The Jordan quiver

:

We deduce that, for eachz2Z.R/, we haveb.z/ ./D .a.z/[ /. Thus, we haveb.z/ .1/D .a.z//. Since andaare surjective, there is an elementzsuch thatb.z/ .1/D1, from which we deduce that .1/is invertible. Furthermore, for eachz; z⁰2Z.R/, we haveb.z/b.z⁰/ .1/D .a.z/[a.z⁰//. Hence, the map is an algebra isomorphism and the diagram below is commutative:

Z R.˛/

a b

k˝kH_G

ƒ

M.ƒ; ˛/;k

Z R^ƒ.˛/

The lemma is proved.

4. The Jordan quiver

We expect that the results above could be generalized to quivers with loops using the generalized quiver Hecke algebras introduced in [17]. In this section, we consider the particular case of the Jordan quiver. In this particular case, the quiver Hecke algebra in [17] is the degenerate affine Hecke algebra of the symmetric group.

From now on, letkbe a commutative domain, letkDkŒ; y₁; : : : ; y_r, and letk⁰ be the fraction field ofk. The ringkisZ-graded with deg.yp/Ddeg./D2. For any k-moduleM, write

M⁰Dk⁰˝_kM:

If M is free Z-graded of finite rank, then let grdim.M / be its graded rank. It is the unique element inNŒt; t¹such that grdim.kŒd /Dt^d and grdim.M ˚N /D grdim.M /Cgrdim.N /.

4.1. The quiver Hecke algebra

For any integern > 0, the quiver Hecke algebra of ranknassociated with the Jordan quiver is the degenerate affine Heckek-algebraR.n/, which is generated by elements 1; : : : ; n1; x1; : : : ; xnwith the defining relations

(a) x_kx_lDx_lx_k,

(b) klDlkifjklj> 1,

(c) _l²D1,

(d) _kx_lx_sk.l/kD.ı_l;kC1ı_l;k/, (e) _kC1kkC1D_kkC1k.

We writeR.0/Dk. For any integerr0, the cyclotomic quiver Hecke algebra of ranknand levelris the quotientR^r.n/ofR.n/by the two-sided ideal generated by the element Qr

pD1.x1yp/. The k-algebras R.n/, R^r.n/ are Z-graded, with deg._k/D0and deg.x_k/D2.

The canonical mapR.n/!R^r.n/yields aZ-gradedk-algebra homomorphism Z.R.n//!Z.R^r.n//. Let Z.R^r.n//^JMbe its image. The canonical inclusionR.n/!R.nC1/ factors to a.R^r.n/; R^r.n//-bilinear mapWR^r.n/!R^r.nC1/. We have the following. yield-ing a.R^r.n/; R^r.n//-bimodule isomorphism

R^r.nC1/DR^r.n/nR^r.n/˚

Parts (a) and (b) are well known. The proof of part (c) is similar to [24, Lemma 5.6.1], where the caseD1is done. Indeed, by part (b) it is enough to show that

By (b), we haveR^r.n/DLr1 kD0

Ln

jD1R^r.n1/.j; n/x_j^k. Sincencommutes with R^r.n 1/ and n.j; n/ D .j; n/.j; n C 1/, we deduce R^r.n/nR^r.n/ D Lr1

kD0

Ln

jD1R^r.n/.j; nC1/x_j^k. Part (d) is proved in [8].

Let us concentrate on (e). First, we prove that Z.R^r.n//^JMis free of finite rank as ak-module. To do that, setk1DkŒy1; : : : ; yrand consider thek1-algebras

R.n/1DR.n/=.1/; R^r.n/1DR^r.n/=.1/:

Then, the assignmentk7!k,xk7!xk˝,yp7!yp˝yields ak-algebra homo-morphism

R^r.n/!R^r.n/1˝_k1kDR^r.n/1Œ: It restricts to an inclusion

Z

R^r.n/JM

Z R^r.n/

Z R^r.n/1

Œ: (39) Now, for anyn-tupleD.1; : : : ; n/of nonnegative integers, let

p.x1; : : : ; xn/DX

x₁¹ x_nⁿ2Z

R^r.n/1

;

where runs over the set of all n-tuples which are obtained fromby permuting its entries. LetPn.r/be the set of all partitionssuch that`./CP

ibi=rc n.

By [8, Theorem 3.2], the canonical map R.n/1 ! R^r.n/1 yields a surjection Z.R.n/₁/!Z.R^r.n/₁/. Further, the elements p.x₁; : : : ; x_n/, where runs over the setPn.r/, form ak1-basis of Z.R^r.n/1/. Therefore, under the inclusion (39), the elementsp.x1; : : : ; xn/˝^jj, where2Pn.r/yield ak-basis of Z.R^r.n//^JM. We deduce that Z.R^r.n//^JMis free of finite rank as ak-module and that

X

n0

grdim

ZR^r.n/JM

q²ⁿDX

n0

X

2P_n.r/

t^2jjq²ⁿ: To compute the right-hand side, note that [8, p. 243] yields a bijection

'Wƒ^C_r .n/!Pn.r/;

whereƒ^C_r .n/is the set ofr-partitionsD.^.1/; : : : ; ^.r//ofn. Further, if'./D, then

jj Drjj .rC1/`./C Xr pD1

p`.^.p//:

We deduce that

X k0, modulo the following definition relations:

(a) ŒD0;lC1; D0;kC1D0,

where the seriesE.z/DP

k0Ekz^kis given by the following formula E.z/DC0CX

There is a unique Liek⁰-algebra anti-involution$ofW such that

$ .Ck/DCk; $ .Dl;k/DDl;k: (41) The Liek⁰-algebraWisZ-graded with deg.Dl;k/D2l and deg.Ck/D0. A rep-resentationV isdiagonalizableif the operatorD0;1=is diagonalizable with integral eigenvalues. ThenV isZ-graded and its degree2ncomponentVnis the eigenspace associated with the eigenvalue n. We will say that a diagonalizable representation is quasifiniteif the degree2ncomponent is finite-dimensional for each n. Finally, we define thecharacter of a quasifinite representationV to be the formal series in NŒŒq; q¹given by

ch.V /.q/DX

n2Z

q²ⁿdim.Vn/:

Given a linear formƒWW0!k⁰and a moduleV, an elementv2V isprimitive of weightƒif W0 acts on vby ƒand $ .W<0/by zero. We callƒ.C0/thelevel ofƒ.

LetM.ƒ/be the Verma module with the lowest weightƒ. It is the diagonalizable module induced from the1-dimensionalW0-module spanned by a primitive vector jƒi. Let V .ƒ/be the top ofM.ƒ/. It is an irreducible diagonalizable module. We callƒthelowest weightandƒ.C0/thelevelofM.ƒ/,V .ƒ/. We calljƒithe highest weight vectors ofM.ƒ/,V .ƒ/.

4.3. The loop operators on the center and the cocenter

For eachr; n2Nwithr ¤0, we setC_n^r DR^r.n/-grproj. WriteC^r DL

nC_n^r. The restriction and induction functors form an adjoint pair.F; E/with

EWR^r.nC1/-grmod!R^r.n/-grmod; N7!N;

F WR^r.n/-grmod!R^r.nC1/-grmod; M 7!R^r.nC1/˝R^r.n/M:

Let"WFE!1andW1!EF be the counit and unit, respectively, of the adjoint pair .F; E/. They are represented by the multiplication map and the canonical map, respectively:

"WR^r.n/˝_Rr.n1/R^r.n/!R^r.n/;

WR^r.n/!R^r.nC1/:

PROPOSITION4.2

(a) The pair.E; F /is adjoint with the counit"OWEF !1and the unitOW1!FE represented by the morphisms "O W R^r.n C 1/ ! R^r.n/, O W R^r.n/ ! R^r.n/˝_Rr.n1/R^r.n/such that".z/O Dp_r1.z/and.1/O D.x_nC1^r /.

(b) The k-algebra R^r.n/ is a symmetric algebra. The symmetrizing form tr;nW R^r.n/!kis the uniquek-linear map sending the elementx₁^r1 xn^rnwto 1 if r1D DrnDr1andwD1and to0otherwise. We havetr;nD O"ı ı O"

(ntimes).

Proof

See [24, Lemma 5.7.2] for (a) and [9, Theorem A.2] for (b).

We have tr.C^r=Z/DL

ntr.R^r.n//. We equip tr.C^r=Z/ with the Z²-grading such that tr.R^r.n//has the weight2nand the order given by the degree of elements of R^r.n/. For eachk2N, we definek-linear operatorsx_k^˙on tr.C^r=Z/DL

ntr.R^r.n//

of weights˙2and order2ksuch that the maps x_k^CWtr

R^r.n/

!tr

R^r.n1/

; x_k Wtr R^r.n/

!tr

R^r.nC1/

are given, for eachf 2EndC^r.a/anda2C^r, by x_k^C

tr.f / Dtr

x^k.a/ı E.f /

; x_k tr.f /

Dtr

x^k.a/ı F .f /

: Herex2End.F /is represented by the right multiplication byxnC1 onR^r.nC1/, andx2End.E/is the left-transposed endomorphism.

THEOREM4.3

The assignmentCk7!pk.y1; : : : ; yr/,D1;k7!x_k^˙defines a representation of level rofW ontr.C^r=Z/⁰.

Proof

First, we check that the operatorsx₀,x₁,x₂, x₃ satisfy relations (c) and (e) for D10,D11,D12,D13in the definition ofW. For eachk2Nandf 2R^r.n/, we have x_k.tr.f //Dtr.f x^k_nC1/, where the elementf on the right-hand side is identified with its image by the map

WR^r.n/!R^r.nC1/:

Thus, using the relations_nC1x_nC2nC1Dx_nC1C_nC1and_nC1² D1, we get Œx₁; x₀

tr.f /

Dtr.f xnC2f xnC1/ Dtr.f x_nC2nC1² f x_nC1/ Dtr.f nC1xnC2nC1f xnC1/ Dtr.f nC1/:

Similarly, using the relations

_nC1x_nC1x²_nC2nC1Dx_nC1² x_nC2Cx_nC1x_nC2nC1;

nC1x³_nC2nC1Dx_nC1³ C.x²_nC2CxnC1xnC2Cx_nC1² /nC1; we deduce that

Œx₂; x₁ tr.f /

Dtr.f xnC1xnC2nC1/;

Œx₃; x₀ tr.f /

Dtr

f .x_nC2² Cx_nC1x_nC2Cx_nC1² /_nC1 : Relation (e) follows from

x₀; Œx₀; x₁ tr.f /

Dtr

f .nC2nC1/ Dtr

f .nC2_nC1² nC1_nC2² / Dtr

f .nC1nC2nC1nC2nC1nC2/ D0:

To prove relation (c), we introduce the element'_lD.x_lx_lC1/_lC. We have '_l²D².xlxlC1/²; 'lxkDxs_l.k/'l; 'llD l'l: And we deduce that

3Œx₂; x₁Œx₃; x₀C²Œx₁; x₀ tr.f / Dtr

f .3xnC1xnC2x_nC2² xnC1xnC2x_nC1^{2 2}/nC1 Dtr

f

.xnC1xnC2/²² nC1 D tr.f '_nC1² nC1/

D tr.f '_nC1nC1'_nC1/ Dtr.f '_nC1² nC1/ D0:

We prove that the operatorsx₀^C,x^C₁,x^C₂,x₃^Csatisfy relations (d) and (e) forD1;0, D1;1,D1;2,D1;3in a similar way.

Next, we prove relations (a), (b), and (f). To do so, for eachl0, consider the elementp_l.x1; : : : ; xn/2Z.R^r.n//given byp_l.x1; : : : ; xn/DPn

iD1x_i^l ifn > 0and 0ifnD0. Letx_lC1⁰ be thek-linear operator on tr.C^r=Z/given by

x_lC1⁰ tr.f /

Dtr

fpl.x1; : : : ; xn/

; 8f 2R^r.n/: (42) Then, under the assignmentD0;kC17!x_kC1⁰ , the defining relation (a) ofW is obvi-ously satisfied. Let us concentrate on (b). We have

Œx_lC1⁰ ; x_k tr.f /

Dtr f

x_nC1^k pl.x1; : : : ; xnC1/x^k_nC1pl.x1; : : : ; xn/ Dx_lCk

tr.f / :

The relationŒx_lC1⁰ ; x_k^CDx_lCk^C is proved in a similar way.

Finally, let us prove relation (f) in the definition of the Lie algebraW. Let O

"WR^r.nC1/!R^r.n/; OWR^r.n/!R^r.n/˝_Rr.n1/R^r.n/

be as above. For eachk2Nwe consider the following elements in Z.R^r.n//:

B_C;n^k D O".x_nC1^r1Ck/;

B_;n^k D 8ˆ ˆ<

ˆˆ :

²_xr1Ck

n ..1//O ifkrC1;

p_rk.x^r_nC1/ if1kr;

1 ifkD0:

We may abbreviateB_˙^k DB_˙;n^k . Consider the formal series B˙.z/DP

Next, a similar computation as in the proof of parts (a) and (c) of LemmaB.1 yields

and the equality in part (b).

The proof of part (c) is similar to PropositionB.5(b), so we briefly indicate the key steps. First, the isomorphism in Proposition4.1(c) represents an isomorphism of functorsWEF 1n!GwithGDFE1n˚1^˚r_n . By Lemma2.3we have

We have

and by (43) its trace is equal to x_lx^C_k C To prove (d), note that, using (b), we get

zd=dzlogB.z/D X

Finally, we concentrate on (e). We have _nx_n^k_nDx^k_nC1 X

Thus, we have

B_C;n1^krC1DB_C;n^krC12

k2X

aD0

.aC1/x_n^aB_C;n^k2arC1: This yields

BC;n1.z/D

1²z².1zxn/²

BC;n.z/:

Hence by (b) we get

B;n.z/D

1²z².1zxn/²

B;n1.z/

forn1. By induction it remains to computeB_;0.z/. Since x₁^rD

Xr aD1

.1/^aea.y1; : : : ; yr/x₁^ra; we have

.x^r₁/D0; B_;0^a D.1/^aea.y1; : : : ; yr/ 8a2Œ1; r; B_;0^a D0 8a > r:

Therefore, we haveB;0.z/DQr

aD1.1zya/.

We can now finish the proof of relation (f) ofW. According to Lemma4.4, the formal seriesE.z/DP

l0Elz^lis given by E.z/Drzd=dzlogY^r

pD1

.1zyp/ Yn kD1

1z.xk/

1z.xkC/

.1zxk/² : Comparing this with formula (42) and the identity

X

k1

.za/^kD z.d=dz/log.1za/; (45)

we deduce that E.z/DrX

k1

Xn pD1

2x_p^k.xp/^k.xpC/^k

z^kCX

k1

Xr pD1

y_p^kz^k: This implies that

E.z/DrCX

k1

.D_0;kC1/C.D_0;kC1/Cp_k.y1; : : : ; yr/ z^k;

(46) t.D_0;kC1/D

Xk pD1

k

p D_0;kpC1t^p1: This finishes the proof of the theorem.

Set Z.C^r=Z/ D L

nZ.R^r.n//. The symmetrizing form t_r D L

nt_r;n on L

nR^r.n/⁰yields ak⁰-bilinear form

Z.C^r=Z/⁰tr.C^r=Z/⁰!k⁰; .a; b/7!tr.ab/;

which induces an isomorphism Z.C^r=Z/⁰'Hom_k⁰.tr.C^r=Z/⁰;k⁰/. Taking the trans-pose with respect to this bilinear form and twisting the action by the anti-involution

$in (41), we get a representation ofW on Z.C^r=Z/⁰of levelr. Letjri 2Z.C^r=Z/⁰ denote the unit of Z.R^r.0//⁰Dk⁰. We define a weight ƒr of level r ofW by the formula

ƒ_r.C_k/Dp_k.y₁; : : : ; y_r/; ƒ_r.D_0;kC1/D0; 8k0: (47)

PROPOSITION4.5 The following hold:

(a) jriis a primitive vector ofZ.C^r=Z/⁰of weightƒr; (b) Z.C^r=Z/⁰is quasifinite of characterQ

j1.1q^2j/^r. Proof

Part (a) follows from formula (42), which implies thatx_lC1⁰ .jri/D0for all l0.

Part (b) follows from Proposition4.1(d).

4.4. The cohomology ring of the moduli space of framed instantons

Let M.r; n/be the moduli space of framed rankr torsion-free sheaves onP² with fixed second Chern classn. SetM.r/DF

nM.r; n/. First, let us review a few basic facts onM.r/(see [34, Section 3] for more details).

The group GL.r/GL.2/ acts on M.r/ in the obvious way : GL.r/ acts by changing the framing and GL.2/via the tautological action onP² which preserves the line at infinity. LetT GL.r/,AGL.2/be the maximal tori, and letCA be thehyperbolictorus¹diag.t; t¹/It2Cº. SetGDTCandGADT A.

We identify the Z-graded k-algebra H_G.;k/ with kin the obvious way. Let hDH_G

A.;k/, and writeh⁰for the fraction field ofh. From now on, letkbe a field of characteristic0.

The GA-variety M.r; n/ is equivariantly formal and smooth of dimension dr;n D 2rn. Thus H_G.M.r/;k/ is a free Z-graded k-module isomorphic to H.M.r/;k/˝k. TheG_A-action yields an˛-partition ofM.r/into affine spaces in the sense of De Concini, Lusztig, and Procesi [11]. We deduce that

X

d;n0

dimH^d

M.r; n/;k q²ⁿt^d DX

n0

grdimH_G

M.r; n/;k q²ⁿ

D Yr pD1

Y1 iD1

.1q²ⁱt^2.riCp1r//¹: (48) In particular, note that the odd cohomology ofM.r/vanishes, and hence thek-algebra H_G.M.r/;k/is commutative. Letjribe the unit ofH_G.M.r; 0/;k/'k. First, we prove the following.

PROPOSITION4.6

(a) There is a representation of the Lie k⁰-algebra W on H_G.M.r/;k/⁰ which is isomorphic to V .ƒr/. This representation is quasifinite of character Q

j1.1q^2j/^r.

(b) There is a unique isomorphism WZ.C^r=Z/⁰!H_G.M.r/;k/⁰of representa-tions ofW⁰which takes the elementjritojri.

(c) For eachn2N, the element .1/2H_G.M.r; n/;k/⁰ is invertible, and the map⁰WZ.R^r.n//⁰!H_G.M.r; n/;k/⁰given by⁰./D .1/¹[ ./is a k⁰-algebra isomorphism.

Proof

For eachn; k2N, we consider the locus

B.r; nCk; n/M.r; nCk/C²M.r; n/

of triples.E; x;F/such thatEF andF=Eis a lengthksheaf supported atx. For each2H_G.B.r; nCk; n/;k/the correspondenceB.r; nCk; n/defines two maps

‚C. /WH_G

M.r; n/;k

!H_G

M.r; nCk/;k

;

‚. /WH_G

M.r; nCk/;k

!H_G

M.r; n/;k0

:

The map‚uses localized equivariant cohomology, because the projectionB.r; nC k; n/!M.r; n/is not proper.

Let_nCk;n,nbe thetautological bundlesonM.r; nCk/M.r; n/,M.r; n/.

Writecifor theith equivariant Chern class. The obvious map

H_G

M.r; nCk/M.r; n/;k

H_G.C²;k/!H_G

B.r; nCk; n/;k is denoted by.a; b/7!a˝b.

Now, we define the action of C_k, D_1;k, D_1;k, D_0;kC1 on an element of H_G.M.r; n/;k/⁰by

D0;1Dn;

CkDpk.y1; : : : ; yr/;

D1;kD ²‚C

c1.nC1;n/^k˝1

; D1;kD.1/^r1‚

c1.nC1;n/^k˝1

; X

k0

D_0;kC2z^kD .d=dz/log

1CX

k1

c_k.n/.z/^k [ :

(49)

The operators D1;k, D1;k, D0;kC1 above are equal to the operators ^kD1;k, ^kD1;k,^kC1D0;kC1 in [37, (3.17)], respectively. The reason for this normalization by powers ofis to give the termin relations (b), (c), and (d) ofW in Section4.2, which does not appear in the corresponding relations in [2].

By [37, Corollary 3.3] and [2], the formulas (49) yield a representation ofh⁰˝k⁰

Won

H_GA

M.r/;k0

Dh⁰˝hH_GA

M.r/;k :

Note that [37] uses equivariant homology rather than equivariant cohomology, but sinceM.r/is smooth, its equivariant homology and cohomology are isomorphic by Poincaré duality. We must check that the formulas (49) give indeed a representation ofWonH_G.M.r/;k/⁰.

The representation ofh⁰˝_k⁰W in [37] depends on parameters y1; : : : ; yr; x; y which are generators of the field extensionh⁰ ofk. Note thaty_p is denoted by the symbolepin [37]. The representation ofWwe consider here is a specialization along the hyperplanexD yDof some integral form of the representation in [37]. See [7] for more details on this integral form. We must check that the representation in [37] specializes effectively.

To prove this, note that the main results of [37] are obtained by explicit computa-tions in theh⁰-basis ofH_G

A.M.r/;k/⁰formed by the fundamental classes of the fixed points ofM.r/under the action of the torusGA. For these computations it is essen-tial that the fixed points are isolated. Now, it is well known, and easy to prove, that the fixed-point setsM.r/^G andM.r/^GA are the same. Indeed, one can easily check that the explicit formulas for the representation ofh⁰˝_k⁰Win [37, Corollary 3.3] in the basis ofH_G

A.M.r/;k/⁰ have no poles along the hyperplanexDy. This follows from formulas (3.17) and (D.1)–(D.3) in [37].

Note, however, that the seriesE.z/ in (40) differs from the corresponding one in [37, (1.70)]. We must check that these formulas are compatible. To do that, let us review quickly the proof in [37, pp. 326–327]. Our setting differs from [37] because there we assumed that both parametersx,yare generic, while here we havexDD ywithgeneric.

We have proved that the formulas in (49) define a representation of W on H_G.M.r/;k/⁰. Now, we must check that this representation is irreducible and is iso-morphic toV .ƒr/.

The irreducibility follows from the main result of [37]. More precisely, it is proved in [37, Theorem 8.33] that the representation ofW onH_G

A.M.r/;k/⁰gives rise to a representation of the W-algebra of the affine Kac–Moody algebrabglr on H_G

A.M.r/;k/⁰ (see, e.g., [1] and [15] for some background onW .bgl_r/). It is also proved there that the W .bglr/-module H_G

A.M.r/;k/⁰ is isomorphic to the Verma module with highest weight and level given, respectively, by

a=x.1Cy=x/ and y=xr: (51)

Here we have setaD.y1; y2; : : : ; yr/andD.0;1; : : : ; 1r/(see also [7]). Then, the irreducibility ofH_G

A.M.r/;k/⁰ as aW .bglr/-module is well known, because a Verma module with a generic highest weight is irreducible. To prove that it is also irreducible as aW-module, use [37, Theorem 8.22] as in [37, Corollary 8.29].

The same argument proves thatH_G.M.r/;k/⁰is irreducible as aW-module.

Next, we must identify the representation ofW onH_G.M.r/;k/⁰ withV .ƒr/.

To do that, it is enough to prove that the elementjriofH_G.M.r/;k/⁰is primitive of weightƒr. The equality [37, (3.9)] yields

c_k.n/[ jri D0; 8k1:

Thus, from (49) we deduce thatD0;kC1.jri/D0for eachk0.

Finally, we must check the character formula in (a). It is well known, and follows easily by counting the (isolated) fixed points inM.r; n/. This finishes the proof of (a).

Now, let us concentrate on (b). Since the elementjriof Z.C^r=Z/⁰is primitive of weightƒrand sinceM.ƒr/has a simple top isomorphic toV .ƒr/, there is a unique surjectiveW-module homomorphism from the submoduleM Z.C^r=Z/⁰generated byjritoH_G.M.r/;k/⁰such thatjri 7! jri. SinceH_G.M.r/;k/⁰and Z.C^r=Z/⁰have the same character, we deduce that

Z.C^r=Z/⁰DM DH_G

M.r/;k0

: Let be the uniqueW⁰-module isomorphism

WZ.C^r=Z/⁰!H_G

M.r/;k0

; jri 7! jri:

Finally, let us prove part (c). By restriction, the map yields ak⁰-linear isomor-phism

WZ R^r.n/0

!H_G

M.r; n/;k0

for eachn2N. We must prove that .1/is invertible inH_G.M.r; n/;k/⁰and the map ⁰WZ

R^r.n/0

!H_G

M.r; n/;k0

; ⁰./D .1/¹[ ./

is ak⁰-algebra isomorphism. To do so, we consider the diagram Z

R.n/0 b⁰ a⁰

H_G

M.r; n/;k0

Z R^r.n/0

The mapb⁰is thek⁰-algebra homomorphism induced by the canonical mapR.n/! R^r.n/. It is surjective by Proposition4.1. The mapa⁰is thek⁰-algebra homomorphism given by

a⁰

ei.x1; : : : ; xn/

Dci.n/; 8i2Œ1; n:

Note that, by the definition of the representation ofW onH_G.M.r; n/;k/⁰, formula (49) yields

¹D_0;kC1Da⁰

p_k.x1; : : : ; xn/

[ onH_G

M.r; n/;k0

: (52)

Next, by the definition of the representation ofW on Z.C^r=Z/⁰, the formula (42) in the proof of Theorem4.3yields

¹D0;kC1Db⁰

pk.x1; : : : ; xn/

on Z R^r.n/0

: (53)

From (52) and (53), since isW-linear, we deduce that b⁰

pk.x1; : : : ; xn/ Da⁰

pk.x1; : : : ; xn/

[ ./: (54) Now, an easy induction using (54) yields

b⁰.z/Da⁰.z/[ .1/; 8z2Z R.n/0

: (55)

We also deduce that

.zz⁰/[ .1/D .z/[ .z⁰/; 8z; z⁰2Z R^r.n/0

: (56)

Now, since b⁰ and are surjective, equality (55) implies that the element .1/ is invertible in the (commutative)k⁰-algebraH_G.M.r; n/;k/⁰. Thus, the map⁰ above is well defined and it is ak⁰-algebra homomorphism by (56). It is clearly bijective because it is injective and both sides are finite-dimensional of the same dimension overk⁰. Further, we have a commutative diagram:

Z R.n/0

b⁰ a⁰

H_G

M.r; n/;k0

Z R^r.n/0 0

(57)

Part (c) of the proposition is proved.

We can now prove the following, which is one of the main results of this paper.

THEOREM4.7

The canonical mapZ.R.n//!H_G.M.r; n/;k/is a surjectivek-algebra homomor-phism. It factors to ak-algebra isomorphismZ.R^r.n//^JM!H_G.M.r; n/;k/.

Proof

We define the mapsaWZ.R.n//!H_G.M.r; n/;k/andbWZ.R.n//!Z.R^r.n//as in the triangle (57) above. Thus, we havea⁰Dk⁰˝aandb⁰Dk⁰˝b. We claim that there is ak-linear mapmaking the following triangle commute:

Z

To prove this, it is enough to check that Ker.b/Ker.a/. Since the triangle (57) commutes, we have Ker.b⁰/Ker.a⁰/. Thus, since Z.R.n//is free as ak-module,

Finally, since Z.R.n//is free as ak-module, the mapx7!1˝x yields an isomor-phism

The claim is proved. Note that, since the mapb⁰ is surjective and the triangles (57) and (58) commute, we have⁰Dk⁰˝. Thus, since⁰is injective, we deduce that is also injective. Now, recall that Proposition4.1and (48) yield

X

The theorem above can be reformulated in the following way. Set k1 D kŒy₁; : : : ; y_rand consider thek1-algebras

R.n/1DR.n/=.1/; R^r.n/1DR^r.n/=.1/:

Recall the inclusion Z.R^r.n//^JMZ.R^r.n/1/Œin (39). By [8], the canonical map R.n/1!R^r.n/1 yields a surjection Z.R.n/1/!Z.R^r.n/1/. Since Z.R.n/1/isN -graded, this yields an increasing separated and exhaustive N-filtration F of Z.R^r.n/1/. Let Rees.Z.R^r.n/1/) be the corresponding Rees algebra; that is,

By construction, the map (39) identifies the k-algebras Z.R^r.n//^JM and Rees.Z.R^r.n/1//. We deduce the following

COROLLARY4.8

There is ak-algebra isomorphismRees.Z.R^r.n/1//'H_G.M.r; n/;k/.

Remark 4.9

In the particular caserD1, the corollary was already known and follows from [38].

Appendices

Appendix A. The symmetrizing form

Fix a dominant weightƒ2PC. Let˛2QCandi; j 2I. SetDƒ˛andiD h˛_i^_; i.

A.1. Bubbles

Assume that˛has the heightn.

Definition A.1

For eachk2NthebubbleB_˙i;^k is the element ofR^ƒ.˛/given by the following:

Ifi0, we set B_Ci;^k D

´"O⁰_i;.x_nC1^i1Cke.˛; i // ifk iC1;

1 ifkD_iD0;

B_i;^k D 8ˆ ˆ<

ˆˆ :

x_n^i1Ck.O⁰_i;.1// ifkiC1;

p_ik.x_nC1ⁱ e.˛; i // if1ki;

1 ifkD0:

Ifi0, we set

B_Ci;^k D 8ˆ ˆ<

ˆˆ : O

"⁰_i;.x_nC1^i1Cke.˛; i // ifk iC1;

x_nⁱ.Q_ik/ if1k i;

1 ifkD0;

B_i;^k D 8<

:

x_n^i1Ck.O⁰_i;.1// ifkiC1;

1 ifkDiD0:

Note thatB_˙i;⁰ D1in all cases. We set by conventionB_˙;^k D0ifk < 0.

LEMMAA.2

The elementsB_˙i;^k are homogenous central elements inR^ƒ.˛/of degree2k.

Proof

The central and homogenous property follows from the fact that"⁰_i,pk,⁰_iare homoge-nous R^ƒ.˛/-bilinear morphisms and that the element xnC1 centralizes R^ƒ.˛/ in R^ƒ.˛C˛i/. The degree is given by an explicit computation.

Remark A.3

In Khovanov and Lauda’s diagrammatic categorification, the elementB_Ci;^k corre-sponds to a clockwisebubblewith adotof multiplicityi1Ck, andB_i;^k corre-sponds to a clockwise bubble with a dot of multiplicityi1Ck.

A.2. A useful lemma

Assume that˛has the heightn1. Let⁰D˛j and⁰_iD h˛_i^_; ⁰i Diaij. Consider the morphisms

X^i;j;WE_i⁰F_i⁰1

E_i⁰⁰_jF_i⁰

E_i⁰E_j⁰F_j⁰F_i⁰1 E_j⁰E_i⁰F_i⁰F_j⁰1 E_j⁰"O⁰

i;0F_j⁰

E_j⁰F_j⁰1;

Ii;j;WE_i⁰F_i⁰1

O

"⁰_i;

1

_j⁰

E_j⁰F_j⁰1:

The morphismXi;j;is represented by the composition

e.˛; i /R^ƒ.˛C˛i/e.˛; i /

j

e.˛; ij /R^ƒ.˛C˛iC˛j/e.˛; ij /

n./n

e.˛; j i /R^ƒ.˛C˛iC˛j/e.˛; j i /

O

"⁰_i;0

e.˛; j /R^ƒ.˛C˛j/e.˛; j /;

andIi;j;is represented by

e.˛; i /R^ƒ.˛C˛i/e.˛; i /

O

"⁰_i;

R^ƒ.˛/

_j

e.˛; i /R^ƒ.˛C˛i/e.˛; i /:

In other words, givena2e.˛; i /R^ƒ.˛C˛i/e.˛; i /, we have X^i;j;.a/D O"⁰_i;0

nj.a/n

; I^i;j;.a/Dj"O⁰_i;.a/:

Note that, sincej WR^ƒ.ˇ/!R^ƒ.ˇC˛j/is the canonical embedding for any ˇ 2QC, we write j.b/Dbe.ˇ; j / or simply j.b/Db for any b2R^ƒ.ˇ/.

Note also that sinceB_˙i;^k 2Z.R^ƒ.˛//, it can be viewed as an element in End.1/.

Thus x^rB_˙i;^s x^t defines an endomorphism of E_i⁰1F_i⁰ D E_i⁰F_i⁰ for each r,s,t2N.

LEMMAA.4 The following hold:

(a) Ifi¤j, thenXi;j;Dc_i;j;aij;0Ii;j;. (b) Xi;i;D Ii;i;CP

g₁Cg₂Cg₃D⁰_i1x^g1B_Ci;^g2 0x^g3.

Note that if⁰_i0the sum overg1; g2; g3is empty, and henceX^i;i;D I^i;i;. Proof

Let us prove part (a). First, assumei> 0. Then aD_n1

.a/

C

_i1

X

kD0

p_k.a/x_n^k

with.a/2R^ƒ.˛/e.˛˛i; i /˝_Rƒ.˛˛_i/e.˛˛i; i /R^ƒ.˛/andp_k.a/2R^ƒ.˛/.

We have

_nj.a/_nD_{nn1ne.˛˛i;ij i /} .a/

C

_i1

X

kD0

p_k.a/_nx_n^ke.˛; ij /_n: The relations (d), (e), and (f) in Definition3.1yield

_{nn1ne.˛˛i;ij i /} .a/

D

._n1n_n1CQi;j .xn1;xn/Qi;j .xnC1;xn/

xn1xnC1 /e.˛˛_i;ij i /

.a/

; nx_n^ke.˛; ij /nDx_nC1^k _n²e.˛; j i /Dx^k_nC1Qj;i.xn; xnC1/e.˛; j i /:

Since ⁰_i Diaij i> 0, we have Xi;j;.a/Dp⁰

i1.nj.a/n/, the coeffi-cient ofx_nC1^iaij1. Since the degree ofxnC1in^Qi;j.xn1_x^{;xn/Qi;j.xnC1;xn/}

n1x_nC1 is at most aij 1, which is less than ⁰_i 1 D i aij 1, and since p⁰

i1

._n1nn1..a///D0, we deduce thatp⁰

i1kills_{nn1ne.˛˛i;ij i /}..a//.

Next, the degree ofxnC1 inx_nC1^k Qj;i.xn; xnC1/Dx_nC1^k Qi;j.xnC1; xn/is less than or equal tokaij with the coefficient ofx_nC1^kaij given byci;j;a_ij;0; therefore,

p⁰ Denote the right-hand side by b. Concretely write aQ DP

raQ_r⁰ ˝ Qa_r⁰⁰ with aQ_r⁰ 2

Qi;j.xn1; xn/Qi;j.xnC1; xn/

SinceQi iD0, the relation (f) in Definition3.1yields _{nn1ne.˛˛i;i}3/

nx_n^kne.˛; i²/Dx^k_nC1n²e.˛; i²/x_n^kx_nC1^k same computation as in the previous lemma yields

_ni.a/_nD

O A computation similar to (61) yields

n1x_n^kn1e.˛˛i; i²/ in the second equality. Finally, recall from (7) that there are elementsQ_`2R^ƒ.˛C

˛_i/e.˛; i /˝_Rƒ.˛/e.˛; i /R^ƒ.˛C˛_i/for0` <sup>0</sup><sub>i</sub>1such that<sub>n</sub>.Q<sub>`</sub>/D0and

In the second equality, we substitutedg3D ⁰_ik. In the third equality, we used the definition ofB_Ci;^g3 0 for0g3 0

iin DefinitionA.1.

Remark A.5

Assume that theQ-cyclicitycondition in [10, (2.4), (2.5)] holds forV^ƒ; that is, the endomorphismsxi2End.E_i⁰/,ij 2End.E_i⁰E_j⁰/are such that

x_i^_D^_xi; _ij^_D^_ij:

See the notation in Section 2.1.4. Then, under the adjunction isomorphism Hom.E_i⁰F_i⁰; E_j⁰F_j⁰/DHom.FjEi; FjEi/, part (a) of the previous lemma gives the second equality in the mixed relation [10, (2.16)]. ForiDj, under the same adjunc-tion, part (b) gives the second equalities in [10, (2.22), (2.24), (2.26)]. The other rela-tions in [10, Section 2.6.3] can be checked similarly. Since the computarela-tions are quite lengthy and will not be needed, we omit the details here. Finally, thefake bubbles relation [10, (2.20)] is proved in LemmaB.2(a) below. Therefore, assuming the Q-cyclicity condition, we have proved thatV^ƒcarries a representation of Khovanov and Lauda’s2-Kac–Moody algebra. TheQ-cyclicity condition can probably be proved by

See the notation in Section 2.1.4. Then, under the adjunction isomorphism Hom.E_i⁰F_i⁰; E_j⁰F_j⁰/DHom.FjEi; FjEi/, part (a) of the previous lemma gives the second equality in the mixed relation [10, (2.16)]. ForiDj, under the same adjunc-tion, part (b) gives the second equalities in [10, (2.22), (2.24), (2.26)]. The other rela-tions in [10, Section 2.6.3] can be checked similarly. Since the computarela-tions are quite lengthy and will not be needed, we omit the details here. Finally, thefake bubbles relation [10, (2.20)] is proved in LemmaB.2(a) below. Therefore, assuming the Q-cyclicity condition, we have proved thatV^ƒcarries a representation of Khovanov and Lauda’s2-Kac–Moody algebra. TheQ-cyclicity condition can probably be proved by

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