vectors, let us use the notation M(v1, w1 ), etc for those, while the notation M means the original M(v, w).

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2(i). Fixed points. We consider the fixed point loci MC , MC0 . The former decomposes as G ∗ M(v1, w1 ) × M(v2, w2 ) (2.1) MC = v=v1 +v2

(see [17, Lemma 3.2]). The isomorphism is given by considering the direct sum of [B 1 , a1 , b1 ] ∈ M(v1 , w1) and [B 2 , a2 , b2 ] ∈ M(v2 , w2) as a point in M. Since quiver varieties M(v1 , w1 ), M(v2 , w2) are connected, ∗ this is a decomposition of MC into connected components. ∗ Let us study the second fixed point locus MC0 . We have a morphism G ∗ σ: M0 (v1 , w1 ) × M0 (v2 , w2) → MC0 v=v1 +v2

given by the direct sum as above. This cannot be an isomorphism unless v = 0 as the inverse image of 0 consists of several points corresponding to various decomposition v = v1S+ v2 . This is compensated by considering the direct limit M0 (w) = v M0 (v, w) if the underlying graph is of type ADE. But this trick does not solve the problem yet in general. For example, if the quiver is the Jordan quiver, and v1 = w1 = v2 = w2 = 1, we have M0 (v1 , w1 ) = M0 (v2 , w2) = C2 , ∗ while MC0 = S 2 (C2 ). The morphism σ is the quotient map C2 × C2 → S 2 (C2 ) = (C2 × C2 )/S2 . Let us study σ further. Using the stratification [14, Lemma 3.27] we decompose M0 = M0 (v, w) as G reg M0 (v0 , w) × M0 (v − v0 , 0), (2.2) M0 = v0

0 0 where Mreg 0 (v , w) is the open subvariety of M0 (v , w) consisting of closed free orbits, and M0 (v − v0 , 0) is the quiver variety associated with W = 0. For quiver varieties of type ADE, the factor M0 (v−v0 , 0) is a single point 0. It is nontrivial in general. For example, if the quiver is the Jordan quiver, we have M0 (v−v0 , 0) = S n (C2 ) where n = v−v0 . Then

Lemma 2.3. (1) The above stratification induces a stratification G ∗ reg 2 1 1 2 0 Mreg MC0 = 0 ( v, w ) × M0 ( v, w ) × M0 (v − v , 0). v0 ,1v,2v v0 =1v+2v

(2) σ is a surjective finite morphism. Thus the factor with W = 0 appears twice in M0 (v1 , w1 )×M0 (v2 , w2 ) ∗ while it appears only once in MC0 .

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0 0 Proof. (1) We consider Mreg 0 (v , w) as an open subvariety in M(v , w) and restrict the decomposition (2.1). Then it is easy to check that 0 (x, y) ∈ M(1v, w1 ) × M(2v, w2) is contained in Mreg 0 (v , w) if and only ∗ reg 1 reg 2 1 2 0 C if x, y are in M0 ( v, w ), M0 ( v, w ) respectively. Thus Mreg = 0 (v , w) reg 1 reg 2 1 2 ∗ M0 ( v, w ) × M0 ( v, w ). Now the assertion is clear as C acts trivially on the factor M0 (v − v0 , 0). (2) The coordinate ring of M0 is generated by the following two types of functions:

• tr(BhN BhN−1 · · · Bh1 : Vo(h1 ) → Vi(hN ) = Vo(h1 ) ), where h1 , . . . , hN is a cycle in our graph. • χ(bi(hN ) BhN BhN−1 · · · Bh1 ao(h1 ) ), where h1 , . . . , hN is a path in our graph, and χ is a linear form on Hom(Wo(h1 ) , Wi(hN ) ). ∗

Then the generators for MC0 are the first type functions and second 1 1 2 2 type functions with χ = (χ1 , χ2 ) ∈ Hom(Wo(h , Wi(h )⊕Hom(Wo(h , Wi(h ). 1) 1) N) N) If we pull back these functions by σ, they become sums of the same types of functions for M0 (v1 , w1 ) and M0 (v2 , w2 ). From this observation, we can easily see that σ is a finite morphism. From (1) it is clearly surjective.  Remark 2.4. Let Z(va , wa) be the fiber product M(va , wa ) ×M0 (va ,wa ) ∗ ∗ M(va, wa ) for a = 1, 2. The fiber product MC ×MC0 ∗ MC is larger than the union of the products Z(v1 , w1 ) × Z(v2 , w2) in general. For example, consider the Jordan quiver variety with v1 = v2 = w1 = w2 = 1. Then M(va , wa ) is C2 . The product Z(v1 , w1 ) × Z(v2 , w2 ) is consisting of points (p1 , q1 , p2 , q2 ) with p1 = q1 , p2 = q2 . On the other ∗ ∗ hand, MC ×MC0 ∗ MC contains also points with p1 = q2 , p2 = q1 . On the other hand, if the quiver is of type ADE, we do not have the factor M0 (v − v0 , 0), and they are the same. 2(ii). Review of [17]. In this subsection we recall results in [17, §3], with emphasis on subvarieties in the affine quotient M0 . We first define the following varieties which were implicitly introduced in [17, §3]: def.

T0 = {x ∈ M0 | lim λ(t)x exists}, t→0

def.

e 0 = {x ∈ M0 | lim λ(t)x = 0}. T t→0

By the proof of [17, Lemma 3.6] we have the following: x = [B, a, b] is e 0 ) if and only if in T0 (resp. T

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2 2 2 • bi(hN ) BhN BhN−1 · · · Bh1 ao(h1 ) maps Wo(h to Wi(h (resp. Wo(h 1) 1) N) 2 to 0 and the whole Wo(h1 ) to Wi(h ) for any path in the doubled N) quiver. e 0 are closed subvarieties From this description it also follows that T0 , T in M0 . ∗ We have the inclusion i : T0 → M0 and the projection p : T0 → MC0 defined by taking limt→0 λ(t)x. The latter is defined as M0 is affine. def. e def. e 0 ). These definitions coincide We define T = π −1 (T0 ), T = π −1 (T with ones in [17, §3]. Note that we do not have an analog of p : T0 → ∗ MC0 for T. Instead we have a decomposition G T(v1 , w1; v2 , w2 ) (2.5) T= v=v1 +v2

into locally closed subvarieties, and the projection (2.6)

p(v1 ,v2 ) : T(v1 , w1 ; v2 , w2) → M(v1 , w1) × M(v2 , w2),

which is a vector bundle. These are defined by considering the limit limt→0 λ(t)x. Note that they intersect in their closures, contrary to (2.1), which was the decomposition into connected components. Since pieces in (2.5) are mapped to different components, p(v1 ,v2 ) ’s do not give a morphism defined on T. As a vector bundle, T(v1 , w1 ; v2, w2 ) is the subbundle of the normal bundle of M(v1 , w1 ) × M(v2 , w2 ) in M consisting of positive weight spaces. Its rank is half of the codimension of M(v1, w1 )×M(v2, w2 ). In fact, the restriction of the tangent space of M to M(v1 , w1)×M(v2 , w2) decomposes into weight ±1 and 0 spaces such that • the weight 0 subspace gives the tangent bundle of M(v1 , w1) × M(v2 , w2), • the weight 1 and −1 subspaces are dual to each other with respect to the symplectic form on M. We define a partial order < on the set {(v1, v2 ) | v1 + v2 = v} defined by (v1 , v2 ) ≤ (v′1 , v′2 ) if and only if v1 ≤ v′1 . We extend it to a total order and denote it also by