quiver varieties and kac-moody algebras - CiteSeerX

Vol. 91, No. 3

DUKE MATHEMATICAL JOURNAL

(C) 1998

QUIVER VARIETIES AND KAC-MOODY ALGEBRAS HIRAKU NAKAJIMA 1. Introduction 2. A modified enveloping algebra 3. Quiver varieties i. Definition ii. Stability condition iii. The projective morphism iv. tm-action v. Stratification of 9J/0 4. Tautological bundle homomorphisms 5. Hecke correspondence i. Definition ii. Fibration 6. A decomposition of the diagonal 7. A Lagrangian subvariety Z 8. Convolution algebra i. Convolution in homology ii. The case when Z is Lagrangian 9. A geometric construction of the algebra U i. Main construction ii. Definition of Ek, Fk, and Hk iii. Integrability iv. Relations 10. Module )ntop(gJ/(v, wx)) i. Operators Ek and Fk ii. Integrable highest weight modules iii. Criterion for the nonemptiness of w/reg 0 iv. Coordinate algebras 11. Intersection form of 9J/(v, w) Appendix i. Proof of Lemma 9.8 ii. Proof of Lemma 9.9 iii. Proof of Lemma 9.10

516 517 519 519 522 525 526 526 529 532 532 536 536 538 539 539 541 542 542 543 543 544 546 546 548 548 551 552 553 553 555 558

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, we defined a new family of hyper-K/ihler manifolds, called quiver varieties, associated to finite graphs without edge loops. The definition was motivated by the ADHM (Atiyah, Drinfel’d, Hitchin, and Manin) description (see [18]), which we can now interpret as the identification of moduli spaces of anti-self-dual connections on ALE (asymptotically locally Euclidean) spaces with quiver varieties of affine type. Quiver varieties can be viewed as higher dimensional analogues of ALE spaces in many respects. We studied their geometric properties which are surprisingly rich. In particular, we constructed integrable highest weight representations of the Kac-Moody algebra on the space of constructible functions on certain Lagrangian subvarieties. This construction was motivated by Ringel and Lusztig’s construction (see [31] and [19]) of the lower triangular part of the Drinfel’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 Lagrangian subvarieties in them. Then we define the convolution product on homology groups, which satisfies the defining relation of the enveloping algebra. It also gives reformulation of our previous construction of representations. We use homology groups instead of constructible functions. In fact, when the underlying graph is not of finite type, it is more natural to relate our convolution algebra to the mo_dified enveloping algebra, which is the specialization (at q 1) of the algebra Uq introduced by Lusztig in [23]. The algebra is a quotient of ll [Jqlq=l by a certain ideal depending on the choice of a highest weight vector w. When w goes to the infinity, the ideal becomes smaller. (The limit is not treated in this paper.) As an application of our construction, we obtain a basis of the quotient of the modified enveloping algebra given by fundamental classes of Lagrangian subvarieties. It is natural to conj_ecture that this basis relates to the specialization of Lusztig’s canonical basis on

Uq (see [23]). Our construction is strongly motivated by Ginzburg’s construction of the universal enveloping algebra U(eln) in terms of homology groups of cotangent bundles of generalized flag varieties (see [7]). Our definition of the convolution product is also borrowed from Ginzburg’s paper. Cotangent bundles of generalized flag varieties are examples of quiver varieties [27, Sect. 7], and our construction is a generalization of his construction. Ginzburg’s proof was given in [4, Chap. 3]. His construction was motivated by Beilinson, Lusztig, and MacPherson’s construction [1] of quantized enveloping algebras, in turn. We remark that our construction can be explained in the gauge theoretic language, as in [28], when the underlying graph is of affine type. During the preparation of this paper, we received an announcement of similar results for general projective algebraic surfaces by Ginzburg, Kapranov, and Vasserot [8]. Their formulation uses functions on rational points of moduli spaces of vector bundles, instead of homology groups. Our proof heavily depends on our description in terms 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 for general surfaces. In the approach of Ginzburg et al., this transversality condition is unnecessary since they work on functions on rational points. The paper is organized as follows. In {}2 we give a quick review of the definition of the universal enveloping algebras of Kac-Moody algebras associated to symmetrizable generalized Cartan matrices and their modified version. In 3 we introduce quiver varieties. Our approach is a little bit different from the approach in [27], and works for arbitrary algebraically closed fields. In 4 we define certain vector bundles on quiver varieties and tautological bundle homomorphisms between them. We also introduce a stratification of quiver varieties in terms of the rank of these homomorphisms. In 5 we define nonsingular Lagrangian subvarieties, called Hecke correspondences, in Cartesian products of quiver varieties. These give us generators of Kac-Moody algebras and play a crucial role in our construction. In 6 we show that the quiver variety is connected. For the proof we use a decomposition of the diagonal, which is of independent interest. In 7 we introduce Lagrangian subvarieties which contain Hecke correspondences as irreducible components. In {}8 we review the definition of the convolution product on homology groups following Ginzburg (see [7]). In {}9 we apply Ginzburg’s theory to the above Lagrangian subvarieties introduced in 7. We give a homomorphism from lI to the convolution algebra. In 10 we identify natural modules of the convolution algebra with integrable highest weight representations of Kac-Moody algebras. We also introduce a certain natural subalgebra to which the homomorphism is surjective. During the proof, we give a geometric analogue of Kashiwara’s operator (see [12]). In 11 we identify the intersection form of the quiver variety with the invariant inner product on the highest weight representation. In particular, we can show that the intersection form is definite. In the appendix, we give proofs of the lemmas in 9. After this work was done, the author learned that Grojnowski announced a similar geometric construction of the affinization of the quantized universal enveloping algebra using the equivariant K-theory on the same Lagrangian subvarieties (see [10]).

Acknowledoements. I am deeply indebted to V. Ginzburg whose lecture at Kyoto (September 1993) is the starting point of the present work. I would like to express my sincere gratitude to M. Kashiwara for a number of interesting discussions. I thank G. Lusztig who pointed out an error in the earlier version of this paper. My thanks go also to R. Hotta, T. Uzawa, K. Hasegawa, and G. Kuroki, who answered a number of my questions on representation theory. 2. A modified enveloping algebra. We give a quick review of the definition for the universal enveloping algebras of Kac-Moody variety associated to symmetrizable generalized Cartan matrices and their modified version following Lusztig 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* (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.

-

Z,

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’]

for h,h’

0

(h, k)ek,

(2.2)

[h, fk]

--(h,k)fk,

(2.3)

6klhk,

(2.4)

1

E (- 1)

1

p

=0

/

where

Ckl

P

p=0 --Cki

(2.1)

[h, ek]

[ek, fl] (_1)

-

P*,

ee-,-p

(k v l),

(2.5)

Ckl 4,p_.1--Ckl--p :0 (k /),

(2.6)

0

JJJlJk

) 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 have the triangle decomposition (see [11, 9.1.1])

U

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

We introduce the modified enveloping algebra sum of 1-dimensional algebras

(2.7)

I1

by replacing U with the


519

Here the multiplication is defined by the following rules:

aaau eka,

aA+kek

(ekfl flek)a.

6xuax

(2.8)

fka aA-otkfk,

(2.9)

6kl(hk, 2)a.

(2.10)

This was the specialization (at q 1) of the modified quantized enveloping algebra introduced by Lusztig in [23, Chap. 23]. Note that U is an algebra without unit, 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 weight

decomposition.

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 notation will be used for the unital U-module. 3. Quiver varieties. The purpose of this section is to review the definition and prove some properties of the quiver varieties introduced in [27]. Almost all properties of quiver varieties given in this section were proved in [27]. Our approach is slightly different from the original one. Instead of using a Kihler quotient by a compact group, we use the geometric invariant theory (GIT) quotient by the eomplexifieation with a certain polarization. This is useful when we compute the convolution product in {}9. Moreover, the definition makes sense over any algebraically closed field IF, not only over E. We replace the original proofs, which used K[ihler metrics, by arguments which are valid even in positive 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 0. This is only for brevity. The modification for 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 G C in [27]). The moment map # used here is the complex part #c of the hyper-Kihler moment map in [26]. The symplectie form co corresponds to the holomorphic sympleetie form coez. The symbols An and ]pn denote, respecdenotes the multiplieative tively, the affine n-space and projective n-space; group.

ez

m

3.i. Definitions. Suppose that a finite graph is given and assume that there are no edge loops (i.e., no edges joining a vertex with itself). Let I be the set of vertices 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 symmetric generalized Cartan matrix C (Ckt) as follows: --Ckt =--Ctk is the number of and Ckk 2. This gives a bijection between the edges joining k and in E if k finite 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 reverse orientation. Choose and fix an orientation fl of the graph; that is, a subset c H such that f u f We assume that f has no cycle. H, 1" c f 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) E(V1, V2 kI

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

-

het

def._.

,2 ,1 Vin(h) @ Hom(Vout(h)

hH

def.__

Vn(h)2 ),

@ Hom( gout(h) Vi)__n(h).hef

For B (Bh) E(V 1, V2) and C plication of B and C by

in(h)=k

(Ch) E(V2, V3), let

us define a multi-

k

Multiplications ba and Ba of a e L(V 1, V2), b L(V2, V3), and B e E(V2, V3) will be defined in an obvious manner. If a L(V 1, V1), its trace tr(a) will be understood 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. The dimension of M is tv(2w + (21- C)v), where I is the identity matrix. The above three 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 enveloping algebra. The dimension vectors will be mapped into the weight lattice in the fol-


lowing way: def.

E

l)kOk,

w

k

-

Aw def.

521

E WkAk, k

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

For a collection S (Sk)keI of subspaces of Vk and B e E(V, V), we say S is 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(V 1,V2), let us denote by eB E(V 1, 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

Gv d=f.

G

H GL(Vk), k

where we use the notation on M by

Gv when we want to emphasize the dimension. It acts

(B, i, j) preserving the symplectic form given by

g(B, i, j) d=f. (gBg_l, gi, jg-1), 09.

The moment map vanishing at the origin is

eBB + ij e L(V, V),

/(B, i, j)

(3.4)

,

where the dual of the Lie algebra of G is identified with L(V, V) via the trace. The moment map is defined only over but the above explicit form makes sense over any 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)) be the coordinate ring of the affine algebraic variety #-1 (0). Then 9J/0 is defined as a variety 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 vectors v and w. By the GIT in [25], this is an affine algebraic variety. It is also known that 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 fix one 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 can define 9Jl

9X(v, w) de_f. Proj @ A(/z-(0)) t’z

(3.6)

n>0

This is what we call a quiver variety.

Remark 3.7. There are many choices of characters of G. The construction of quiver varieties can be done for any character. The stability condition, introduced in the next subsection, will be changed (see [29]). Our choice seems most convenient. 3.ii. Stability condition. In this subsection, we shall give a description of the quiver 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 relation on 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) Then the geometric points of the GIT quotient 9J/ can be described as

s.

#-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. A point above conditions.

(B, i, j)e t-l(o)

is said to be stable if it satisfies the


523

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

Proof of Lemma 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 1parameter subgroups 2 {m G. Let S (Sk) be a collection of subspaces that violates the stability condition. in Vk, we define a 1-parameter subgroup 2 Taking complementary subspaces Then j-l(A(t)) is a positive power as follows: 2(t) t on Sk and 2(t) 1 on of t, and hence lim_0 z-l(2(t)) --0. On the other hand, it is easy to check that the limit of 2(t)(B,i, j) exists. This means (B,i, j) lies in the complement of

S-

S-.

/./-1 (0)s. To show the converse, let us suppose that 2(t)((B, i, j), z) converges to 0. If V as V’ is the weight space decomposition with respect to 2, the existence of the limit implies

(B, P, j, 0)

m

nh(Vont(h)) c n>m O V(h), jk(V) =0 form> 0. Then $ (m>0 vn)k violates the stability condition. But z-l(2(t)) does not 0 for m > 0. Hence S cannot be zero. converge to zero if

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)

nonsin#ular subvariety of dimension tv(2w + (I- C)v). Proofi For (1), suppose # (/k) G stabilizes (B,i, j). Then a collection of subspaces Im(gk- 1) is B-invariant and contained in Kerj. Hence we have is a

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

gk

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

Im is B-invariant and contained in Ker j. Hence

must be zero by the stability

condition.

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 symplectic form 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-orbit through (B, i, j). is a geometric quotient (see [23, COROLLARY 3.12. The quiver variety 0.6]) of/z-l(0) by G and a nonsingular variety of dimension tv(2w- Cv). In par-

ticular, the set of geometric points of the quiver variety 93l consists of

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

.

The tangent space

Ker d#/Im

of 9Jl at the point corresponding to (B, i, j) can be identified with

Proof. As we remarked above, the stability condition implies the triviality of the stabilizer. Hence the dimension of stable orbits is equal to dimG. Thus any stable 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. In particular, all orbits are closed in/z-l(0) Hence the stability and semistability conditions coincide, and our assertion follows.

s.

Remark 3.13. When W 0, the scalar group ltm acts on M trivially, and hence we have the residual action of G/tm on M. Choosing a character Z:G/ffJm (m appropriately and introducing a stability condition, we can modify 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 symMarsden-Weinstein plectic quotient (or reduction) (see [24]). Hence we have the following.

COROLLARY 3.14 (Base field IF

). The symplecticform o9 in (3.3) descends

to 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 as a 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 will be denoted also by [B, i, j]. Remark 3.16. The original definition of the quiver variety given in [27] is different from the one given here, and is given in terms of the hyper-Kihler moment map. In particular, the quiver variety has a natural hyper-Kihler metric. It was


525

proved in [27, 3.5] that (B, i, j) is stable in the above sense if and only if the Gorbit intersects with a level set of the Kihler moment map. Hence our 9J/coincides with the previous one. The following theorem was proved in [27, 4.2]. TrmORM 3.17 (Base field IF IE). The quiver variety 9Jl is diffeomorphic to an

affine al#ebraic manifold. In fact, this can be proved easily by using the hyper-K/ihler structure in our is affine algebraic with respect to another complex structure J. This situation: result is not used in this paper. 3.iii. The projective morphism natural projective morphism

7r"

Ro. From the definition, we have

Jt

n:

,,

(3.18)

9J/o.

,

a

If 7r([B, i, j]) [B j0], then G(B i0, j0) is the unique closed orbit contained in the 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. Suppose that (B,i,j)eM and

a B-invariant increasing

filtration

O=V (-)V ()

V (N)=V

c V () are given. Then set grm V grm V. v(m)/v (m-l) and gr V Let grmB denote the endomorphism that B induces on grinV. For m 0, let gr0i e L(W, V ()) be such that its composition with the inclusion V () c V is i, and let gr0 j be the restriction of j to V (). For m # 0, set grmi 0 and grm j 0. Let gr(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 that gr(B, i, j) is a representative ofy on grV.

when Im

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

,,

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

,,

(B

j0).

Decompose V into A-weight spaces V )Vm, where Vm is the space of weight m. 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 (), and 0. The stability condition for (B, i, j) implies V (-) 0. Moreover, the limit (B io, j0) is equal to the restriction of (B, i, j) to the direct summand

j(V (-))

,

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 gian subvariety which is homotopic to 9Jl.

7g-l(0)t::: 9J

is a

Lagran-

We do not use this result except in 11. One can give the proof of being Lagrangian by modifying arguments in Theorem 7.2. (It is different from the original 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 component Eft(V, V)@ L(V, W) is unchanged; it commutes with the G-action. The symplectic form co on M transforms as co- tco. Moreover, #-1(0) is invariant under 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, thanks to the assumption that f has no oriented cycles. Hence the fixed points are contained in the Lagrangian subvariety

n-l(0).

3.v. Stratification ofgJlo. We study a stratification of 9210 subsection. We first introduce an open stratum of 9J/0 (possibly empty), def. 0

Iz-l(O)//G in this

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

PROPOSITION 3.24. If [B i, j] isomorphism n-l(gj/eg) mreg ’0

flreg *o

then it is stable. Moreover,

(3.23) induces an

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

527


limt_oA(t)(B,i,j) exists and its stabilizer contains Hk GL(Sk). But the orbit G(B, i, j) is closed and the limit lies in the same orbit. Hence the assumption implies S 0. This proves the first statement. The inverse image of [B, i, j] under 7 is the union of all those orbits whose closure contains (B, i, j). But the dimension of the orbit G(B, i, j) is maximal when the 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 0 may be empty. In fact, we shall give a criterion for 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 by n(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 definition. 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] there is a direct sum decomposition V

V0

(V 1) e ()

(])

(Vr) e

(gJ/0)(d)

with nontrivial

Then

and times

Thus, after replacing the representative (B, i, j), if necessary, we have (1) each summand is invariant under B; (2) the restriction of B to (V i) e IFO’7 (R) V is of the form 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) 0 is not excluded); (though the case V (6) the subgroup l-Ik GL(V) meets the stabilizer only in the scalar subgroup

(Ik GL(Vki))

c0

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 V i.

Proof Since the orbit G(B, i, j) G/ is closed, it is an affine variety. Hence the stabilizer ( is reductive (see [30]). Let g (Ok) be a nontrivial semisimple 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 stabilizer of the restriction of B to V(2) for 2 1 is larger than m, we further decompose V(2). Thus we have a B-invariant direct sum decomposition V

V @

V’@ V"@

...,

such that and (a) the image of is contained in V and j is zero on V’ @ V" @ (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 declaring V’~ V" if and only if there exists an isomorphism V’ V" which commutes with B. Collecting equivalence classes of summands, we have the desired decomposition. 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

529


contributes by

2- tviCvi. The term 2 appears because we make the quotient by This completes the proof of the lemma.

GlUm. (See Remark 3.13.)

If V’ (Vkt)kei is a collection of the subspaces of V ural inclusion map

(Vk)kei, we have a nat-

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, the 0. This can be proved as follows. If the restriction of B to V is zero for variety 9J/0(vi, 0) contains points with stabilizer m, its dimension is given by 2- tvCv. Since C is positive definite when the graph is of finite type, we have dim 9J/0(vi, 0) 0. But this is impossible unless 9J/0(vi, 0) (0), since 9J/0(vi, 0) is a cone. In particular, we have

U lf?reg(vO’ W) 0

_

0(, W)

vO 0, then the first case does not occur. Otherwise, we see that

Thus ro

9J/k;o(V, w) of rank

is nonempty and we should have a vector bundle tek(w C), which is negative by assumption.

Qk(V, w) gJ/k;o (V, w) 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

for any k I.

Vk

(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 complementary subspace. Define a 1-parameter subgroup A’tm G as follows: A(t) 1 on Sk -1 on and (1 # k), 2(t) S-. Then the limit limt_0 2(t)(B, i, j) exists and has a stabilizer containing GL(Sk). But (B, i, j) has a closed orbit, and hence the limit also has the trivial stabilizer. This means S 0. Hence we have shown the first statement of the lemma. If there is a vertex k with tek(w- Cv) < 0, (4.8) does not hold by Corollary lq 4.6. Hence we have w- Cv Z0.

We also have the following lemma. LEMMA 4.9. Let [B] e 9Jl0(v, 0) (i.e., w

m.

0). Assume the stabilizer orB in Gv is

Then we have either

(1) ’in(h)=k Im Bh Vk for any k I, or k (and hence (2) there exists a vertex k such that Vk IF and Vt 0 for B 0). Moreover, in the first case we have (a) -Cv >o, and the support (b) 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. The last 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 element 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 author does not know a criterion. When the underlying graph is affine, such a v is a multiple of the imaginary root given in [11, 5.5]. One can show that 0(v, 0) contains a point with the stabilizer if and only if v

m

.

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

Definition. Let

533


,

1 _+_ ek. In other words, av2 av + tk in Convention 3.2. Choose collections of vector spaces W, V and V2, with dim W w and dim Vi= vi. As in {}4, we consider V and V2 as holomorphic vector bundles over 9Jl(v w) x 9J/(v2, w). Taking a point ([B 1, 1, jl], [B2, 2, j2]) from the product 9Jl(v 1, w) x 9J/(v2, w), we define a 3-term sequence of vector bundles V2

L(V V2)

where

-

,

tEu(V V2) tL(W, V2)

E6(V

,

V2)

L(V W)

/

tL(V V2)

tiF,

(5.1)

(B2 B 1) @ (_il) j2, b) (eB2C + eCB’ + i2b + aft) (tr(ilb) + tr(aj2)).

o’()

z(C

a

Here tmv is the notation in Notation 4.1. This is a complex; that is, za 0, thanks to the ADHM equation and the equation tr(ij 2) tr(ilj2). Notice the similarity with 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-invariant and 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 between L(V2, V 1) () 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. Hence we must have Ker 0, but this is impossible since the dimension of V is less than V2. Thus we have 2 0. Then, applying the stability condition for B to the 0. image of we have

,

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

rank(Ker z/Im a)

-

tv2(Cvl -I- w) tvlw- 1 (1/2) (dim 9Jt(v 1, 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(BIB 1) tr(B2B2) 0. The point ([BI,i 1, jl], [B2, i2, j2]) is contained in the zero locus Z(s) of s if and only 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 subspace of V2 with dimension v 1, which is B2-invariant and contains Im 2. Moreover, 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 ek. Similar varieties were studied by Lusztig in [19] and by the author in [26].

1

2

Definition 5.6. We call Z(s) the Hecke correspondence, and denote it by 2, w). Introducing a connection V on Ker /Im a, we can define the differential Vs: T 9Jl(v 1, w)

T (v2, w)

Ker z/Im a

of the section s. Its restriction to Z(s)= k(V2, W) is independent of the connection. Let us introduce the symplectic form on the product space 9Jl(v 1, w)x 9J/(v2, w) by changing the sign on the second factor; that is, to x (-to).

differential Vs is surjective over k(2, w). Hence k(2, W) of 9J(v 1, w) x 9Jl(v2, w). Moreover, k(2, W) is Lalrangian when the underlying field lF is ff. Proof. Take a point ([B 1, il, jl], [B2, i2, j2]) in k(2, w). There is of E(V 1, V2) 1) we define a section L(V 1, V2), as in (5.5). Fixing L(W, V2) L(V 1, W) by THEOREM 5.7.

The

is a nonsingular subvariety

,

(/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 (B 1, 1, jl, B2, i2, j2), and gives a lift of s. Take a tangent vector (fiB 6i 6j 6B2, 6i2, 6j 2) of lYt(v w) x 9Yt(v2, w), and

535


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

fiB2 )

( 6B

6i2)

( 6i

(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,C t, b’, a’) (mod tL( V2, V2)). (5.9) The surjectivity of Vs follows from the claim below.

CLAIM 5.10. t(Vs) is injective. Proof. Suppose that (C’,a’, b (mod Imtz) is in the kernel of t(Vs) There exist yl L(V V), y2 L(V2, V2) such that

,

3C t

IB

b

])1il,

-a’

B1])1,

e,C’ ])2B2 B2])2, (5.11)

{b’ _at= _j2])2.

Combining with (5.5), we find that the image of ])1 ])2 is Bl-invariant and lies in the kernel of j 1. 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 complementary subspace, 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 can define k" by k ]) 2 id. For k, define l" Vt by

V V

V

536

HIRAKU NAKAJIMA

Substituting into (5.11), we get

C’= e((B2 B 1), a’= j2( ( + 2 id), b’= (( + 2 id)i2. This shows that claim.

(C’,a’,b’)= tz(

2). We have completed the proof of the

Proof of Theorem 5.7, continued. In order to prove the second statement of Theorem 5.7, it is enough to show that the tangent space to 3k(V2, w) is involutive, since we already know that 3k(V2, w) is half-dimensional by (5.3). The involutivity 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) tion. Let 9Jk;r(Vi, W) be as in (4.3).

fj(i, W) (i-- 1, 2) be the natural projec-

LEMMA 5.12. (1) Let E be the vector bundle by the restriction of

of rank tek(w Cv 1) + r obtained

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 bundle

P(E). (2) Let F be the vector bundle of rank r obtained by the restriction of

(V/

Im

/ in(h)=k

B + Im i

9Jk;r(2, W). Here v means the dual vector bundle. Then pl(k;r(V2 W)) can be identified with the projective bundle P(F). Proof (See also [27, 10.10]). For (1), since the proof is almost identical to tO

Proposition 4.5, we omit it. For (2), the fiber p-I ([B2, i2, j2]) is isomorphic to the variety of all codimension1 subspaces Sk of containing + Im ik2. Hence we have the assertion.

V

EIm B

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

537


W. Then the quiver variety 9J/can be defined as a hyper-Kihler quotient of M by 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(V 1, V2)

)

tL(W, V2)

where

O’()

z(C ) a

b)

,

tL(V V2),

(6.1)

L(V W)

(n2- B 1) t0) (_il) tj2 (eB2C + eCB + i2b + aft).

Here we used the convention (4.1) to make the complex equivariant. As in Lemma 5.2, tr is injective and z is surjective. In this section, we do not need to take care about 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 homomorphism

,

over 9J/x 92/

L(V V2)

L(V 1, V2)

by

(1 ( 2

)

-

E(V 1, V2) t L(W, V2)

L(V x, W)

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 diagonal 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(V 1, V2) IE. Let L denote the tautological line bundle. The composite L ’-, p* (L(V 1, 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(V 1, V2) ) L(W, V2) ) L(V 1, W)). Its zero locus is isomorphic to the diagonal via the projection p, and it is transversal as in Theorem 5.7.

538

HIRAKU NAKAJIMA

-,

Suppose that 9J/ has two components fJ/1 and 9J/2, and take X1 ff 9Jl and x2 93/2. We consider the restriction of P to 9J/x {x} 9T/, which we can get by over x in the above definition. Choosing an isoreplacing V2 with the fiber

V

V

and V2:, we have an identification for the restrictions morphism between for x and x2. We denote this identified restriction of p:P--, 9J/x 9J/ by p" P’ ---, and the restriction of L* (R) p* (E(V V2) (9 L(W, V2) (9 L(V 1, 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 {x2}). Let fl be the Thom class of F. Since si does not vanish outside p’- (xi), the pullback s’f is a cohomology with compact support. (Note that the Euler class of F, which is the pullback of f by the zero 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 s2 is bounded and the norm Ilslll is proper, (1 t)sl + ts2 does not vanish outside a compact set. In particular, sf2 and sD are equal in the cohomology with compact support H(P’). But the integrals over p’-lgJ11 are different. We have

,

p-llgj

s’n

1

This is a contradiction.

Remark 6.3. (1) When the underlying graph is of type An, Theorem 6.2 is proven with a totally different method in [26]. (2) When 93l is an ALE space, the description of the diagonal plays an essential role in the ADHM description for instantons (see [18]).

7. A Lagrangian subvariety Z. Let V 1, 2, and w be dimension vectors, and also let V 1, V2, and W be the corresponding collections of vector spaces. Consider quiver varieties ffJ/(v 1, 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(X 1) /g(X2)}.

.

_

-

(7.1)

We use the notation Z unless we need to be conscious of dimensions. Let Z reg zreg(v 1, v2; w) c Z(v 1, v2; w) be the complement of the closure of the set of all points for any v

(X

X

2) such that 7g(X 1) 7g(X2) eg(0, W) J(1 2, W)

The subvariety Z reg 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 change the sign of the symplectic form on the second factor as in 5.)

THEOREM 7.2.


539

Proof. The following proof was suggested to us by G. Lusztig. The original proof 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 inverse image by n in 0/(v, w), where v is either v or v2. It is enough to show that 9J/(v,w)x has pure dimension of (dim 9J/(v,w)- dim 9Y/(v,w))/2, and that the pullback of the symplectic form vanishes on 0/(v, W)x. These statements can be shown 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 we are done. Otherwise, by Proposition 3.20, there exists a filtration 0 V (-1) c c V (N) V () c V such that dim V () v. Moreover gr(B, i, j) is a representative 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.

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

X

de__f,

p(X t’3 9Jk;r( W))

9Jk;O(

(7.4) 8k(X), we define

rek, w),

where p is as in (4.4). Recall that there is a vector bundle Qk(V- rek, w) of rank tek(w- Cv)+ 2r over 9Y/k;0(v- rek, w). Then p is the Grassmann bundle of 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 hence the 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 are done. Moreover, if v and w are tangent to the X c 9J/k;r(v, w), the evaluation of the symplectic form satisfies to(v, w) to(p,v, p.w). Here to in the right-hand side is the 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 in the homology given by Ginzburg in [7].

540

HIRAKU NAKAJIMA

For a locally compact topological space X, let H.(X) denote the homology group of possibly infinite singular chains with locally finite support (the Borel-Moore homology) with rational coefficients. If X is embedded in an ndimensional oriented manifold M as a closed subset, then we have the Poincar6 duality 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).

Y is an open embedding, there is a pullback homomorphism

If i" U

i*: Hi(Y)

If j" Y\ U sequence

Hi(U).

Y denotes the embedding of the complement, there is a long exact 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, we have 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 product 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 M 1, 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 M 3,

Hi+a3 (p{2 Z) (R) Hj+dl (pl Z t)

Hi+j-a (pl Z p-lZ" 1, 23

Assume that the map P3

PZ

p-lZ, 23

-

M x M3

di

dim Mi.

.


541

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), (P2 c P3C’),

(8.3)

where p2 c stands for c x [M3], and so on. This makes sense for disconnected manifolds (possibly variable dimensions), as well. Let M be an oriented manifold, N a topological space, and 7" M---, N a proper continuous map. One can define Z as before, and the convolution makes H,(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 proper morphisms, 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 subsection, we further suppose the following: (1) M 1, M2, and M3 are holomorphic symplectic manifolds of complex dimension dl, d2, and d3, respectively; (2) Z (resp., Z ) is a nonsingular Lagrangian subvariety of M x M2 (resp., M2 x M3); n-lz! denoted by is a submanifold of p-I(z (3) The intersection piZ r3 ) 23 Z o Z is a fiber bundle. (4) P13 The fundamental classes [Z] and [Z ] have degrees dl-+-d: and d: / d3, respectively, 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 the fiber, respectively. Then we have

[Z] [Z’]

(- 1)de[Z o Z’].

Proof. We calculate the convolution using the fact that the fundamental class of a complex submanifold is the Poincar6 dual of the top Chern class of the normal bundle. Let us consider the fiber square p21Z p2._31Z!

j12

P-1223

(8.6)

p{21 z

i12

M1 X M2 X M3

542

HIRAKU NAKAJIMA

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

By the symplectic form, it is the dual of the orthogonal complement of T(p21Z) + T(p-IZ J. Since Z and Z’ are Lagrangian, the orthogonal comple23 -1 ment is Ker dpl3 c3 T(p-2 Z) c T(p23 Z’), which is equal to the relative tangent bundle Tf. Hence we get

’

(P13), (PE[Z] P’23 [Z’])

(P13), (C(dl+d2)/2(N’) P3 [Z’])

(ca(r.tY (Pla),(cd(T

(P13),

ca c(a+a2)/2-a(N)

t p*23

[Z’])

[p{EZ P23-1Z’]j

(--1)de[Z o Z’]. 9. A geometric construction of the algebra U. We assume that the base field is 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/(v 1, w) x 9J/(v2, w) be the subvariety introduced in (7.1). When we have three dimension vectors v 1, v2, and v3, it holds Z(v 1, v2; w) o Z(v2, v3; w) c Z(v 1, 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(v 1, 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 of Z(v v2; w). It has a natural basis {[Z]}, where Z runs over all irreducible components of Z(v 1, v2; w). Note that the degree top may differ for different v and v2’s since the dimensions are changing. Take x [.)v0 9J/0( w). As in 7, we consider the projection r as a map from 9J/(v, w) to v0 9J/0( w). Let 9J/(v,W)x denote the inverse image of x in 9J/(v, w). Note that, for x 0, 9J/(v, w)x is the Lagrangian variety (v, w). 9.1. The PROPOSITION makes the convolution direct sum x, into a Q-algebra, and )vH,(gJ/(v,w)x) is a left

,

Jv

v,

v,


H*(Z(vl’ 2, w))-module. Moreover, the top degree part O a subalgebra, and is a 1, v2;

543

Htop(Z(v 1, 2; W)) w))-stable submodule.

is

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

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

Htop(Z(v 1, 2; w))

[A(V, W)]" V

V

._

--

O2 ntop(Z(, 2; w)) G Htop(Z(vl’ v; w)). V

When 2 1 ek, the Hecke correspondence k(2, W) is an irreducible component of Z(v 1, v2; w). Define Ek I-I ntop(Z( vl, v2; w)) as the formal sum of the sheaf [!13k(V2, w)]:

(v w)]. Let

09"

lf)(2, w) x 9J( 1, w)

factors. Set

Fk

-

2

9J( 1, w) x 9"J(2, w) be the exchange of the

two

r(v2’w) [(O(k(V2, W))], E(-1) 2

where r(v2, w) 1/2(dim 9X(v 1, w) dim 9J/(v2, w)) --tek(w Cv2) 1. Though Ek and Fk are not in Htop(Z(vl,v2;w)), in general, the multiplications Ek’, .Ek, and so on, are linear operators on Htop(Z(v 1, v2; w)), and Ek[A(v, w)], [A(v, w)]Fk, and so on, are elements in Htop(Z(v 1, V2; W)). The following relations are easy to check.

G

@

O

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

[A(v, w)] [A(v’, w)]

w)], ek, w)]Ek,

Ek[A(v, w)]

[A(v

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 that EkN V FNk v O. If we consider H,(Z(v 1, v2; w)) as an H,(Z(v 1, rE; w))-

544

HIRAKU NAKAJIMA

module by the multiplication from the are locally nilpotent.

Proof

left or right,

then the operators Ek and Fk

The assertion follows from the statement if tekv

< 0 or tek(w- Cv + v) < 0, then 93l(v, w)

,

v and E F map any element into homology of quiver varieties with the above dimension condition, provided N is sufficiently large. since

The first case of the statement is obvious, and the second follows from the injectivity of

G

nhjk" Vk

out(h)=k

0

Vin(h) t) l/l,

out(h)=k

which is deduced from the stability condition. 9.iv. Relations. From now on, we study the top degree part of the homology group. For dimension vectors v and w, we associate elements v and Aw of the weight 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

dp(a )=

(eka,t)

[A(v, w)] /f 2 Aw- v, o if there is no such v, Ek(a,), dp(af)

(a,)Fk.

The relations (2.8) and (2.9) are already checked in Proposition 9.2. It is known 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. The followin9 relation holds in

(E Fk, Fk, Ek) [A(v, w)]

Htop(Z(v 1, 2; W))."

6k,k,(hk, Aw v>[A(v, w)].

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

(9.7)

545


LEMMA 9.8. Let v 1, v2, and

V3

be dimension vectors such that V2 V --ek v + e k’. Let (x x2, x3) 9J/(v w) x 9J/(v2, w) x 9J/(v3, w) be a point in the inter-1 section of 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 k I, 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 v 1, 2, and V3 be dimension vectors such that 2 V ek’ v3- ek. Let (x x2, x 3) 9J/(v 1, w) x 9J/(2, w) x 9J(v3, w) be a point in the intersection of P1-Elog(3k,(Vl,W)) and PE-alk(va,w). Let U be as above, and assume (x 1, x3) U. Then the intersection is transverse at (x x2, x3).

,

._

LEMMA 9.10. Take v and 3 and let vE=vl+ek, isomorphism

k’ tO be dimension vectors with 3 1 ek v4=v 1-e k’. Let U be as in Lemma 9.8. There is an

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.

Proofof Proposition 9.6. Let v 1, v2, v3, and v4 be as in Lemma 9.10. We set C

def.

(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(v 1, W) 9J(V 3, W) be the inclusion, and j the inclusion of the complement (i.e., the diagonal or the empty set). There is an exact sequence

Htop(Z( 3;w)\U)

Htop(Z(1,v3; w))

i*

Htop(U z(vl,3; w)).

From the three previous lemmas, we deduce i*(c) -0. Hence c is in the image of 0 if k k’. We assume k k’, and hence v v3 hereafter. Since the support of c is contained in Z(v 1, v3; w)\ U A(v 1, w), it is a multiple of [A(v w)]. Let us assume tek(w Cv 1) > 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,

j,. In particular, c

,

(9.12)

546

HIRAKU NAKAJIMA

holds outside a proper subvariety. Since we already know that c is a multiple of [A(v 1, w)], it is enough to show that the restriction of both sides of (9.10) to the open set Jt(v w) x 9Ttk;0(v3, w) is equal. From now on, we consider everything on the open set and use the same notation for the restriction. In particular, the term FkEk[A(v w)] vanishes, and we need to calculate only the term EkFk[A(v3, w)]. Let Y be the intersection p-1 pcOk,(V2, W). If ([B 1, j], 12 3k( rE, w) [BE, 2, j2 ], [/3, 3, j3]) lies in the Y, we have

,

,

V= in(h)=k ImB+Imi-- V by (9.12). Hence Pl3(Y) is the diagonal A(vl,w). Moreover, Y is isomorphic to k(V2,W) with P13 corresponding to px" k(V2,W) 93(v1,w). Hence Y is the projective 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. We get c

(-1)(a2-dl)/2(p13), (p2[k(V2, w)] p3 [Ogk(V2, W)]) ((d2 dx)/2 + 1)[A(v 1, w)],

where di dim 93/(vi, w). Since we have proved (9.11) 10. Module ) Htop(gJ(v,W)x

(d2 d)/2

.

and Fk. Fix a dimension vector v Recall that 9J/0reg (v0 w) 10.i. Operators is the set of points in 9J/0(v,w) with trivial stabilizer (3.23). Suppose

k

X

fforeg "’*0

(0 W).

We 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 point x 9Jeg(v w) 9Yt0(v, w) under the projection n" 9Jr(v, w) 0(v, w). Let X be an irreducible component of 9Yt(v, w)x. Take r- ek(X) as (7.3), consider the 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 consider the Grassmann bundle of (r-1)-planes in Qk(V-rek, w) * restricted to X’. By Proposition 4.5, we can regard it as a subset of 9Ytk;_l (v- ek, w). We denote its closure by X", which is an irreducible component of 9Jt(v- ek, w)x by the same

,

-


547

reason as above. Then define an operator Ek by

k[X] de=r. [X"] 0

if r

> 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 closure

as S’, we define

if r

> tek(w- Cv) d- 1,

otherwise.

It is clear that -k[X]

[X"] if and only if k[X m] LEMMA 10.1. If ek(X) r > O, then we have

[X].

+_ r[X] +

FkEk[X]

cx,[x’] ek(x’)>r

for some constants cx,. Proof. Since 9J/k; p + 1, the ith term of (10.6) is nonnegative, since w Cv, w Cv 7z n>0 and tekCek 2. For < p, we rewrite the ith term of (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 >o and ff/v v >o, the second term is nonnegative. Thus the case

w) since the restriction of < p cannot occur. Then we have (9910)(d) mreg(v o the 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 Im Bh + Im ik

c

VO)

in(h)=k

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

COROLLARY 10.8. Under the same assumption as in Proposition 10.5, is nonempty if and only if w Cv Zo and Aw v is a weight of the

9J/eg(v, w)

integrable 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

HIRAKU NAKAJIMA

FIGURE 1. An example for 931(v, w) \ of weight assertion.

Aw v. Since 9T/(v, W)o #

v, 71:-lreg( v’,’’’0 w)

if and only if 9J/(v, w) #

,

we have the

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

,

,

Counterexample 10.10. When the assumption in Proposition 10.5 is not satand w Cv ,n>0" We even 9J/(v, w) # isfied, we may have 9J/eg(v, w) 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. Hence 0 or jo 0. In either case, we multiply the scalar t to V 1, V2, and V3, and 0 in the limit. Since (B make jo 0 or jo) has a closed orbit, we must jo 0. In particular, 9J/o does not contain any flDreg(V’, W). have Taking into account the Gv-action, one can show that the image n(931(v, w)) is isomorphic to 2/7z3, that is, the simple singularity of type A2. 9J/(v, w) is its minimal 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 projective morphism r 9Jl(v, w) --. 7(gJl(v, w)) is semismall with respect to the stratification n(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 dimension of the fiber of 7t over (gYt0)(d) is given by

(dim 9J/- dim(gTto) )


Since we have an open stratum get the assertion.

(gJ/0)(d)

551

of dimension dim 99l by Lemma 10.4, we [

l O.iv. Coordinate al#ebras. In general, the homomorphism defined in Theorem 9.4 is not surjective. We shall take a certain quotient of ) ntop(Z(v v2; w)) to which maps surjectively. Let zreg(v 1, v2; w) c Z(v 1, v2; w) be the complement of the closure of the set of all points (x x2) such that r(x ll/reg(v0, w) for any v (cf. 7). The open embedding zreg(vl, v2; w) Z(v 1, 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 have Z(v 1, 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.

zreg(v 1, v2; w)

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

-

O2 Htop(Zreg(vl, v2; w)) OIt0 L(A,

V

a,o)v (R) L(Aw

where v runs over the the set of dimension vectors such that empty.

a,o),

(10.14)

lflreg(0, W)

is non-

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 the linear 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 components contained in the closure of r-l(gj/eg(v w)) zreg(vl,v2;w). Then (10.14) induces a linear map

Htop(Zreg(v 1, 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 the irreducible components of 9X(v w) and that of 9Y(v2, w), where x 931oreg (v0 w). The dimension of the left-hand side is the number of irreducible components of z-a(gxg(v w)) c zreg(v, w), which is smaller than or equal to the above product. Hence the surjeetive homomorphism (10.14) must be an isomorphism.

,

;

Note that we can determine all

v’s

such that 10.8 under the assumption in Proposition 10.5.

,

9xeg(v w)

#

by Corollary

Remark 10.17. It seems natural to conjecture that the basis given by the fundamental classes of irreducible components of Z( 1, v2; w) is related to Lusztig’s canonical basis in [23] of (Jq. (See Remark 10.3(1_).) It is also desirable to have a geometric 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 subvariety introduced 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 of the variety 9X(v, W)x considered in 10.ii. Hence Htop((v, w)) is the integrable highest 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 intersection form, which is defined as follows. First by the Poincar6 duality, we map nia(gJ/(v,w)) to the cohomology with compact supports na(gJ/(v,w)). We compose it with the natural homomorphism to the ordinary cohomology ncmid(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)),


553

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

THEOREM 11.1.

The normalized intersection form

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

on

Htop((, w))

gives the unique nondegenerate contravariant bilinear form on the integrable highest weight 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

-

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

,

L( V V2)

tr- E(V

V2)

L(W, V2)

L(V W)

L(V3, V2)

a-- E(V3, V2)

L(W, V2)

L(V3, W)

L( V V2) i) I,

L(V3, V2)

where we put suffixes 12 and 32 to distinguish endomorphisms. We have sections s 12 and s 32 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(s 12) k(V2, w) x 9Jl(v3, w) and Z(s 32) 9J/(v w) x og(k, (v2, w)). As in the proof of Theorem 5.7, we consider the transposed homomorphisms of Vs 12 and Vs 32 via the symplectic form. Their sum gives a vector bundle endomorphism

t(Vs12

_

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

-

TgJ/(v w)

TgJ(v2, w)

TgJ(v3, w).

32 is zero at It is enough to show that the kernel of t(Vs2) + (Vs) (x x2, x3). a Take representatives (B",ia, fl) of x (a 1,2, 3). Then we have 2 and which 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’c 32)

32

554

HIRAKU NAKAJIMA

is in the kernel. Then there exist

/3Ct12 12 b n2

,

a

1B

atl2 12

B

ff

L(Va, Va) (a

,

1, 2, 3) such that

/3Ct32 32

),3 B3 B3 y3,

bt32 )3 3, a/32 32

_jl 1,

_j3 ,3, (A.1)

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 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 ofj 2. Here and mean the inverse of as follows:

12

32

(2)-1" Im 2 The stability condition for

--o

V

and

-1

(32) -1" Im 32 __, V2.

(B2, 2, j2) implies on Im

Choosing a complementary subspace (Im 2 and an endomorphism (12 k "Vk -* Vk SO that

12 12kk -I-

(12)-1 and (32)

,12

/

12

(3

Im 32.

(A.2)

12)-1- in V, we define a scalar 212 on Im

2,

on Im

2.

Terms on the fight-hand side are equal on the intersection Im 12k olmk32 by 2 32 2 and (A.2), and the sum Im 12 k + Im k spans Vk by the assumption. Hence 2 (2k are well defined. For v k, define (2 by

555


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 t(Vs12) t(vsa2) is injective.

Z

32. Hence

9.9. The set fD(k,(Vl,w))(v3,W) (resp., described as a zero locus Z(s 21) (resp., Z($23)) of a certain section s 21 (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. Take and 23 as in (5.5). representatives (Ba, a, ja) of xa (a- 1, 2, 3). Then we have (The notation will be an obvious modification of that in the proof of the previous lemma, so we do not repeat the definition.) Suppose that A.ii. Proof of Lemma (1, W) X k(V3, W)) can be

21

(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

/i21 Ct21 ])1 B B ])1, 21 b,21 ])1 il, at21 _jl ])1

23 Ct23 ])3 B3 23 b’23 ])3 i3, at23

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

a,23 23

1, 2, 3) such that B 3 ])3,

_j3 ])3, B2 ])2,

(A.3)

_j2])2.

Hence we have

B(]) 21 21])2) (])1 21 21])2)B2 d-/i 21C’23:23, (A.4)

jl (])1 21

21])2) __j3 ])3 :23.

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)

is contained in Im 23. Then we can composite ),3 B3 23 to 12, and have e C ’23

(23) -1

(A.5)

to the restriction of

23l2 (23)- (73 B3 23 B y3 23)l2 (23)-1),3 B3 231122 B2 (23) -1 ),3 23[122" 3

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.

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

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 The above implies l’2

C::

0 on 12.

l2 by definition. Similarly, we have 2 C:: 1,7"2, and hence

,2= [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 to Im ,1, we can define a scalar 221 and a homomorphism (2,1k Vk Vk2, so that CLAIM A.6.

557


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,b 123) is in the image of 1723. Hence t(Vs21) t(Vs23) is injective. We have thus proved Lemma 9.9. A.6. By definition, l?/2 V/2 if # k, and //2 V/2 if Proof2 of Claim l? l?2 we are done if k k’. So we assume k k’ from now on.

Since

Take an edge h with out(h) have 21 ?iln(h) in(h)

"2 Vin(h

k. Then in(h) # k, so

-21

2

in(h))’in(h)

2 Vn(h

# k’.

Hence we

( )-1 in(h) in(h)"

21 23 --in(h) in(h)

3

23

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

--in(h) (in(h))

k 1/ 1-21

-21 2x

k "k)

Jkkk

Suppose

3

--J k k

V

l?k2=21 l?k2 # V.

,121 k k (V0) Im k

(A.7)

.’3 3 23

such that Then there is an element vo and ’k,323 k (V0) Im 23 k Hence we have a direct sum decom-

position

V

1.21

Im 21 k

q-kgk

2 k21 k(V0),

Vk3 Im Ck23 Let us define /k

klim.

l/’

Vk3 by

3 (1) -1

212 k

3 23

(vo)

By (A.7) we have

( )-1 B

23 21 Bh in(h) in(h)

/k,

j j3 gk. This shows [B 1, il, jl] and with the assumption.

[B3, i3, j3]

define the same point, and hence contradict

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,i 1, fl) and 2 (B i2,j2) (resp., (B3, i3, j3) and (B2, i2,j2)) as in (5.5). The intersection

2

Im 32 is B2oinvariant and contains the image of 2. By definition of U, this has dimension v -e’= v4. Hence the restriction of (B2, i2, j 2) to Im 2c Im 32 gives a point in 9J/(v4, w). This correspondence defines a map,

Im

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,(V 1, w)) J/(v3, w) f3 ffJ/(v 1, w) k(V3, W). 3 There exist 4:V4V and 43:V4V as above. Let us take complementary subspaces (Im 41)+/- and (Im 43)+/- and consider (Im 21)/ ) (Im 23)/.

V4

(A.8)

We identify V and V3 with the subspaces V4( (Im 41)+/- and V4( (Im respectively, 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 required inverse.

Suppose S is a B2-invariant subspace contained in the kernel of jE. Its intersection with V4 (Im 41)4 V is Bl-invariant and contained in Kerjl; hence it is zero by the stability of (B 1, il, fl). In particular, S has a nonzero component only on the vertex k’. Similarly S V4 (Im 43)+/- 0. In particular, S 0 if k k ’. So we assume k k from now on. Write a nonzero element in Sk as a b according to the decomposition Since c S VI= (A.8). 0, we must have b 0. Similarly we V4 (Im 41)+/have b 0. Since S is B2-invariant, we have B(v a) + B3(0 b) 0. Define a homomorphism gk: Vk3 by

v

V

-v’/b -a/b

)

Vk3 Vk2

(Im 43)+/-

Vk3

Vk2

(Im 41

+/-


559

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

B

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