varieties and kac-moody - semantic scholar...groups, which satisfies the defining relation of the...

Vol. 91, No. 3 DUKE MATHEMATICAL JOURNAL (C) 1998

QUIVER VARIETIES AND KAC-MOODY ALGEBRAS

HIRAKU NAKAJIMA

1. Introduction 5162. A modified enveloping algebra 5173. Quiver varieties 519

i. Definition 519ii. Stability condition 522

iii. The projective morphism 525iv. tm-action 526v. Stratification of 9J/0 526

4. Tautological bundle homomorphisms 5295. Hecke correspondence 532

i. Definition 532ii. Fibration 536

6. A decomposition of the diagonal 5367. A Lagrangian subvariety Z 5388. Convolution algebra 539

i. Convolution in homology 539ii. The case when Z is Lagrangian 541

9. A geometric construction of the algebra U 542i. Main construction 542

ii. Definition of Ek, Fk, and Hk 543iii. Integrability 543iv. Relations 544

10. Module )ntop(gJ/(v, wx)) 546i. Operators Ek and Fk 546

ii. Integrable highest weight modules 548iii. Criterion for the nonemptiness of w/reg 5480iv. Coordinate algebras 551

11. Intersection form of 9J/(v, w) 552Appendix 553

i. Proof of Lemma 9.8 553ii. Proof of Lemma 9.9 555

iii. Proof of Lemma 9.10 558

Received 30 May 1995. Revision received 11 December 1996.Author’s work supported in part by Grant-in-Aid for Scientific Research number 05740041, the

Ministry of Education, Science and Cul.ture, Japan, and also by the Inamori Foundation.

515

516 HIRAKU NAKAJIMA

1. Introduction. This paper is a sequel of [27]. In the previous paper, wedefined a new family of hyper-K/ihler manifolds, called quiver varieties, asso-ciated to finite graphs without edge loops. The definition was motivated by theADHM (Atiyah, Drinfel’d, Hitchin, and Manin) description (see [18]), which wecan now interpret as the identification of moduli spaces of anti-self-dual con-nections on ALE (asymptotically locally Euclidean) spaces with quiver varietiesof affine type. Quiver varieties can be viewed as higher dimensional analogues ofALE spaces in many respects. We studied their geometric properties which aresurprisingly rich. In particular, we constructed integrable highest weight repre-sentations of the Kac-Moody algebra on the space of constructible functions oncertain Lagrangian subvarieties. This construction was motivated by Ringel andLusztig’s construction (see [31] and [19]) of the lower triangular part of theDrinfel’d-Jimbo quantized enveloping algebra Uq in terms of quivers.The purpose of this paper is to try to construct the whole enveloping algebra.

We consider Cartesian products of quiver varieties and introduce Lagrangiansubvarieties in them. Then we define the convolution product on homologygroups, which satisfies the defining relation of the enveloping algebra. It alsogives reformulation of our previous construction of representations. We usehomology groups instead of constructible functions.

In fact, when the underlying graph is not of finite type, it is more natural torelate our convolution algebra to the mo_dified enveloping algebra, which is thespecialization (at q 1) of the algebra Uq introduced by Lusztig in [23]. Thealgebra is a quotient of ll [Jqlq=l by a certain ideal depending on the choice ofa highest weight vector w. When w goes to the infinity, the ideal becomessmaller. (The limit is not treated in this paper.) As an application of our con-struction, we obtain a basis of the quotient of the modified enveloping algebragiven by fundamental classes of Lagrangian subvarieties. It is natural to con-j_ecture that this basis relates to the specialization of Lusztig’s canonical basis onUq (see [23]).Our construction is strongly motivated by Ginzburg’s construction of the uni-

versal enveloping algebra U(eln) in terms of homology groups of cotangentbundles of generalized flag varieties (see [7]). Our definition of the convolutionproduct is also borrowed from Ginzburg’s paper. Cotangent bundles of general-ized flag varieties are examples of quiver varieties [27, Sect. 7], and our con-struction is a generalization of his construction. Ginzburg’s proof was given in[4, Chap. 3]. His construction was motivated by Beilinson, Lusztig, and Mac-Pherson’s construction [1] of quantized enveloping algebras, in turn.We remark that our construction can be explained in the gauge theoretic lan-

guage, as in [28], when the underlying graph is of affine type. During the prepa-ration of this paper, we received an announcement of similar results for generalprojective algebraic surfaces by Ginzburg, Kapranov, and Vasserot [8]. Theirformulation uses functions on rational points of moduli spaces of vector bundles,instead of homology groups. Our proof heavily depends on our description interms of quivers. The essential point is to show that Hecke correspondences

QUIVER VARIETIES AND KAC-MOODY ALGEBRAS 517

intersect transversely (see appendix). It seems difficult to prove such results forgeneral surfaces. In the approach of Ginzburg et al., this transversality conditionis unnecessary since they work on functions on rational points.The paper is organized as follows. In {}2 we give a quick review of the defini-

tion of the universal enveloping algebras of Kac-Moody algebras associated tosymmetrizable generalized Cartan matrices and their modified version. In 3 weintroduce quiver varieties. Our approach is a little bit different from the approachin [27], and works for arbitrary algebraically closed fields. In 4 we define certainvector bundles on quiver varieties and tautological bundle homomorphismsbetween them. We also introduce a stratification of quiver varieties in terms of therank of these homomorphisms. In 5 we define nonsingular Lagrangian sub-varieties, called Hecke correspondences, in Cartesian products of quiver varieties.These give us generators of Kac-Moody algebras and play a crucial role in ourconstruction. In 6 we show that the quiver variety is connected. For the proof weuse a decomposition of the diagonal, which is of independent interest. In 7 weintroduce Lagrangian subvarieties which contain Hecke correspondences as irre-ducible components. In {}8 we review the definition of the convolution product onhomology groups following Ginzburg (see [7]). In {}9 we apply Ginzburg’s theoryto the above Lagrangian subvarieties introduced in 7. We give a homomorphismfrom lI to the convolution algebra. In 10 we identify natural modules of the con-volution algebra with integrable highest weight representations of Kac-Moodyalgebras. We also introduce a certain natural subalgebra to which the homo-morphism is surjective. During the proof, we give a geometric analogue of Kashi-wara’s operator (see [12]). In 11 we identify the intersection form of the quivervariety with the invariant inner product on the highest weight representation. Inparticular, we can show that the intersection form is definite. In the appendix, wegive proofs of the lemmas in 9.

After this work was done, the author learned that Grojnowski announceda similar geometric construction of the affinization of the quantized universalenveloping algebra using the equivariant K-theory on the same Lagrangian sub-varieties (see [10]).

Acknowledoements. I am deeply indebted to V. Ginzburg whose lecture atKyoto (September 1993) is the starting point of the present work. I would liketo express my sincere gratitude to M. Kashiwara for a number of interestingdiscussions. I thank G. Lusztig who pointed out an error in the earlier versionof this paper. My thanks go also to R. Hotta, T. Uzawa, K. Hasegawa, andG. Kuroki, who answered a number of my questions on representation theory.

2. A modified enveloping algebra. We give a quick review of the definitionfor the universal enveloping algebras of Kac-Moody variety associated to sym-metrizable generalized Cartan matrices and their modified version followingLusztig in [23] and Kashiwara in [13]. Here we treat only the symmetric case,for the algebras defined via quiver varieties are symmetric.

518 HIRAKU NAKAJIMA

Suppose that the following data are given:(1) P: free g-module (weight lattice),(2) P* Homz(P, Z) with a natural pairing (,) P (R) P* - Z,(3) k P (k 1, 2,..., n) (simple roots),(4) hk P* (k 1, 2,..., n) (simple coroot),(5) a symmetric bilinear form (,) on P.

Those are required to satisfy(a) (hk,2) (Ok,/) for k 1,2,... ,n and 2 P,(b) C ((hk, Ol))k,l is a symmetric generalized Cartan matrix, that is, (hk, tk)

2, and (hk, Ol) (hi, Ok) < 0 for k l,(C) {0k)= are linearly independent,(d) there exists Ak e P (k 1, 2,..., n) such that (he, Ak) 6kl.The universal enveloping algebra U of the Kac-Moody algebra is the -algebra generated by ek, fk (k 1, 2,..., n), h P* with relations

[h,h’] 0 for h,h’ P*, (2.1)

[h, ek] (h, k)ek, (2.2)

[h, fk] --(h,k)fk, (2.3)

[ek, fl] 6klhk, (2.4)

(_1)1 Ckl ee-,-p 0 (k v l), (2.5)

p=0 P

--Cki

E (-1)1 Ckl 4,p_.1--Ckl--p /), (2.6)

=0 p JJJlJk :0 (k

/

where ) is the binomial coefficient. The equations (2.5) and (2.6) are called Serre

relations.Let U+ (resp., U-) be the -subalgebra of U generated by elements ek’s (resp.,

fk’S). Let U be the -subalgebra generated by elements h P*. Then we havethe triangle decomposition (see [11, 9.1.1])

U U+ (R) U (R) U-. (2.7)

We introduce the modified enveloping algebra I1 by replacing U with thesum of 1-dimensional algebras


Here the multiplication is defined by the following rules:

aaau 6xuax (2.8)

eka, aA+kek fka aA-otkfk, (2.9)

(ekfl flek)a. 6kl(hk, 2)a. (2.10)

This was the specialization (at q 1) of the modified quantized enveloping alge-bra introduced by Lusztig in [23, Chap. 23]. Note that U is an algebra withoutunit, in general.A U-module M is said to be unital (see [23, 23.1.4]) if(a) for any m M we have axm 0 for all but finitely many 2 P,(b) for any m e M we have Y’], a,m m.

If M is a unital I-module, it can be considered as a U-module with weightdecomposition.For a dominant integral weight A, the integrable highest weight U-module

with highest weight A (see [23, 3.5.6]) is denoted by L(A). The same notationwill be used for the unital U-module.

3. Quiver varieties. The purpose of this section is to review the definition andprove some properties of the quiver varieties introduced in [27]. Almost allproperties of quiver varieties given in this section were proved in [27]. Ourapproach is slightly different from the original one. Instead of using a Kihlerquotient by a compact group, we use the geometric invariant theory (GIT) quo-tient by the eomplexifieation with a certain polarization. This is useful when wecompute the convolution product in {}9. Moreover, the definition makes senseover any algebraically closed field IF, not only over E. We replace the originalproofs, which used K[ihler metrics, by arguments which are valid even in posi-tive characteristic. (Results concerning the sympleetie form are exceptions.)Another difference is that we do not introduce the deformation parameter ez,and we touch only on the ease ez 0. This is only for brevity. The modificationfor the general case is straightforward.The notation is also slightly different from [27]. In [27] the group G was a

compact group, but here G means the eomplexification of the compact group(denoted by GC in [27]). The moment map # used here is the complex part #c ofthe hyper-Kihler moment map in [26]. The symplectie form co corresponds tothe holomorphic sympleetie form coez. The symbols An and ]pn denote, respec-tively, the affine n-space and projective n-space; m denotes the multiplieativegroup.

3.i. Definitions. Suppose that a finite graph is given and assume that thereare no edge loops (i.e., no edges joining a vertex with itself). Let I be the set ofvertices and E be the set of edges. We number the vertices and identify I with

520 HIRAKU NAKAJIMA

{1,2,...,n}, where n #I. We associate with the graph (I,E) a symmetricgeneralized Cartan matrix C (Ckt) as follows: --Ckt =--Ctk is the number ofedges joining k and in E if k and Ckk 2. This gives a bijection between thefinite graphs without edge loops and symmetric Cartan matrices.

Let H be the set of pairs consisting of an edge together with its orientation.For h H, we denote by in(h) (resp., out(h)) the incoming (resp., outgoing)vertex of h. For h H we denote by h the same edge as h with the reverseorientation. Choose and fix an orientation fl of the graph; that is, a subsetc H such that f u f H, 1" c f . We assume that f has no cycle.Let V (Vk)kt be a collection of finite-dimensional vector spaces for any

vertex k I. The dimension of g is a row vector

dim V t(dim V1, dim Vn) n

If V and V2 are such collections, we define vector spaces by

L(V’ V2) a_=r. @ Hom(V Vk2)kI

E(V1, V2 def._. @ Hom(Vout(h),1 Vin(h),2hH

Ef(vl, V2 def.= ( Hom( Vout(h)1 Vn(h)2 ),het

def.__ @ Hom( Vi)__n(h).-gout(h)hef

For B (Bh) - E(V1, V2) and C (Ch) E(V2, V3), let us define a multi-plication of B and C by

in(h)=k k

Multiplications ba and Ba of a e L(V1, V2), b L(V2, V3), and B e E(V2, V3) willbe defined in an obvious manner. If a L(V1, V1), its trace tr(a) will be under-stood as k tr(ak).For two collections V and W of vector spaces with v=dim g and

w dim W, we consider the vector space given by

M M(v, w) de__f. E(V, V) L(W, V) (9 L(V, W), (3.1)

where we use the notation M unless we want to specify dimensions of V, W. Thedimension of M is tv(2w + (21- C)v), where I is the identity matrix. The abovethree components for an element of M will be denoted by B, i, and j, respectively.An element of M will be called an ADHM datum.

Convention 3.2. In 9, we shall relate the quiver varieties to the envelopingalgebra. The dimension vectors will be mapped into the weight lattice in the fol-


lowing way:

def. E def. El)kOk, w - Aw WkAk,k k

where v t(1)l, On) and w t(Wl,..., w,). Since (0k} and {Ak} are both linearlyindependent, these maps are injective.

For a collection S (Sk)keI of subspaces of Vk and B e E(V, V), we say Sis B-invariant if lh(Sout(h) c Sin(h). The orientation f defines a function e"

H-{_+I} bye(h)=lifhef and bye(h)=-lifh. For BE(V1,V2),let us denote by eB E(V1, V2) the data given by (eB)h e(h)Bh for h H.Let us also define a symplectic form o on M by

m((B, i, j), (B’, i’, j’)) dL-f" tr(eB B’) + tr(ij’ i’j). (3.3)

Let G be the algebraic group defined by

G Gv d=f. H GL(Vk),k

where we use the notation Gv when we want to emphasize the dimension. It actson M by

(B, i, j) g(B, i, j) d=f. (gBg_l, gi, jg-1),

preserving the symplectic form 09. The moment map vanishing at the origin isgiven by

/(B, i, j) eBB + ij e L(V, V), (3.4)

where the dual of the Lie algebra of G is identified with L(V, V) via the trace. Themoment map is defined only over , but the above explicit form makes sense overany field. Let #-1(0) be an affine algebraic variety (not necessarily irreducible)defined as the zero set of/. The equation/ 0 will be called the ADHM equation.We want to consider two types of quotients of #-1 (0) by the group G. The first

one is the affine algebro-geometric quotient given as follows. Let A(#-I(0)) bethe coordinate ring of the affine algebraic variety #-1 (0). Then 9J/0 is defined as avariety whose coordinate ring is the invariant part of A(#-I(0)):

9Jo --- 9"o(v, W) de./,,t_l(o)//G Spec A(#-I(0))G. (3.5)

522 HIRAKU NAKAJIMA

As before, we use the notation 9X0 unless we need to specify the dimension vectorsv and w. By the GIT in [25], this is an affine algebraic variety. It is also knownthat the geometric points of 9J/0 are closed G-orbits.For the second quotient we follow King’s approach in [16]. For this, we need

to introduce a character of G. There are lots of choices of characters, but we fixone of them given by ;t(g) Hk det g-I for g (gk). Set

A(.u_l(O))O,z,, de__f. {f ; A(#_(O))lf(g(B,i, j)) Z(g)nf(B,i, j) }.

The direct sum with respect to n e Z>0 is a graded algebra, and hence we candefine

9Jl 9X(v, w) de_f. Proj @ A(/z-(0))t’zn>0

(3.6)

This is what we call a quiver variety.

Remark 3.7. There are many choices of characters of G. The construction ofquiver varieties can be done for any character. The stability condition, intro-duced in the next subsection, will be changed (see [29]). Our choice seems mostconvenient.

3.ii. Stability condition. In this subsection, we shall give a description of thequiver variety 92/that is easier to deal with. We again follow King’s work [16].We lift the G-action on #-1(0) to the trivial line bundle #-1(0)x IF by

g(B, i, j, z) (g(B, i, j), (g)-lz). Define

,t/_l(0)s de=ff. {(n,i,j) #-l(O)lG(n,i,j,z c (#-1(0) x {0}) for z = 0},

where the overline means the closure. Let us introduce an equivalence relationon g- (0) as follows. (B, i, j) (B’, i’, j’) if and only if G(B, i, j) and G(B’, i’, j’)intersect in t-l(0)s. Then the geometric points of the GIT quotient 9J/ can bedescribed as

#-l(o)S/,

LEMMA 3.8. Let (B, i, j) e/g-l(0). Then the following are equivalent:(1) (B, i, j) is contained in #-1 (0)s;(2) if a collection S of subspaces is B-invariant and contained in Kerj, then

S-0.

Definition 3.9.above conditions.

A point (B, i, j)e t-l(o) is said to be stable if it satisfies the


A point whose orbit closure does not contain the origin is called semistable inGIT literature. The stability condition usually means the closed orbit and finitestabilizer. But stability and semistability coincide in our situation as we shall seelater in Corollary 3.12. Thus we adapt this terminology. Note that the stabilitycondition is invariant under the G-action.

ProofofLemma 3.8 (See also [16, Proposition 3.1].) By the Hilbert criterion(see [23, Chap. 2]), it is enough to study the closure of 2(m)(B,i, j) for 1-parameter subgroups 2 {m G.

Let S (Sk) be a collection of subspaces that violates the stability condition.Taking complementary subspaces S- in Vk, we define a 1-parameter subgroup 2as follows: 2(t) t on Sk and 2(t) 1 on S-. Then j-l(A(t)) is a positive powerof t, and hence lim_0 z-l(2(t)) --0. On the other hand, it is easy to check thatthe limit of 2(t)(B,i, j) exists. This means (B,i, j) lies in the complement of/./-1 (0)s.To show the converse, let us suppose that 2(t)((B, i, j), z) converges to

(B, P, j, 0) as 0. If V m V’ is the weight space decomposition withrespect to 2, the existence of the limit implies

nh(Vont(h)) c O V(h),n>m

jk(V) =0 form> 0.

Then $ (m>0 vn)k violates the stability condition. But z-l(2(t)) does notconverge to zero if 0 for m > 0. Hence S cannot be zero.

LEMMA 3.10. Suppose that (B, i, j) //-1(0) is stable. Then(1) the stabilizer of (B, i, j) in G is trivial, and(2) the differential dl M L(V, V) is surjective at (B, i, j), and hence #-(0)

is a nonsin#ular subvariety ofdimension tv(2w + (I- C)v).

Proofi For (1), suppose # (/k) G stabilizes (B,i, j). Then a collection ofsubspaces Im(gk- 1) is B-invariant and contained in Kerj. Hence we havegk 1 by the stability condition.For (2), suppose ( L(V, V) is orthogonal to the image of d# with respect to

the inner product given by the trace. Then we have

B--B, i=0, j=0.

Im is B-invariant and contained in Ker j. Hence must be zero by the stabilitycondition.

If (B, i, j) e/- (0), we can consider the following complex

L(V, V) E(V, V) L(W, V) L(V, W) L(V, V), (3.11)

524 HIRAKU NAKAJIMA

where is given by

t() (B B) (-i) ( j.

If we identify E(V, V) L(W, V) 9 L(V, W) with its dual via the symplecticform 09, then is the transpose of d/. The above lemma shows is injective and d#is surjective if (B, i, j) is stable. The image of is the tangent space of the G-orbitthrough (B, i, j).

COROLLARY 3.12. The quiver variety is a geometric quotient (see [23,0.6]) of/z-l(0) by G and a nonsingular variety ofdimension tv(2w- Cv). In par-ticular, the set ofgeometric points of the quiver variety 93l consists of

{(B,i, j) #-I(O)I(B, j) is stable}/G.

The tangent space of 9Jl at the point corresponding to (B, i, j) can be identified withKer d#/Im .

Proof. As we remarked above, the stability condition implies the triviality ofthe stabilizer. Hence the dimension of stable orbits is equal to dimG. Thus anystable orbit cannot be contained in the closure of another orbit. This means(B, i, j) (B’, i’, j’) if and only if (B, i, j) and (B’, i’, j’) lie in the same orbit. Inparticular, all orbits are closed in/z-l(0) s. Hence the stability and semistabilityconditions coincide, and our assertion follows.

Remark 3.13. When W 0, the scalar group ltm acts on M trivially, andhence we have the residual action of G/tm on M. Choosing a characterZ:G/ffJm (m appropriately and introducing a stability condition, we canmodify all of the above arguments. The detail was explained in [29].

When the underlying field IF is , the quiver variety can be viewed as a sym-plectic quotient (or Marsden-Weinstein reduction) (see [24]). Hence we have thefollowing.

COROLLARY 3.14 (Base field IF ). The symplecticform o9 in (3.3) descendsto a symplectic form on the quiver variety 9Jl.

We denote this symplectic form also by o9, and hope this causes no confusion.

Notation 3.15. For a stable point (B, i, j) #-1(0), its G-orbit, considered asa geometric point in the quiver variety 9J/, is denoted by [B, i, j]. If (B, i, j)e#-1(0) has a closed G-orbit, then the corresponding geometric point in 93/0 willbe denoted also by [B, i, j].

Remark 3.16. The original definition of the quiver variety given in [27] is dif-ferent from the one given here, and is given in terms of the hyper-Kihler momentmap. In particular, the quiver variety has a natural hyper-Kihler metric. It was


proved in [27, 3.5] that (B, i, j) is stable in the above sense if and only if the G-orbit intersects with a level set of the Kihler moment map. Hence our 9J/coin-cides with the previous one.The following theorem was proved in [27, 4.2].

TrmORM 3.17 (Base field IF IE).affine al#ebraic manifold.

The quiver variety 9Jl is diffeomorphic to an

In fact, this can be proved easily by using the hyper-K/ihler structure in oursituation: is affine algebraic with respect to another complex structure J. Thisresult is not used in this paper.

3.iii. The projective morphism 7r" Jt Ro.natural projective morphism

From the definition, we have a

n: 9J/o. (3.18)

If 7r([B, i, j]) [B, , j0], then G(B, i0, j0) is the unique closed orbit contained inthe closure of G(B,i, j). In order to explain the relation between [B,i,j] and[B0, 0, j0], we need the following notion.

Definition 3.19.filtration

Suppose that (B,i,j)eM and a B-invariant increasing

O=V(-)V() V(N)=V

when Im c V() are given. Then set grmV v(m)/v(m-l) and gr V grmV.Let grmB denote the endomorphism that B induces on grinV. For m 0, letgr0i e L(W, V()) be such that its composition with the inclusion V() c V is i, andlet gr0 j be the restriction ofj to V(). For m # 0, set grmi 0 and grm j 0. Letgr(B, i, j) be the direct sum of (grmB grmi grm j) considered as data on grV.

PROPOSITION 3.20. Suppose r(x) y. Then there exists a representative (B, i, j)of x and a B-invariant increasing filtration V(*), as in Definition 3.19, such thatgr(B, i, j) is a representative ofy on grV.

Proof (See also [27, 5.9] for the case y 0). If n(x) y, then by a version ofthe Hilbert criterion (see [2] and [15, Theorem 1.4]) there exist representatives(B, i, j) of x and (B, , j0) of y and a 1-parameter subgroup 2:m G suchthat

lim 2(t)(B, i, j) (B, , j0).

Decompose V into A-weight spaces V )Vm, where Vm is the space of weightm. We also have the induced decomposition of M(v, w). Set

v(m) de. @ vn"n>m

526 HIRAKU NAKAJIMA

Since the limit for 2(t)(B, i, j) exists, V(m) is B-invariant, Imi V(), andj(V(-)) 0. The stability condition for (B, i, j) implies V(-) 0. Moreover, thelimit (B, io, j0) is equal to the restriction of (B, i, j) to the direct summand

E(Vm, Vm) L(W, V) L(V, W).

It can be considered ADHM data on grV, as in the assertion.

The following is proved in [27, 5.8].

TI-mOlM 3.21 (Base field IF rE). The subvariety 7g-l(0)t::: 9J is a Lagran-gian subvariety which is homotopic to 9Jl.

We do not use this result except in 11. One can give the proof of beingLagrangian by modifying arguments in Theorem 7.2. (It is different from theoriginal proof.) This Lagrangian subvariety will be denoted by (v, w).

3.iv. m-action. Let us consider a tl3m-action on M defined by

(Bh, ik, jk) (t(l+e(h))/2Bh, tik, jk), tm.

Namely, the component Ea(V, V))L(W, V) is multiplied by t, while the com-ponent Eft(V, V)@ L(V, W) is unchanged; it commutes with the G-action. Thesymplectic form co on M transforms as co- tco. Moreover, #-1(0) is invariantunder the tl3m-action. In particular, we have the following proposition.

PROPOSITION 3.22. There are tm-actions on 9Jl and 9Jlo such that(1) the symplectic form transforms as co tco,(2) the projection n 9Jl ---, 9Jlo is equivariant.

It is not difficult to show that the fixed point in 93/0 is only the origin, thanksto the assumption that f has no oriented cycles. Hence the fixed points are con-tained in the Lagrangian subvariety n-l(0).

3.v. Stratification ofgJlo. We study a stratification of 9210 Iz-l(O)//G in thissubsection.We first introduce an open stratum of 9J/0 (possibly empty),

def.0 ([B, i, j] 01(B, i, j) has the trivial stabilizer in G}. (3.23)

flreg then it is stable. Moreover, induces anPROPOSITION 3.24. If [B i, j] *oisomorphism n-l(gj/eg) mreg

’0

Proof (See also [27, 4.1]). Suppose that S is a collection of subspaces that isB-invariant and contained in Ker j. Choosing complementary subspaces S, wetake a 1-parameter subgroup 2, as in the proof of Lemma 3.8. Then the limit


limt_oA(t)(B,i,j) exists and its stabilizer contains Hk GL(Sk). But the orbitG(B, i, j) is closed and the limit lies in the same orbit. Hence the assumptionimplies S 0. This proves the first statement.The inverse image of [B, i, j] under 7 is the union of all those orbits whose clo-

sure contains (B, i, j). But the dimension of the orbit G(B, i, j) is maximal whenthe stabilizer is trivial and cannot be contained in the closure of another orbit.Hence is an isomorphism on 7-l(2r/eg). 1--!

Remark 3.25 In general, fDreg may be empty. In fact, we shall give a criterion0for the nonemptiness of meg in lO.iii.

Now we go to general stratum. We introduce the following definition.

Definition 3.26 (cf. Sjamaar-Lerman [32]). For a subgroup 0 of G, denote byn(d the set of all points in M whose stabilizer is conjugate to G. A point[(B,i,])] eJl0 is said to be of G-orbit type () if its representative (B,i,]) is in

n(d). The set of all points of orbit type (G) is denoted by (l)(d).The stratum (gJl0)(1) corresponding to the trivial subgroup 1 is reg.,.,,0 by defi-

nition. We have the following decomposition of [R0:

(d)

where the summation runs over the set of all conjugacy classes of subgroups of G.

L.MMA 3.27 (cf. [27, 6.5]). Suppose [B, i, j] (gJ/0)(d) with nontrivial . Thenthere is a direct sum decomposition

V V0 (V1)e () (]) (Vr)e and

times

Thus, after replacing the representative (B, i, j), if necessary, we have(1) each summand is invariant under B;

(2) the restriction ofB to (Vi) e IFO’7 (R) V is of theform lr (R) BI,,;(3) the image of is contained in V and j is zero on V ( V;(4) if # j, there is no isomorphism V --, vJ that commutes with B;(5) the restriction of (B,i, j) to V has the trivial stabilizer in I-[ GL(V)

(though the case V 0 is not excluded);(6) the subgroup l-Ik GL(V) meets the stabilizer only in the scalar subgroup

(Ik GL(Vki)) c 0 Gin,"

528 HIRAKU NAKAJIMA

(7) the stabilizer of (B, i, j) is conjuoate to the suboroup

Gig is ofform gi (R) lv, on (Vi)Ce for some gi GL() };

(8) the dimension of the stratum (gJ/0)(d) is given by

tv(2w- Cv) + (2 tviC,i),i=1

where v dim Vi.

Proof Since the orbit G(B, i, j) G/ is closed, it is an affine variety. Hencethe stabilizer ( is reductive (see [30]). Let g (Ok) ; be a nontrivial semi-simple element. If

(R)

is the eigenspace decomposition of Vk with respect to Ok, then we have

nh(Vout(h)()) c l/]n(h)(,), ik(14) c Vk(1), jk(Vk(2)) 0, unless 2-- 1.

In particular, the collection V(2) (Vk(2))k is B-invariant.If the stabilizer of the restriction of (B, i, j) to V(0) is nontrivial, or the stabil-

izer of the restriction of B to V(2) for 2 1 is larger than m, we furtherdecompose V(2). Thus we have a B-invariant direct sum decomposition

V V @ V’@ V"@ ...,such that

(a) the image of is contained in V and j is zero on V’ @ V" @ ..., and(b) the restriction of B to V""’ satisfies conditions (5) and (6) for the stabilizers

by replacing V(i) with V"".Define an equivalence relation on the summands V’, and so on, by declaringV’~ V" if and only if there exists an isomorphism V’ V" which commuteswith B. Collecting equivalence classes of summands, we have the desired decom-position.For the dimension of the stratum, the contribution of the restriction of (B, i, j)

to V is given by

dim fforeg(0 w) dim 9Jl(v, w) tv(2w Cv),

where we have used Proposition 3.24 for the first equality. The restriction of B to


contributes by

2- tviCvi.

The term 2 appears because we make the quotient by GlUm. (See Remark 3.13.)This completes the proof of the lemma.

If V’ (Vkt)kei is a collection of the subspaces of V (Vk)kei, we have a nat-ural inclusion map

9 0(v’, w) 9 0(v, w).

The image is the union of the strata such that the restriction of B to V" is zero,where V V’ V".

Remark 3.28 (See also [27, 6.7]). When the graph is of finite type, therestriction of B to V is zero for 0. This can be proved as follows. If thevariety 9J/0(vi, 0) contains points with stabilizer m, its dimension is given by2- tvCv. Since C is positive definite when the graph is of finite type, we havedim 9J/0(vi, 0) 0. But this is impossible unless 9J/0(vi, 0) (0), since 9J/0(vi, 0) isa cone. In particular, we have

U lf?reg(vO’ W) 0(, W)0vO<v

4. Tautological bundle homomorphisms. Recall that the projection #-1 (0)s__

9Jl(v, w) is a principal G-bundle. Since Vk is a representation space of G, we candefine an associated vector bundle

/-l(0)s X G Vk

over 9(v, w). We lift the m-action to this bundle by letting t act on the Vk-factortrivially. For brevity, we denote this vector bundle also by Vk.

Notation 4.1. For an equivariant vector bundle V, we denote by tmv thebundle tensored with the 1-dimensional m-module of weight m.

Let us consider an equivariant sequence of vector bundles over 9J/(v, w):

Vk Bhjk, 0 t(l+e(h))/2Vout(h) l/Vk ’(h)Bh+ik) tVk. (4.2)in(h)=k

These homomorphisms are analogues that we call tautological bundle homo-morphisms in [18, p. 270].

530 HIRAKU NAKAJIMA

The stability condition implies that the first homomorphism is injective (cf. theproof of Lemma 54), and the ADHM equation implies that the composition iszero. Let Q/c(v, w) denote the the homology sheaf of the middle

in(h)=k out(h)=k

Following [21, 12.2], we introduce the following subset of 9J/(v, w)"

W) do__f. { i, j] W)

fJk; <r(V, W) def. U fk;s(V, W).

--r

(4.3)

Since k;<r(, W) is an open subset of 9J/(v, w), fk;r(, W) is a locally closed sub-variety. The restriction of Qk(V,W) to 0lk;(v,w) is a vector bundle of rankek(w- Cv) + r, where ek denotes the dimension vector such that the kth entry is1 and the other entries vanish. Note that tek(w- Cv) is equal to (hk, Aw v) inConvention 3.2.

Replacing Vk by -i(h)=k Im Bh + Im ik, we have a natural map

p" 9Jk;r(V, W) -- 9Jk;O(V- rek, w). (4.4)

PROPOSITION 4.5. Let Qk(V- re/c, w)lgY/k;0(v- rek, w) denote the vector bundleof rank tek(w--Cv)+2r obtained by the restriction of Qk(V--rek, w) to

9J/k;O(V- rek, w). Then the map p identifies $ff/k;r(v, w) with the Grassmann bundleoft-planes in Qk(V- rek, w)]gJ/k;o(V- rek, w). In particular, we have

dim 9Jlk;r(V, W) dim 9Y/k;o(V- rek, w) + r(tek(w- Cv) + r)dim Ol(v, w) r(tek(w- Cv) + r).

Proof We have a surjective homomorphism

p*Qk(v rek, w) Qk(V, w)

of codimension r over fJk;r(V, w). This gives a morphism from lk;r(V W) to theGrassmann bundle.

Conversely suppose a point in the Grassmann bundle is given. Let[B’, i’, j’] e 9J/k;0(v rek, w) denote the image of under the projection. We take


ADHM data (B, i, j) so that

in(h)=k out(h)=k

corresponds to tp. We want to show [B, i, j] e Jk;r(, w). The only thing in doubtis the stability condition. Suppose there exists a B-invariant subspace S containedin the kernel ofj. Define S by

Sl if v k,S[

Sk Im(’ e(h)Bh + ik) if l-- k.

Then the collection S’ is B’-invariant and contained in the kernel of j’. Thus thestability condition for (B’, i’, j’) implies S’-O. In particular, S has a nontrivialcomponent only on the vertex k. Then we must have )Bh ] jk(Sk) 0 since S isB-invariant. Since we have taken )Bh ]) jk tO be injective, Sk is also zero.Taking Gv-action into consideration, we have obtained a morphism from the

Grassmann bundle to 9J/k;r(v, w). It is the inverse of the previous morphism. Thelast equality in the assertion follows from the formula

dim 9Jl(v, w) dim 93l(v rek, w) 2r( ’ek(w Cv) + r).

COROLLARY 4.6. On a nonempty open subset of gJ/(v, w),

Proof. Let r0 min{ rlgJ/k;r(v w) - }. Then Jlk;ro (V, W) is a nonemptyopen subset in 9J/(v, w). Hence we have

0 dim 9J/(v, w) dim 9J/k;o (v, w) ro(tek(w Cv) + to).

Thus ro 0 or ro -tek(w- Cv).If--tek(w- Cv) > 0, then the first case does not occur. Otherwise, we see that

9J/k;o(V, w) is nonempty and we should have a vector bundle Qk(V, w) gJ/k;o (V, w)of rank tek(w C), which is negative by assumption. I--]

532 HIRAKU NAKAJIMA

In the end of this section we give a lemma which will be needed in 10.ii.LEMA 4.7. Let [B, i, j] e 9J/eg(v, w). Then we have

Im Bh d- Im ik Vk for any k I. (4.8)in(h)=k

Moreover, we have w- Cv Z>0.

Proof Let Sk be the left-hand side of (4.8), and let S- be a complementarysubspace. Define a 1-parameter subgroup A’tm G as follows: A(t) 1 on Skand (1 # k), 2(t) -1 on S-. Then the limit limt_0 2(t)(B, i, j) exists and has astabilizer containing GL(Sk). But (B, i, j) has a closed orbit, and hence the limitalso has the trivial stabilizer. This means S 0. Hence we have shown the firststatement of the lemma.

If there is a vertex k with tek(w- Cv) < 0, (4.8) does not hold by Corollary4.6. Hence we have w- Cv Z0. lq

We also have the following lemma.

LEMMA 4.9. Let [B] e 9Jl0(v, 0) (i.e., w 0). Assume the stabilizer orB in Gv is

m. Then we have either(1) ’in(h)=k Im Bh Vk for any k I, or(2) there exists a vertex k such that Vk IF and Vt 0 for k (and hence

B 0).Moreover, in the first case we have

(a) -Cv >o, and(b) the support of v (i.e., the subdia#ram which consists of the vertices k with

tekv O, and of all edges joining these vertices) is connected.

The proof is almost the same as the above lemma, and hence is omitted. Thelast assertion (b) follows from the observation that the stabilizer is the product of(the number of connected components of the support of v)-copies of m.The set of all nonnegative vectors with properties (a) and (b) is introduced in

[11, 5.4]. Every positive imaginary root can be moved into this set by an ele-ment of the Weyl group.

Question. When v satisfies conditions (a) and (b) in Lemma 4.9, does I/0(v, 0)contain a point with the stabilizer m? In general, the answer is no. The authordoes not know a criterion. When the underlying graph is affine, such a v is amultiple of the imaginary root given in [11, 5.5]. One can show that 0(v, 0)contains a point with the stabilizer m if and only if v .

5. Hecke correspondence

5.i. Definition. Let ek be the dimension vector such that the kth entry is 1and the other entries vanish. Take dimension vectors w, v1, and vz so that


V2 1 _+_ ek. In other words, av2 av + tk in Convention 3.2. Choose collectionsof vector spaces W, V, and V2, with dim W w and dim Vi= vi. As in {}4, weconsider V and V2 as holomorphic vector bundles over 9Jl(v, w) x 9J/(v2, w).

Taking a point ([B1, 1, jl], [B2, 2, j2]) from the product 9Jl(v1, w) x 9J/(v2, w),we define a 3-term sequence of vector bundles

tEu(V V2) E6(V V2) /L(V V2) -- tL(V V2) tiF,

tL(W, V2) L(V, W) (5.1)

where

o’() (B2 B1) @ (_il) j2,

z(C a b) (eB2C + eCB’ + i2b + aft) (tr(ilb) + tr(aj2)).

Here tmv is the notation in Notation 4.1. This is a complex; that is, za 0, thanksto the ADHM equation and the equation tr(ij2) tr(ilj2). Notice the similaritywith complex (3.11).

LEMMA 5.2. a is injective and z is surjective.

Proof. Suppose is in the kernel of a. Then the image of is B2-invariantand contained in the kernel of j2. Thus the stability condition for B2 implies

0. Hence a is injective.Next consider the surjectivity of z. Suppose that 2 L(V2, V) IF is

orthogonal to the image of z with respect to the natural pairing betweenL(V2, V1) () IF and L(V V2) () IF. Then we have

B B2, (i2 -k- 2i 0, jl( + ,j2 0.

If 2 is not zero, then the kernel of violates the stability condition for B2. Hencewe must have Ker 0, but this is impossible since the dimension of V is lessthan V2. Thus we have 2 0. Then, applying the stability condition for B to theimage of , we have 0.

By Lemma 5.2, the quotient Ker /Im a is a vector bundle, and we have thefollowing formula for its rank:

rank(Ker z/Im a) tv2(Cvl -I- w) -- tvlw- 1

(1/2) (dim 9Jt(v1, w) + dim 99l(v2, w)). (5.3)

534 HIRAKU NAKAJIMA

Let us take a section s of Ker /Im a given by

s=(O(-fl)fl) (modIma), (5.4)

where x(s)= 0 follows from the ADHM equation and tr(BIB1) tr(B2B2) 0.The point ([BI,i1, jl], [B2, i2, j2]) is contained in the zero locus Z(s) of s if andonly if there exist L(V1, V2) such that

B =B2, i =i2, jl =j2. (5.5)

Moreover, Ker is zero by the stability condition for B2. Hence Im is a sub-space of V2 with dimension v1, which is B2-invariant and contains Im 2. More-over, such a is unique if it exists. Hence we have an isomorphism between Z(s)and the variety of all pairs (B, i, j) and S (modulo Gv2-action) such that

(a) (n, i, j) e #- (0) is stable, and(b) S is a B-invariant subspace containing the image of with dim S

1 2 ek.Similar varieties were studied by Lusztig in [19] and by the author in [26].

Definition 5.6.2, w).

We call Z(s) the Hecke correspondence, and denote it by

Introducing a connection V on Ker /Im a, we can define the differential

Vs: T 9Jl(v1, w) T(v2, w) Ker z/Im a

of the section s. Its restriction to Z(s)= k(V2, W) is independent of theconnection.

Let us introduce the symplectic form on the product space 9Jl(v1, w)x 9J/(v2, w)by changing the sign on the second factor; that is, to x (-to).THEOREM 5.7. The differential Vs is surjective over k(2, w). Hence k(2, W)

is a nonsingular subvariety of 9J(v1, w) x 9Jl(v2, w). Moreover, k(2, W) isLalrangian when the underlying field lF is ff.

Proof. Take a point ([B1, il, jl], [B2, i2, j2]) in k(2, w). There isL(V1, V2), as in (5.5). Fixing , we define a section of E(V1, V2) 1)

L(W, V2) L(V1, W) by

(/1, ’1, .1,/2, ’2, j2) de__f. (/1 __/2) (’1 ’2)I (j1 __]2)

for (/1, ’1, j1,/2, ’2, j2)6 fl-l(o X fl-l(o) c:: M(1, w) x M(2, w).

This section is contained in Ker , vanishes at (B1, 1, jl, B2, i2, j2), and gives a liftof s. Take a tangent vector (fiB 6i 6j 6B2, 6i2, 6j2) of lYt(v w) x 9Yt(v2, w), and


consider it an element in Ker d#/Im t, as in Corollary 3.12. Differentiating andtaking its projection to Ker z/Im tr, we can compute the differential Vs in thisdirection as

( 6B fiB2 ) ( 6i 6i2) (6jl 6j2 ) (mod Im tr).

Identifying the cotangent space with the tangent space by the symplectic form,we consider the transposed homomorphism of Vs

t(Vs)" Ker a/Im r, TgJ]( w) ( TgJl(v2, w).

Taking the transposed homomorphism of (5.8), we find

t(Vs)((C’, a’, b’) (mod Im tz))

(eC’, b’, -a’) (mod/L(V V1)) ] (e,Ct, b’, a’) (mod tL(V2, V2)).(5.9)

The surjectivity of Vs follows from the claim below.

CLAIM 5.10. t(Vs) is injective.

bProof. Suppose that (C’,a’, (mod Imtz) is in the kernel of t(Vs) Thereexist yl L(V, V), y2 L(V2, V2) such that

3Ct IB B1])1,b ])1il,

-a’

e,C’ ])2B2 B2])2,{b’_at= _j2])2.

(5.11)

Combining with (5.5), we find that the image of ])1 ])2 is Bl-invariant and liesin the kernel ofj1. Hence we have ])1 ])2 by the stability condition.

In the kth component of V2, we have the codimension-1 subspace Im k,which is stable under ]) by the above consideration. Taking a complementarysubspace, we can write ]) in the matrix form

where 2 is considered as a scalar. Then ]) 2 id has the image in Im k, so we candefine k" V V by k ]) 2 id. For k, define l" V - Vt by

536 HIRAKU NAKAJIMA

Substituting into (5.11), we get

C’= e((B2 B1),a’= j2( ( + 2 id),

b’= (( + 2 id)i2.

This shows that (C’,a’,b’)= tz( 2). We have completed the proof of theclaim.

Proof of Theorem 5.7, continued. In order to prove the second statement ofTheorem 5.7, it is enough to show that the tangent space to 3k(V2, w) is involu-tive, since we already know that 3k(V2, w) is half-dimensional by (5.3). The invo-lutivity follows that the image of t(Vs) given by (5.9) lies in the kernel of Vs.

5.ii. Fibration. Let Pi: k(2, W) fj(i, W) (i-- 1, 2) be the natural projec-tion. Let 9Jk;r(Vi, W) be as in (4.3).

LEMMA 5.12. (1) Let E be the vector bundle of rank tek(w Cv1) + r obtainedby the restriction of

Ker(in(h)=ke(h)B +i)/Im(out(h)=kB J)tO 9Jk;r(Vl,w). Then p?l(gJk;r(1,W)) can be identified with the projective bundleP(E).

(2) Let F be the vector bundle ofrank r obtained by the restriction of

(V/ Im B + Im i/ in(h)=k

tO 9Jk;r(2, W). Here v means the dual vector bundle. Then pl(k;r(V2 W)) can beidentified with the projective bundle P(F).

Proof (See also [27, 10.10]). For (1), since the proof is almost identical toProposition 4.5, we omit it.For (2), the fiber p-I ([B2, i2, j2]) is isomorphic to the variety of all codimension-

1 subspaces Sk of V containing EIm B + Im ik2. Hence we have the assertion.

6. A decomposition of the diagonal. In this section, we fix dimension vectorsv and w, and use the notation 9Jl instead of 9Jl(v, w). The base field IF is assumedto be II in this section. We put hermitian inner products on vector spaces V and


W. Then the quiver variety 9J/can be defined as a hyper-Kihler quotient of Mby the compact group I-[k U(Vk) as in [27].

Let us take collections 1,1 and 1/’2 of vector spaces with dim 1,1 dim 1,2 v.As in 5, we construct a complex of vector bundles

tEn(V V2) ) Efi(V V2)L(V1, V2) - ) - tL(V, V2),

tL(W, V2) L(V, W) (6.1)

where

O’() (n2- B1) t0) (_il) tj2

z(C ) a b) (eB2C + eCB + i2b + aft).

Here we used the convention (4.1) to make the complex equivariant. As in Lemma5.2, tr is injective and z is surjective. In this section, we do not need to take careabout the 3m-action, so we omit the notation m from now on.The hermitian inner product on the vector space 1, induces hermitian fiber

metrics on vector bundles V and V2. So we define a vector bundle homo-morphism over 9J/x 92/

L(V, V2) L(V1, V2) - E(V1, V2) t L(W, V2) L(Vx, W)by

(1 ( 2 ) 0"(1) + zf(2) + (0 ) (_i2) () jl),

where zt is the hermitian adjoint of z.The same argument as in 5 shows that is not injective exactly on the diag-

onal A.We have a geometric application of the above description of the diagonal.

THEOREM 6.2. The quiver variety is connected.

Proof. Let p: P 9J/x 9J/ be the projective bundle of L(V, V2) t)L(V1, V2) IE. Let L denote the tautological line bundle. The composite

L ’-, p* (L(V1, V2) L(V’, V2) ) p* (E(V’, V2) L(W, V2) ) L(V’, W))

induces a section, denoted by s, of L* (R) p* (E(V1, V2) ) L(W, V2) ) L(V1, W)).Its zero locus is isomorphic to the diagonal via the projection p, and it is trans-versal as in Theorem 5.7.

538 HIRAKU NAKAJIMA

Suppose that 9J/ has two components fJ/1 and 9J/2, and take X1 ff 9Jl andx2 93/2. We consider the restriction of P to 9J/x {x} - 9T/, which we can get byreplacing V2 with the fiber V over x in the above definition. Choosing an iso-morphism between V and V2:, we have an identification for the restrictionsfor x and x2. We denote this identified restriction of p:P--, 9J/x 9J/ byp" P’ ---, , and the restriction of L* (R) p* (E(V, V2) (9 L(W, V2) (9 L(V1, W))by F.Hence we have two sections Sl and s2 of F, corresponding to the restrictions of

s to p-l(gJ {X1}) and p-llgj {x2}). Let fl be the Thom class of F. Since sidoes not vanish outside p’- (xi), the pullback s’f is a cohomology with com-pact support. (Note that the Euler class of F, which is the pullback of f by thezero section, does not have compact support, and, in fact, is equal to zero.)

Consider the homotopy (1 t)Sl + ts2 (0 < < 1). Since the difference Sl s2is bounded and the norm Ilslll is proper, (1 t)sl + ts2 does not vanish outside acompact set. In particular, sf2 and sD are equal in the cohomology with com-pact support H(P’). But the integrals over p’-lgJ11 are different. We have

s’n 1

This is a contradiction.

Remark 6.3. (1) When the underlying graph is of type An, Theorem 6.2 isproven with a totally different method in [26].

(2) When 93l is an ALE space, the description of the diagonal plays an essentialrole in the ADHM description for instantons (see [18]).

7. A Lagrangian subvariety Z. Let V1, 2, and w be dimension vectors, andalso let V1, V2, and W be the corresponding collections of vector spaces. Con-sider quiver varieties ffJ/(v1, w) and 9J/(v2, w) and the projections 7r 93l(vi, w) --9310(vi, w) (i- 1, 2). As in 3.v, we have an inclusion 9J/0(vi, w) c 9310(v + v2, w).Then 7r is regarded as a map to 9Jl0(v + v2, w). Define

z(vl,v2;w) =_ Z de--f" {(X1,X2) U (V1,W) X 9J(V2, W) ITg(X1) /g(X2)}. (7.1)

We use the notation Z unless we need to be conscious of dimensions.Let Zreg zreg(v1, v2; w) c Z(v1, v2; w) be the complement of the closure of

the set of all points (X X2) such that 7g(X1) 7g(X2) eg(0, W) J(1_2, W)

for any v.THEOREM 7.2. The subvariety Zreg is purely (1/2)dim 9J(1,W)X 9J(V2, W)

dimensional. Moreover, Z is Lagrangian when the base field lF is (E. (We changethe sign of the symplectic form on the secondfactor as in 5.)


Proof. The following proof was suggested to us by G. Lusztig. The originalproof has an error.

Let us take a point x e 9J/eg(v, w) for some v, and let 9J/(v, w)x be its inverseimage by n in 0/(v, w), where v is either v or v2. It is enough to show that9J/(v,w)x has pure dimension of (dim 9J/(v,w)- dim 9Y/(v,w))/2, and that thepullback of the symplectic form vanishes on 0/(v, W)x. These statements can beshown by the induction.

Let X be an irreducible component of 9J/(v, W)x. Taking a generic element of[B, i, j] X, we set

(k I). (7.3)

If v v, then 9J/(v, W)x consists of one point by Proposition 3.24. Hence weare done. Otherwise, by Proposition 3.20, there exists a filtration 0 V(-1) cV() c c V(N) V such that dim V() v. Moreover gr(B, i, j) is a repre-sentative of x, and hence we have grmB 0 for any m -0 by the assumption

mreg(v, w). In particular, we have

B(V(m)) V(m-l) for m > 1. (7.4)

Thus we have 8k(X) > 0 for some vertex k.Let IJk;r( W) be the subset introduced in 4. Setting r 8k(X), we define

X de__f, p(X t’3 9Jk;r( W)) 9Jk;O( rek, w),

where p is as in (4.4). Recall that there is a vector bundle Qk(V- rek, w) ofrank tek(w- Cv)+ 2r over 9Y/k;0(v- rek, w). Then p is the Grassmann bundleof r-planes in Qk(v-rek,w). In particular, we have dim(fiber)1/2(dim 9J/(v, w) dim 9J/(v rek, w)). The map r factors through p, and hencethe closure of X’ is an irreducible component of (v- rek, w). By induction,this has pure dimension of (dim 9Y/(v- rek, w)- dim (v, w))/2. Thus we aredone.

Moreover, if v and w are tangent to the X c 9J/k;r(v, w), the evaluation of thesymplectic form satisfies to(v, w) to(p,v, p.w). Here to in the right-hand side isthe symplectic form on dim 9J/(v- rek, w). Thus by the induction hypothesis,o(v, w) 0.

8. Convolution algebra

8.i. Convolution in homolo#y. Let us first recall the convolution product inthe homology given by Ginzburg in [7].

540 HIRAKU NAKAJIMA

For a locally compact topological space X, let H.(X) denote the homol-ogy group of possibly infinite singular chains with locally finite support (theBorel-Moore homology) with rational coefficients. If X is embedded in an n-dimensional oriented manifold M as a closed subset, then we have the Poincar6duality isomorphism

Hi(X) - Hn-i(M, M\X), (8.1)

where the fight-hand side is the relative singular cohomology group.If f" X - Y is a proper map, there is a pushforward homomorphism

f, Hi(X) Hi(Y).

If i" U Y is an open embedding, there is a pullback homomorphism

i*: Hi(Y) Hi(U).

If j" Y\U Y denotes the embedding of the complement, there is a long exactsequence

i*"-> Hi+I(U)-- Hi(Y\U)- Hi(Y)-- Hi(U)---> Hi-I(Y\U)’-+ "".

If X and Y are closed subsets of an n-dimensional oriented manifold M, wehave the cup product in the relative cohomology group

Hn-i(M, M\X) (R) Hn-J(M, M\Y) H2n-i-J(M, M\(X c Y)).

By the Poincar6 duality isomorphism in (8.1), it can be transfered to the cap prod-uct in the Borel-Moore homology group

Hi(X) (R) Hi(Y) --, Hi+j-n(X Y). (8.2)

Note that this product depends on the ambient space M.Let M1, M2, and M3 be oriented manifolds, and pq:Mlx M2x M3 ----M x MJ be the natural projection. Let Z c M x M2 and Z c M2 x M3 be

closed subsets. By (8.2) we have the cap product in M x M2 x M3,

p-lZ"Hi+a3 (p{2 Z) (R) Hj+dl (plZt) Hi+j-a (plZ 23 1, di dim Mi.

Assume that the map

p-lZ, -- M M3P3 PZ 23 x


is proper. Let us denote its image by Z o Z. We define a convolution by

*" Hi(Z) ( Hj(Z’) Hi+j-d2 (Z o Z’), c * c’ (P13), (P2c P3C’), (8.3)

where p2c stands for c x [M3], and so on. This makes sense for disconnectedmanifolds (possibly variable dimensions), as well.

Let M be an oriented manifold, N a topological space, and 7" M---, N aproper continuous map. One can define Z as before, and the convolution makesH,(Z) a -algebra. The fundamental class of the diagonal is the unit.For x N, consider the fiber Mx -l(x). We have Z o Mx Mx, and the

convolution makes H,(Mx) an H,(Z)-module.Remark 8.4. The convolution product can be defined on any theory which

has operations "pullback" for smooth morphisms, "pushforward" for propermorphisms, and "intersection" (e.g., the Chow rings, the equivariant K-theory,the linear space of constructible functions, and so forth (cf. [8])).

8.ii. The case when Z is Lagrangian. Under the setting of the previous sub-section, we further suppose the following:

(1) M1, M2, and M3 are holomorphic symplectic manifolds of complexdimension dl, d2, and d3, respectively;

(2) Z (resp., Z) is a nonsingular Lagrangian subvariety of M x M2 (resp.,M2 x M3);

n-lz!(3) The intersection piZ r3 denoted by , is a submanifold of p-I(z)23(4) P13 Z o Z is a fiber bundle.The fundamental classes [Z] and [Z] have degrees dl-+-d: and d: / d3, re-

spectively, and the construction of the previous section gives [Z], [Z] e

Ha+a (Z o Z). More precisely, we have the following lemma.

LEMMA 8.5. Let d and e be the complex dimension and the Euler number of thefiber, respectively. Then we have

[Z] [Z’] (- 1)de[Z o Z’].

Proof. We calculate the convolution using the fact that the fundamental classof a complex submanifold is the Poincar6 dual of the top Chern class of thenormal bundle.

Let us consider the fiber square

p21Z p2._31Z!j12

P-1223

p{21z i12 M1 M2 M3X X

(8.6)

542 HIRAKU NAKAJIMA

where morphisms are natural inclusions. Let us denote the normal bundle of thefirst (resp., second) column by N (resp., N’). The restriction of N’ to Z’ contains Nas a subbundle, and we let E be the quotient bundle (i.e., the excess normal bundlein [5, 6.3]). It can be identified with the quotient bundle

By the symplectic form, it is the dual of the orthogonal complement ofT(p21Z) + T(p-IZ’ Since Z and Z’23 J. are Lagrangian, the orthogonal comple-ment is Ker dpl3 c3 T(p-2Z) c T(p-123 Z’), which is equal to the relative tangentbundle Tf. Hence we get

(P13), (PE[Z] P’23 [Z’]) (P13), (C(dl+d2)/2(N’) P3 [Z’])

(P13), (ca(r.tY ca c(a+a2)/2-a(N) p* [Z’])t 23

(Pla),(cd(T [p{EZ P23-1Z’]j(--1)de[Z o Z’].

9. A geometric construction of the algebra U. We assume that the base fieldis tE in this section. From now on, we omit the symbol for multiplication.Throughout this section, we fix w while v moves.

9.i. Main construction. We apply Ginzburg’s construction to quiver varieties.Let Z(v, v2; w) c 9J/(v1, w) x 9J/(v2, w) be the subvariety introduced in(7.1). When we have three dimension vectors v1, v2, and v3, it holdsZ(v1, v2; w) o Z(v2, v3; w) c Z(v1, v3; w). Hence we have the convolution product

t/, (Z(v w)) (Z(v v3; w)) (Z(v v3; w))

by (8.3).Let Htop(Z(v1, v2; w)) denote the top degree part of H,(Z(v v2; w)), that is, the

subspace spanned by the fundamental classes of irreducible components ofZ(v, v2; w). It has a natural basis {[Z]}, where Z runs over all irreducible com-ponents of Z(v1, v2; w). Note that the degree top may differ for different v andv2’s since the dimensions are changing.Take x [.)v0 9J/0(v, w). As in 7, we consider the projection r as a map from

Jv 9J/(v, w) to v0 9J/0(v, w). Let 9J/(v,W)x denote the inverse image of x in9J/(v, w). Note that, for x 0, 9J/(v, w)x is the Lagrangian variety (v, w).

PROPOSITION 9.1.x,

The convolution makes the direct suminto a Q-algebra, and )vH,(gJ/(v,w)x) is a left


OH*(Z(vl’ 2, w))-module. Moreover, the top degree part Htop(Z(v1, 2; W)) isa subalgebra, and Htop(gJ/(v, w)x is a Htop(Z(v1, v2; w))-stable submodule.

9.ii. Definition of Ek, Fk, and Hk. Let A(v,w) denote the diagonal in9J/(v, w) x 9X(v, w). Its fundamental class [A(v, w)] is in Htop(Z(v, v; w)). The leftand right multiplication by [A(v, w)] define projections

[A(, w)]." ( Htop(Z(1, 2; w)) -- O ntop(Z(, 2; w))12 2

[A(V, W)]" Htop(Z(v1, 2; w)) -- G Htop(Z(vl’ v; w)).V V V

When 2 1 ._ ek, the Hecke correspondence k(2, W) is an irreducible com-ponent of Z(v1, v2; w).

Define Ek I-I ntop(Z(vl, v2; w)) as the formal sum of the sheaf [!13k(V2, w)]:

(v w)].2

Let 09" lf)(2, w) x 9J(1, w) -- 9J(1, w) x 9"J(2, w) be the exchange of the twofactors. Set

Fk E(-1)r(v2’w) [(O(k(V2, W))],2

where r(v2, w) 1/2(dim 9X(v1, w) dim 9J/(v2, w)) --tek(w Cv2) 1.Though Ek and Fk are not in G Htop(Z(vl,v2;w)), in general, the multi-

plications Ek’, .Ek, and so on, are linear operators on O Htop(Z(v1, v2; w)), andEk[A(v, w)], [A(v, w)]Fk, and so on, are elements in @ Htop(Z(v1, V2; W)).The following relations are easy to check.

PROPOSITION 9.2. The followin# relations hold in @ Htop(Z(v 2; w)):

[A(v, w)] [A(v’, w)] w)],

Ek[A(v, w)] [A(v ek, w)]Ek,

Fk[A(v, w)] [A(v + ek, w)]Fk.

9.iii. Integrability

LEMMA 9.3. The operators Ek and Fk are locally nilpotent on H,(gJ/(v, w)x).Namely, for any v H,(gJ/(v, W)x), there exists a positive integer N such thatN NEk V Fk v O. If we consider H,(Z(v1, v2; w)) as an H,(Z(v1, rE; w))-

544 HIRAKU NAKAJIMA

module by the multiplication from the left or right, then the operators Ek and Fkare locally nilpotent.

Proof The assertion follows from the statement

if tekv < 0 or tek(w- Cv + v) < 0, then 93l(v, w) ,since Ev and F map any element into homology of quiver varieties with theabove dimension condition, provided N is sufficiently large.The first case of the statement is obvious, and the second follows from the

injectivity of

G nhjk" Vk 0 Vin(h) t) l/l,out(h)=k out(h)=k

which is deduced from the stability condition.

9.iv. Relations. From now on, we study the top degree part of the homologygroup. For dimension vectors v and w, we associate elements v and Aw of theweight lattice P, as in Convention 3.2. We have (hk, Aw- v)= tek(w- Cv).Note that Aw tv determines v once w is fixed.The main result of this section is the following theorem.

THEOREM 9.4. There exists the unique algebra homomorphism

(I)" O Htop(Z(vl, v2; w)) (9.5)V ,V

such that

[A(v, w)]dp(a )=

o

(eka,t) Ek(a,),

/f 2 Aw- v,

if there is no such v,

dp(af) (a,)Fk.

The relations (2.8) and (2.9) are already checked in Proposition 9.2. It isknown that the Serre relation follows from the integrability and (2.4) (cf. [11,3.3]). Hence it is enough to show the following proposition.

PROPOSITION 9.6. Thefollowin9 relation holds in Htop(Z(v1, 2; W))."

(E Fk, Fk, Ek) [A(v, w)] 6k,k,(hk, Aw v>[A(v, w)]. (9.7)

The rest of this subsection is devoted to proving (9.7).


LEMMA 9.8. Let v1, v2, and V3 be dimension vectors such that V2 V --ekv + ek’. Let (x x2, x3) 9J/(v w) x 9J/(v2, w) x 9J/(v3, w) be a point in the inter-section of -1

PiE k(v2, w) and p-alcO(k,(V2, w)). Let U 9J/(v w) x 9J(v3 w)denote the outside of the diagonal when k kI, and the whole set otherwise.Assume (X1,X3) is contained in U. Then the intersection is transverse at(X1,X2, X3).LEMMA 9.9. Let v1, 2, and V3 be dimension vectors such that 2 V ek’

v3- ek. Let (x x2, x3) 9J/(v1, w) x 9J/(2, w) x 9J(v3, w) be a point in the inter-section of P1-Elog(3k,(Vl,W)) and PE-alk(va,w). Let U be as above, and assume(x1, x3) U. Then the intersection is transverse at (x, x2, x3).LEMMA 9.10. Take v and 3 tO be dimension vectors with 3 1 ._ ek k’

and let vE=vl+ek, v4=v1-ek’. Let U be as in Lemma 9.8. There is anisomorphism

U c3 plk(V2 w) c3 plco(k, (V2, W)) --, U c3 plco(k,(V W)) c3 p2-lk(V3, W).

The proofs of these three lemmas will be given in the appendix.

ProofofProposition 9.6. Let v1, v2, v3, and v4 be as in Lemma 9.10. We set

def.C (Ek fk, fk, Ek)[A(v3, w)].

We want to show

c 6k,k,(hk, Aw Ov,)[A(v3, w)]. (9.11)

Take U as in Lemma 9.10. Let i" U - 9J(v1, W) 9J(V3, W) be the inclusion,and j the inclusion of the complement (i.e., the diagonal or the empty set). Thereis an exact sequence

i*Htop(Z( 3;w)\U) -- Htop(Z(1,v3; w)) Htop(U z(vl,3; w)).

From the three previous lemmas, we deduce i*(c) -0. Hence c is in the image ofj,. In particular, c 0 if k k’.We assume k k’, and hence v v3 hereafter. Since the support of c is con-

tained in Z(v1, v3; w)\U A(v1, w), it is a multiple of [A(v, w)]. Let us assumetek(w Cv1) > 0. The proof for the other case tek(w Cv) < 0 is similar.By Corollary 4.6, 9J/k;0(v3, w) is an open subvariety of 9J/(v3, w); that is,

(9.12)

546 HIRAKU NAKAJIMA

holds outside a proper subvariety. Since we already know that c is a multiple of[A(v1, w)], it is enough to show that the restriction of both sides of (9.10) to theopen set Jt(v w) x 9Ttk;0(v3, w) is equal. From now on, we consider everything onthe open set and use the same notation for the restriction. In particular, the termFkEk[A(v, w)] vanishes, and we need to calculate only the term EkFk[A(v3, w)].

Let Y be the intersection p-112 3k(rE, w) pcOk,(V2, W). If ([B1, , j],[BE, 2, j2 ], [/3, 3, j3]) lies in the Y, we have

V= ImB+Imi-- Vin(h)=k

by (9.12). Hence Pl3(Y) is the diagonal A(vl,w). Moreover, Y is isomorphic tok(V2,W) with P13 corresponding to px" k(V2,W) 93(v1,w). Hence Y is theprojective bundle of

Ker(in(h)=ke(h)Bq-i)/Im(out(h)=kB)J)by Lemma 5.12. Thus we are in the situation where Lemma 8.5 is applicable. Weget

c (-1)(a2-dl)/2(p13), (p2[k(V2, w)] p3 [Ogk(V2, W)])

((d2 dx)/2 + 1)[A(v1, w)],

where di dim 93/(vi, w). Since (d2 d)/2we have proved (9.11)

10. Module )Htop(gJ(v,W)xreg 010.i. Operators k and Fk. Fix a dimension vector v. Recall that 9J/0 (v w)

is the set of points in 9J/0(v,w) with trivial stabilizer (3.23). Supposefforeg W).X "’*0 (0We introduce a geometric analogue of Kashiwara’s operators (see [12]). We

follow Lusztig’s idea in [20]. Recall 9J(v, w)x is the inverse image of a pointx 9Jeg(v, w) 9Yt0(v, w) under the projection n" 9Jr(v, w) - 0(v, w). LetX be an irreducible component of 9Yt(v, w)x. Take r- ek(X) as (7.3), considerthe Grassmann bundle p’931k;(v,w)--, 9Jk;O(v--rek, w) (4.4), and set X’--p(X c 93k;(v, w)) as in the proof of Theorem 7.2. Suppose r > 0, and considerthe Grassmann bundle of (r-1)-planes in Qk(V-rek, w)* restricted to X’. ByProposition 4.5, we can regard it as a subset of 9Ytk;_l (v- ek, w). We denote itsclosure by X", which is an irreducible component of 9Jt(v- ek, w)x by the same


reason as above. Then define an operator Ek by

k[X] de=r. [X"] if r > 0,

0 if r=0.

By definition, we have 13k(St’ ,k(X) 1.If r > tek(w Cv) + 1, we regard the Grassmann bundle of (r + 1)-planes in

Qk(V rek, w)* restricted to X’ as a subset of 9J/k;r+l (v + ek, w). Setting its closureas S’, we define

if r > tek(w- Cv) d- 1,

otherwise.

It is clear that -k[X] [X"] if and only if k[Xm] [X].LEMMA 10.1. If ek(X) r > O, then we have

FkEk[X] +_ r[X] + cx,[x’]ek(x’)>r

for some constants cx,.

Proof. Since 9J/k;<r(v, w) is an open set, we have a homomorphism

i*" H, (gJl(v, w)x -- H,(gJ/k;<r(v, w) 99/(V, W)x).

It is enough to show i*(FkEk[X]) -F ri*[X].Let Pi be the projection in 9J/(v, w) x 9J/(v- ek, w) (i-- 1, 2). If we restrict the

projection

Pl coPk (v, w) c p-1X,, -- Xto the open set 9J/k;<r(V, W)X 9"J/(- ek, w), it becomes a fiber bundle with thefiber isomorphic to IPr-1. This is an almost the same situation as in Lemma 8.5.The only difference is that X" is not Lagrangian, in general. However, we canshow that the normal bundle of X" in 9J/(v ek, w) can be identified with

(Ker dn c TX")v

via the symplectic form by Theorem 7.2. Then one can modify the proof ofLemma 8.5 to get

i* (Fkk[X]) (-- 1)r(v’w)+r- ri*[X].

548 HIRAKU NAKAJIMA

lO.ii. Integrable highest weight modules. Again, fix a dimension vector v andtake x ff?reg(0,.,.,0 W) and consider ffJ/(v, W)x.TrIEOREU 10.2. As a -module, GvHtop(/(v, W)x is the irreducible inteora-

ble hiohest weight module with the hiohest weioht vector [gJ/(v, W)x]. (Hence thehighest weight is Aw avO.)

Proof (Compare with [22, 3.6]). The integrability was shown in Lemma 9.3.By [11, Corollary 10.4] it is enough to show that Htop(gJ/(v, W)x is the high-est weight module. By Lemma 4.7, we have, w)x] 0 for keI..

Thus it is enough to show Htop(gJ/(v, W)x c l[l[[Y/(v0, W)x].We shall give the proof by induction on the dimension vector v. When v v,

the result is trivial. Let X be an irreducible component of 9Y/(v, w)x. If ek(X) 0for all k, X must be the point 9Y/(v, w)x by (7.4). Suppose ek(X)> 0. By thedescending induction on ek, we may assume that [X’] is contained in I[l[gJ/(v, w)xif ek(X’) > 8,k(S). Then [X] is also in the image of lI[8l/(v, W)x by Lemma 10.1.

Remark 10.3. (1) After this work was done, Kashiwara and Saito proved thatthe set of all irreducible components of 92/(v, W)x with operators Ek, Fk is iso-morphic to the crystal of the highest weight module in [14].

(2) The author does not know much about the structure of Htop(gJ/(v, W)xfor general x not necessarily contained in I/eg(v, w).

?reg10.iii. Criterion for the nonemptiness ofLEMMA 10.4 (cf. [27, 4.1]). Let n" 931-- 9Jlo be the projective morphism

defined in (3.18). Then there exists a stratum (gJ/0)(d) in rr(gJl) such that dim 93/=dim(gJ/0) (d).

Proof Let (gJ/0)(d) be a stratum contained in the image r(gJ/). Take a pointx (gJ/0)(d). By [27, 6.10], H.(rr-l(x)) is isomorphic to the homology groupof an affine algebraic manifold of dimension dim gJ/-dim(gY/0)(d). Hencedim 7g-1 ((9"J0)()) < dim 9J/, unless we have dim 9J/= dim(gJ/0)(d). [-l

Following [11, 5.3], we set

K def. {2 n>0\{0}l- CA e 7Z0 and the support of 2 is connected}.

PROPOSITION 10.5. Assume that the dimension vector w satisfies

tAw>2 for all 2 K.

Then ureg(v,’’’o w) is nonempty if and only/fw Cv 7/.o and ffYt(v, w) is nonempty.


Proof Suppose that lreg(,’’0 w) is nonempty. Then we have w- Cv Z0 byLemma 4.7, and 9J/(v, w) is nonempty by Proposition 3.24.To show the converse, suppose w Cv Z0 and 9J/(v, w) is nonempty. Take

a stratum (gJ/0)(d) in the image of n" 9J/(, w) 9J0(v, w) with dim 9J/(, w)dim(gJ/o)().

Let us decompose V as in Lemma 3.27. Since v v + -]i ff/vi, we have

0 dim 9J/(v, w) dim(gJ/o)()

tv(Ew Cv) -tv(Ew Cv) (2 -tviCv)i=1

(tvi(w Cv) + tvi(w Cv)) (2 tviCvi).i=1

(10.6)

By Lemma 4.9 we have either(1) -Cv 7z0 and the support of v is connected, or(2) v ek for some vertex k

for each i. Assume that the first case occurs for 1,..., p and the second caseoccurs for i= p + 1,... ,r. For i> p + 1, the ith term of (10.6) is nonnegative,since w Cv, w Cv 7zn and tekCek 2. For < p, we rewrite the ith term of>0(10.6) as

2(if/tviw- 1) tviC(ff/(v + v) vi). (10.7)

From the assumption on w, the first term of (10.7) is positive. Since we have-CviZ and ff/v v>o >o, the second term is nonnegative. Thus the case

mreg(v w) since the restriction of< p cannot occur. Then we have (9910)(d) othe ADHM data to V must be zero for > p.Take [B, i, j] 9J(v, w). Since 7r([B, i, j]) 9Jt0(v, w), we have

E Bh + ik c VO)Im Imin(h)=k

in the notation in Proposition 3.20. But the left-hand side of the above must be Vkfor genetic [B, i, j] by Corollary 4.6. Thus we must have 0 and mreg(v, w) is’0nonempty.

COROLLARY 10.8. Under the same assumption as in Proposition 10.5,9J/eg(v, w) is nonempty if and only ifw Cv Zo and Aw v is a weight of theintegrable highest weight module with highest weight Aw.Proof By Theorem 10.2, (vHtop(gJ(v, w)0 is the integrable highest weight

module of highest weight Aw. The component Htop(gJ(v, w)0 is the weight space

550

FIGURE 1.

HIRAKU NAKAJIMA

An example for 931(v, w) \ v, 71:-lreg(v’,’’’0 w)

of weight Aw v. Since 9T/(v, W)o # if and only if 9J/(v, w) # , we have theassertion.

Remark 10.9. If the graph is of finite type, the set K is empty and theassumption in Proposition 10.5 is vacuous. If the graph is of affine type, thenK ;E>0di, where di is the imaginary root given in [11, 5.6]. Hence t6w > 2 isenough. In general, the assumption holds if w is sufficiently large (e.g., Wk > 2 foreach k).

Counterexample 10.10. When the assumption in Proposition 10.5 is not sat-isfied, we may have 9J/eg(v, w) , even 9J/(v, w) # , and w Cv ,n We>0"give an example.Take the graph of affine type 2. Let v t(1,1,1) and w t(1,0,0). We

consider the ADHM data given as in Figure 1, where (zl,z2) e2\{O},jl \{0}. They satisfy the ADHM equation (with a suitably chosen orientation)and the stability condition. Hence 9J/(v, w) #- .

If [B, i0, jo] e 9J/o, then we have iojo= 0 from the ADHM equation. Hence0 or jo 0. In either case, we multiply the scalar t to V1, V2, and V3, and

make jo 0 or 0 in the limit. Since (B, , jo) has a closed orbit, we mustflDreg(V’, W).have jo 0. In particular, 9J/o does not contain any

Taking into account the Gv-action, one can show that the image n(931(v, w)) isisomorphic to 2/7z3, that is, the simple singularity of type A2. 9J/(v, w) is itsminimal resolution. (In fact, this is the case studied by Kronheimer in [ 17] .)

COROLLARY 10.1 1. Suppose the graph is of finite type. Then the projectivemorphism r 9Jl(v, w) --. 7(gJl(v, w)) is semismall with respect to the stratificationn(gYt(v, w)) (90)(d) and all strata are relevant (see [3]). Namely, we have

2 dim 7-1(x) codim(gJl0)(d) for x (gJlo)(d).Proof In the proof of Theorem 7.2, we have already shown that the dimen-

sion of the fiber of 7t over (gYt0)(d) is given by

(dim 9J/- dim(gTto) )


Since we have an open stratum (gJ/0)(d) of dimension dim 99l by Lemma 10.4, weget the assertion. [

l O.iv. Coordinate al#ebras. In general, the homomorphism defined inTheorem 9.4 is not surjective. We shall take a certain quotient of) ntop(Z(v v2; w)) to which maps surjectively.

Let zreg(v1, v2; w) c Z(v1, v2; w) be the complement of the closure of the set ofll/reg(v0, w) for any v (cf. 7). The openall points (x x2) such that r(x

embedding zreg(vl, v2; w) Z(v1, v2; w) induces a homomorphism of algebras

) H,(Z(v v2; w)) -- )H, (zreg(v v2; (10.12)

Remark 10.13. When the underlying graph is of finite type, we havezreg(v1, v2; w) Z(v1, v2; w). This follows from Remark 3.28. For general graphs,the equality does not hold in general. The counterexample is given in 10.10.

Let L(A) denote the integrable highest weight module with a dominant inte-grable weight A as highest weight. Then by Theorem 10.2 we have a homo-morphism of -algebras

O Htop(Zreg(vl, v2; w)) -- OL(A, a,o)v (R) L(Aw a,o),V 2 It0

(10.14)

lflreg(0, W) is non-where v runs over the the set of dimension vectors such thatempty.

The following is one of the main results in our paper.

THEOREM 10.15. The homomorphism (10.14) gives an isomorphism.

Proof. Let us consider the compositions of the homomorphisms in (9.5),(10.12), and (10.14):

O - () L(Aw ,o) v (R) L(A, a,o). (10.16)

Since L(Aw- ,o) is a quotient of the Verma module, it is easy to check that thelinear map

U+ (R) (ah,,-a,o) (R) U- -, L(Aw- avo) v (R) L(A,-

is surjective. Hence the composition (10.16) and thus the homomorphism (10.14)are surjective.For v P, we define the weight spaces by

L(A, avo)v de__f. (V e L(A, avo)lhv (h, v)v for all h e U}.

552 HIRAKU NAKAJIMA

Let Htop(Zreg( 2;W))v0 be the linear subspace spanned by irreducible compo-nents contained in the closure of r-l(gj/eg(v, w)) zreg(vl,v2;w). Then (10.14)induces a linear map

Htop(Zreg(v1, V2; W))vo "-- L(A. avO)._,t (R) L(A. av0)A._%2.

The dimension of the fight-hand side is equal to the product of the number of thereg 0irreducible components of 9X(v w) and that of 9Y(v2, w), where x 931o (v w).

The dimension of the left-hand side is the number of irreducible components ofz-a(gxg(v, w)) c zreg(v,; w), which is smaller than or equal to the aboveproduct. Hence the surjeetive homomorphism (10.14) must be an isomorphism.

Note that we can determine all v’s such that 9xeg(v, w) # by Corollary10.8 under the assumption in Proposition 10.5.

Remark 10.17. It seems natural to conjecture that the basis given by the fun-damental classes of irreducible components of Z(1, v2; w) is related to Lusztig’scanonical basis in [23] of (Jq. (See Remark 10.3(1_).) It is also desirable to have ageometric construction of the canonical basis of Uq similar to our construction.

11. Intersection form of 9J/(v,w). Let 2(v, w) be the Lagrangian subvarietyintroduced in Theorem 3.21. It is homotopic to 9J/(v, w), and hence Htop((v, w))is isomorphic to the middle degree ordinary homology group Hemid(gJ/(v, w)) (i.e.,homology offinite singular chains). Moreover, (v, w) is the special case x 0 ofthe variety 9X(v, W)x considered in 10.ii. Hence @ Htop((v, w)) is the integrablehighest weight module with highest weight vector [(0, w)].

In the set-up of {}8, put MI= M3= point, M2= 9X(v,w), and Z Z=(v, w). Then the convolution defines a bilinear form

(’, ")" Htop((, w)) () Htop((, w)) .If we identify Htop((v, w)) with HCmid(gJ(, w)), the above is the ordinary inter-section form, which is defined as follows. First by the Poincar6 duality, we mapnia(gJ/(v,w)) to the cohomology with compact supports na(gJ/(v,w)). Wecompose it with the natural homomorphism to the ordinary cohomologyncmid(gJ/(v, w)) nmid(gJ/(v, w)), and denote the composition by 9. Then we have

(c, c’) (c, for c, c’ Hid(gJ/(v w)),

where (., .) is the natural pairing between the homology and the cohomology.The following is clear from the definition

(Ekc, c’) (-- 1) (c, Fkc’) for c e Htop((, w)), c Htop(( e/c, w)),


where r 1/2(dim 9J/(v,w)-dim 9J/(v-ek, w)). Hence by [11, 9.4] we obtainthe following.

THEOREM 11.1. The normalized intersectionform

(- 1)dim [II(v,w)/2 (., .) on Htop((, w))

gives the unique nondegenerate contravariant bilinearform on the integrable highestweight representation ( Htop (t(v, w)).The unitarity of the bilinear form is known (see [11, 11.5]), and hence the

above theorem leads to the following geometric consequence.

COROLLARY 1 1.2. The intersection form of gJ/(v, w) is definite.

APPENDIX

A.i. ProofofLemma 9.8. The proof will be similar to that of Theorem 5.7.As in 5, we consider complexes of vector bundles

L(V V2) tr- E(V V2) L(W, V2) L(V, W) -- L(V V2) i) I,

L(V3, V2) a-- E(V3, V2) L(W, V2) L(V3, W) -- L(V3, V2)

where we put suffixes 12 and 32 to distinguish endomorphisms. We have sectionss 12 and s32 of Ker z12/Im tr 12 and Ker z32/Im o"32, respectively.

Identifying these vector bundles and sections with those of pullbacks to(V I’) X IJ(V2, I’) X (V3, W), we consider their zero loci Z(s12) k(V2, w) x9Jl(v3, w) and Z(s32) 9J/(v w) x og(k, (v2, w)).As in the proof of Theorem 5.7, we consider the transposed homomorphisms

of Vs12 and Vs32 via the symplectic form. Their sum gives a vector bundle endo-morphism

t(Vs12__

t(Vs32 KerttrlE/im tz12 l Kerta32/Im tz32

-- TgJ/(v w) TgJ(v2, w) TgJ(v3, w).32It is enough to show that the kernel of t(Vs2) + (Vs) is zero at (x x2, x3).

Take representatives (B",ia, fl) of xa (a 1,2, 3). Then we have 2 and 32which satisfy (5.5) corresponding to (Ba, a, fl). Suppose that

(C’12, a’12, b’12) (mod Im t’c12) (C’32, a’32, b’32) (mod Imt’c32)

554 HIRAKU NAKAJIMA

is in the kernel. Then there exist a ff L(Va, Va) (a 1, 2, 3) such that

/3Ct12 12 1 B B ,bn2 ,

atl2 12 _jl 1,

/3Ct32 32 ),3 B3 B3 y3,

bt32 )3 3,a/32 32 _j3 ,3,

3( 12 C/12 -I-- 32 C,32) 2 B2 B2 ,2,

12 b,12 -t- 32 b/32 3 i2,a,12 a,32 j22

(A.1)

Consider Im 12 Im 32 c V2. Then, using (A.1), we can check that

(n2 12T1 (12)-1_323 (32)-1)(im 12 63 Im 32)

is a B2-invariant subspaee contained in the kernel ofj2. Here (12)-1mean the inverse of 12 and 32 as follows:

and (32) -1

(2)-1" Im 2 --o V and (32) -1" Im 32 __, V2.

The stability condition for (B2, 2, j2) implies

on Im 12 (3 Im 32. (A.2)

Choosing a complementary subspace (Im 12)-1- in V, we define a scalar 212and an endomorphism (12 2

k "Vk -* Vk SO that

12 12kk -I- ,12 / on Im 2,

on Im 2.Terms on the fight-hand side are equal on the intersection Im 12k olmk32 by(A.2), and the sum Im 12 32 2

k + Im k spans Vk by the assumption. Hence 22 and(2k are well defined. For v k, define (2 by


Then we have

CI12 t3((12/2 8 (12),a,12 j2(12 (12 + ,12 id),

b,12 ((1212 + ,12 id)i2.

This shows that (C,12, a’12, b’12) tz12((12 ( 212).A similar argument shows that (C,32, a,32, b,32) is in the image of Z32. Hence

t(Vs12) -- t(vsa2) is injective.

A.ii. Proof of Lemma 9.9. The set fD(k,(Vl,w))(v3,W) (resp.,(1, W) X k(V3, W)) can be described as a zero locus Z(s21) (resp., Z($23)) ofa certain section s21 (resp., $23) of a vector bundle Ker zE1/Im 0"21 (resp.,Ker zE3/Im 0"23). It is enough to show that t(Vs21)+ t(Vs23) is injective. Takerepresentatives (Ba, a, ja) of xa (a- 1, 2, 3). Then we have 21 and 23 as in (5.5).(The notation will be an obvious modification of that in the proof of the previouslemma, so we do not repeat the definition.)

Suppose that

(C’21, a’21, b’21) (mod Im t1721) () (C’23, a’23, b’23) (mod Im tz23)

is in the kernel. Then there exist ])a U L(Va, Va) (a 1, 2, 3) such that

/i21 Ct21 ])1 B B ])1,

21 b,21 ])1 il,

at21 _jl ])1

23 Ct23 ])3 B3 B3 ])3,

23 b’23 ])3 i3,

at23 _j3 ])3,

/3(C,21 21 _+_ C,23) 23 ])2 B2 B2 ])2,bt21 ._[_ bt23 ])3 i2,

a,21 21 a,23 23 _j2])2.

(A.3)

Hence we have

B(]) 21 21])2) (])1 21 21])2)B2 d-/i 21C’23:23,

jl (])1 21 21])2) __j3 ])3 :23.(A.4)

Let

1,2 de2" {v e V21])3 23(v) e Im 23}.

556 HIRAKU NAKAJIMA

For v

)3 B3 23 (v) B3 ),3 23 (/)) _[_/ 23 C/23 23 (v) (A.5)

is contained in Im 23. Then we can composite (23) -1

),3 B3 23 to 12, and haveto the restriction of

e C’23 23l2 (23)- (73 B3 23 B3 y3 23)l2

(23)-1),3 B3 231122 B2 (23) -1 ),3 23[122"

Substituting into (A.4), we find

Bx(x 2x 2y2 + 2 (23)-3 23)l(1 21 21),2 q_ 21 (23)-1),3

j1(),1 21 212 _[_ 21 (23)-13 23l 0.

Since ),32B2(2) ),3B323(I’2) is contained in Im 23 by (A.5), (]yl 21 21),221 (23)-3 23)(12) is Bl-invariant and contained in Kerfl. Hence we have

1 21 212 .+. 21 (23)-1),3 23 0 on l’2.

Similarly, for

2 def.

we have

)3 23 23),2 .+_ 23 (21)-1),1 21 0 on 12.

The above implies l’2 C:: l2 by definition. Similarly, we have 2 C:: 1,7"2, and hence,2=

CLAIM A.6. [2._ [2__ V2.We shall complete the proof of Lemma 9.9, assuming the claim. The claim

means Im 21 is stable under ),1. Hence, taking a complementary subspace toIm ,1, we can define a scalar 221 and a homomorphism (2,1k Vk Vk2, so that


For # k’, let

/21 /21 -1 ,21), ) (

Then we have (C’21,at21,b’21) t1721((21 ),21).A similar argument shows that (Ct23, a123,b123) is in the image of 1723. Hence

t(Vs21) -- t(Vs23) is injective. We have thus proved Lemma 9.9.

Proof of Claim A.6. By definition, l?/2 V/2 if # k, and //2 V/2 if # k’.Since l?2 l?2 we are done if k - k’. So we assume k k’ from now on.

"2 2Take an edge h with out(h) k. Then in(h) # k, so Vin(h Vn(h Hence wehave

?iln(h) 21 -21 2 21 ( 23 )-1 3 23in(h) in(h))’in(h) --in(h) in(h) in(h) in(h)"

Combining with (A.4) and (A.5), we have

k --in(h) (in(h)) 3

1/ 1-21 -21 2x .’3 3 23Jkkk k "k) --J k k

(A.7)

Suppose l?k2= l?k2 # V. Then there is an element vo V such that

,121 21 ,323 23k k (V0) Im k and ’k k (V0) Im k Hence we have a direct sum decom-position

V Im 21 1.21 21 2k q-kgk k k(V0),

Vk3 Im Ck23

Let us define /k l/’ Vk3 by

klim. 3(1)-1 212 3 23 (vo)k

By (A.7) we have

( )-1Bh 21 23in(h) in(h) B /k,

j j3 gk.

This shows [B1, il, jl] and [B3, i3, j3] define the same point, and hence contradictwith the assumption.

558 HIRAKU NAKAJIMA

A.iii. Proof of Lemma 9.10 (See also [27, Proof of 10.11]). Take ([Bl,il,jl],[B2, i2,j2],[B3, i3,j3]) U k(V2, W) 9Y/(V3,W) CgJI(vl,w) 09(k,(V2, W)).There exists 12: V V2 (resp., 32" V3 V2) which relates (B,i1, fl) and(B2 i2,j2) (resp., (B3, i3, j3) and (B2, i2,j2)) as in (5.5). The intersectionIm2Im 32 is B2oinvariant and contains the image of 2. By definition ofU, this has dimension v -e’= v4. Hence the restriction of (B2, i2, j2) toIm 2c Im 32 gives a point in 9J/(v4, w). This correspondence defines a map,which turns out to be the required isomorphism.To show the map is an isomorphism, we construct the inverse. Let ([B

[B4, 4, j4], [B3, 3, j3]) 6 U 02(k,(V1, w)) J/(v3, w) f3 ffJ/(v1, w) k(V3, W).There exist 4:V4V and 43:V4V3 as above. Let us take comple-mentary subspaces (Im 41)+/- and (Im 43)+/- and consider

V4 (Im 21)/ ) (Im 23)/. (A.8)

We identify V and V3 with the subspaces V4( (Im 41)+/- and V4( (Imrespectively, and define

B2 de=f. B4 + Bll(im 41)1 "- B3l(im 43)1,

j2 de=f. j4 _[.. fl I(Im 41)1 -+- j3 I(im 43)+/-.

It is clear that (B2, E, j2) satisfies the ADHM equation.

CLAIM A.9. (BE, 2, jE) is stable.

Once we have this claim, it is clear that this correspondence gives the requiredinverse.

Suppose S is a B2-invariant subspace contained in the kernel of jE. Its inter-section with V4 (Im 41)4 V is Bl-invariant and contained in Kerjl; henceit is zero by the stability of (B1, il, fl). In particular, S has a nonzero componentonly on the vertex k’. Similarly S V4 (Im 43)+/- 0. In particular, S 0 ifk k’. So we assume k k from now on.

Write a nonzero element in Sk as v a b according to the decomposition(A.8). Since S c V4 (Im 41)+/- VI= 0, we must have b 0. Similarly wehave b 0. Since S is B2-invariant, we have B(v a) + B3(0 b) 0. Definea homomorphism gk: Vk3 V by

-v’/b ) 2 Vk3 Vk (Im 41 +/-

-a/b Vk3 Vk (Im 43)+/- 2


where we identify (Im 43)+/- with C and consider b as scalar. Then gk is invertibleand B Bgk for any h with out(h) k. Hence [B1, 1, jl] and [B3, 3, j3] are thesame point. This contradicts with the assumption. Hence we prove the claim.

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GRADUATE SCHOOL OF MATHEMATICAL SCIENCES, UNIVERSITY OF TOKYO, KOMABA, MEGURO-KU,TOKYO 153, JAPAN

CURRENT: DEPARTMENT OF MATHEMATICS, KYOTO UNIVERSITY, KYOTO 606-01, JAPAN;[email protected]

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