ON THE CENTER OF QUIVER HECKE ALGEBRAS P. SHAN, M. VARAGNOLO, and E. VASSEROT

P. SHAN, M. VARAGNOLO, and E. VASSEROT

Abstract

We compute the equivariant cohomology ring of the moduli space of framed instan- tons over the affine plane. It is a Rees algebra associated with the center of cyclotomic degenerate affine Hecke algebras of typeA. We also give some related results on the center of quiver Hecke algebras and the cohomology of quiver varieties.

Contents

1. Introduction . . . . 1005

2. Generalities . . . . 1008

3. The center of quiver Hecke algebras . . . . 1015

4. The Jordan quiver . . . . 1052

Appendices . . . . 1069

Appendix A. The symmetrizing form . . . . 1069

Appendix B. Relations . . . . 1080

Index of notations . . . . 1096

References . . . . 1098 1. Introduction

A few years ago, the cohomology ring of the Hilbert scheme of points on C² was computed in [27] and [38], motivated by some conjectures of Chen and Ruan on orbifold cohomology rings, which were later proved in [14]. One of the main moti- vations of the present work is to compute a larger class of cohomology ring of quiver varieties. More precisely, for each pair of positive integersr; n, one can consider the moduli spaceM.r; n/of framed instantons with second Chern class nonP². This is a smooth quasiprojective variety (overC) which can be viewed as a quiver variety attached to the Jordan quiver. An.rC2/-dimension torus acts naturally onM.r; n/

and it contains an.rC1/-dimensional subtorus whose action preserves the symplec- tic form. One of our goals is to compute the equivariant cohomology ring ofM.r; n/

DUKE MATHEMATICAL JOURNAL

Vol. 166, No. 6, © 2017 DOI10.1215/00127094-3792705 Received 1 November 2014. Revision received 13 July 2016.

First published online 10 February 2017.

2010Mathematics Subject Classification. Primary 06B15; Secondary 20C08.

1005

with respect to this subtorus. SinceM.r; n/ is equivariantly formal, one can easily recover the usual cohomology ring from the equivariant one. ForrD1, that is, the case of the Hilbert scheme, it can be easily deduced from [38] that the equivariant cohomology ring we are interested in is the Rees algebra associated with the center of the group algebra of the symmetric group with respect to the age filtration. Here we obtain a similar description for arbitraryr with the group algebra of the symmet- ric group replaced by a levelr cyclotomic quotient of the degenerate affine Hecke algebra of typeA.

This result was conjectured soon after [38] was written. However, its proof requires two new ingredients which were introduced only very recently. An impor- tant tool used in [38] is Nakajima’s action of a Heisenberg algebra on the cohomol- ogy spaces of the Hilbert scheme. A similar action on the cohomology ofM.r/D F

n0M.r; n/was introduced by Baranovsky a few years ago, but it is insufficient to compute the cohomology ring. What we need in fact is the action of (a degenerate version of) a new algebraWwhich is much bigger than the Heisenberg algebra. This action was introduced recently in [37] to give a proof of the AGT conjecture of Alday, Gaiotto, and Tachikawa for pureN D2gauge theories for the groupU.n/. Here, we define a similar action ofWon the center (or the concenter) of cyclotomic quotients of degenerate affine Hecke algebras. Then we compare it with the representation of Won the cohomology ofM.r/to obtain the desired isomorphism. To do this, we use categorical representation theory.

Categorical representations of Kac–Moody algebras have captured a lot of inter- est recently since the work of Khovanov and Lauda [21]–[23] and Rouquier [36]. It was observed recently by Beliakova, Habiro, Guliyev, Lauda, and Zivkovic [3], [4]

that a categorical representation gives rise to some interesting structures on the center (or cocenter) of the underlying categories and not only on their Grothendieck group.

More precisely, Khovanov and Lauda define a 2-Kac–Moody algebra in [23] which is a 2-category satisfying certain axioms. Their idea is that the trace of this 2-category has a natural structure of an associative algebraLgwhich should be some kind of loop algebra over the underlying Kac–Moody algebrag. Thesl2-case was worked out in [4], and thesln-case in [3]. Naturally, the center (or cocenter) of2-representations of the2-Kac–Moody algebra gives rise to representations ofLg.

In this paper, we first use a similar idea to investigate the center and cocenter of cyclotomic quiver Hecke algebras associated with a Kac–Moody algebragof arbi- trary type. The category of projective modules over these algebras provides mini- mal categorical representations ofg, in the sense of Rouquier [36]. We compute the representation ofLgon the center (or cocenter) of these minimal categorical repre- sentations. Whengis symmetric of finite type, we identify theseLg-modules with

thelocal and global Weyl modules, which can be realized in the (equivariant) Borel–

Moore homology spaces of quiver varieties by [33].

Then, in order to compute the cohomology ring ofM.r; n/, we consider another situation wheregis replaced by a Heisenberg algebra. On the categorical level this corresponds to theHeisenberg categorificationswhich have also been studied recently.

But once again, instead of considering a2-Heisenberg algebra, we focus on the par- ticular categorical representation given by the module category of degenerate affine Hecke algebras of typeA. The analogue of Lgin this case is the algebra W men- tioned above. Probably one can generalize both situations (the Kac–Moody one and the Heisenberg one) using [17], where categorifications of some generalized (Borcherds–)Kac–Moody algebras are considered. Here, we do not go further in this direction.

Another motivation for this work is to check if the center of the module cate- gory of cyclotomic quiver Hecke algebras is positively graded with a1-dimensional degree0component. These two conditions are difficult to check for minimal categor- ical representations. Using the representation ofLg, we prove that the first condition holds. The second one is more subtle. It is equivalent to the indecomposability of the weight subcategories of the minimal categorical representations. In other words, each of these categories should have a single block. This is well known in type A and in affine type A by the work of Brundan [8] and Lyle and Mathas [29]. We can prove it in some new cases, using the fact that quiver varieties are connected (proved by Crawley-Boevey). But the general case is still unknown.

Now, let us describe more precisely the structure and the main results of the paper.

Fix a symmetrizable Kac–Moody algebragand a dominant integral weightƒofg.

In Section2, we give some generalities on the centers and cocenters of linear categories. In Section3, we introduce the cyclotomic quiver Hecke algebra of type gand levelƒover a fieldk(of any characteristic). It is a symmetric algebra which decomposes as a direct sumR^ƒDL

˛2Q_CR^ƒ.˛/, where˛runs over the positive part of the root lattice ofg. To gwe can attach another Lie algebra, Lg, given by generators and relations. It coincides with the loop algebra of g in finite types A, D,E. The first result is the following.

THEOREM1

Assume that gis symmetric and that condition (11) is satisfied. Then we have the following:

(a) There is aZ-graded representation ofLgontr.R^ƒ/.

(b) Ifgis of finite type andkis of characteristic zero, thentr.R^ƒ/is isomorphic, as aZ-gradedLg-module, to the Weyl module with highest weightƒ.

Note that there are two different notions of Weyl modules for loop Lie algebras used in the literature (the local and the global ones). Both versions can indeed be recovered (see Theorem3.37for more details). Note also that the proof of part (b) involves the geometrical incarnation, given by Nakajima, of Weyl modules ofLgvia the equivariant cohomology of a quiver varietyM.ƒ/attached toƒ.

THEOREM2

Assume thatkis of characteristic0. For any˛2QCthe following hold:

(a) The trace and the center ofR^ƒ.˛/are positively graded.

(b) Ifgis symmetric of finite type, then the dimension of the degree0subspace of Z.R^ƒ.˛//is1-dimensional.

The proof uses a reduction tosl2. Part (b) relies on the geometrical interpretation of Weyl modules. It also uses an identification between the center Z.R^ƒ.˛//and the dual of the trace tr.R^ƒ.˛//ofR^ƒ.˛/given by the symmetrizing form.

Finally, in Section4we focus on the Jordan quiver. In this case, instead of the cyclotomic quiver Hecke algebra, we consider a levelr cyclotomic quotientR^r.n/

of the degenerate affine Hecke algebra ofSn defined overkŒ. LetR^r.n/1 be its specialization atD1. The center ofR^r.n/1has a natural filtration defined in terms of Jucy–Murphy elements. Let Rees.Z.R^r.n/₁// be the corresponding Rees alge- bra. SetR^rDL

n2NR^r.n/, and let tr.R^r/⁰ be a localization of tr.R^r/. Consider the equivariant cohomologyH_G.M.r; n/;k/of the quiver varietyM.r; n/ relatively to an.rC1/-dimensional torusGwith coefficient ink.

THEOREM3 The following hold:

(a) There is a levelrrepresentation ofW intr.R^r/⁰.

(b) Assume thatkis of characteristic0; then there is aZ-graded algebra isomor- phism

Rees Z

R^r.n/1 'H_G

M.r; n/;k :

The proof of this theorem uses the representation of W on a localization H_G.M.r/;k/⁰introduced in [37].

After our paper appeared in arXiv, A. Lauda informed us that there is some over- lap between our results and his ongoing projects with collaborators.

2. Generalities

Letkbe a commutative Noetherian ring.

2.1. The center and the trace of a category 2.1.1. Categories

All categories are assumed to be small. Ak-linearcategory is a category enriched over the tensor category ofk-modules, and ak-categoryis an additivek-linear cate- gory. For anyk-linear categoryC and anyk-algebrak⁰, letk⁰˝_kC be thek⁰-linear category whose objects are the same as those ofC but whose morphism spaces are given by

Hom_k⁰˝_kC.a; b/Dk⁰˝_kHomC.a; b/; 8a; b2C:

We denote the identity of an objectaby1aor by1if no confusion is possible. All the functorsF onC are assumed to be additive and/ork-linear. An additive andk-linear functor is called ak-functor. Let End.F /be the endomorphism ring ofF. We may denote the identity element in End.F /byF; 1F, or1, and the identity functor ofC by1C or1. The center ofCis defined as Z.C/DEnd.1C/. A composition of functors EandF is written asEF, while a composition of morphisms of functorsyandx is written asyıx.

An additive categoryC will be always equipped with itstrivialexact structure;

that is, the admissible exact sequences are the split short exact sequences. Therefore, a Serre subcategoryIC is a full additive subcategory which is stable under taking direct summands, and the quotient additive categoryBDC=Iis such that

HomB.a; b/DHomC.a; b/=X

c2I

HomC.c; b/ıHomC.a; c/; 8a; b2C: Ashort exact sequenceof additive categories is a sequence of functors which is equiv- alent to a sequence0!I!C!B!0as above.

Fix an integer`. By an.`Z/-graded k-category we will mean a k-categoryC equipped with a strictk-automorphismŒ`, which we call ashift of the grading. Unless specified otherwise, a functorF of.`Z/-gradedk-categories is always assumed to be graded; that is, it is ak-functorF with an isomorphismF ıŒ`'Œ`ıF. For each integer k2N\.`Z/we will abbreviateŒkDŒ`ıŒ`ı ıŒ`(jk=`j times) and ŒkDŒk¹.

LetC=`Zbe the category enriched over the tensor category of.`Z/-gradedk- modules whose objects are the same as those ofC but whose morphism spaces are given by

HomC=`Z.a; b/DM

k2`Z

HomC

a; bŒk :

Note that the center Z.C=`Z/is a graded ring whose degreekcomponent is equal to Hom.1; Œk/.

Given aZ-gradedk-moduleM, letM^dD ¹x2MIdeg.x/Ddºfor eachd 2Z. For any integer `, the `-twist of M is the .`Z/-graded k-module M<sup>Œ`</sup> such that .M^{Œ`</sup>/<sup>d</sup> DM<sup>d=`} if `jd and 0 otherwise. Then, for eachZ-gradedk-categoryC there is a canonical .`Z/-gradedk-categoryC<sup>Œ`</sup> called the `-twist of C such that C<sup>Œ`</sup>DC as ak-category and the shift of the gradingŒ`inC^Œ` is the same as the shift of the gradingŒ1inC. We have

Hom_CŒ`=`Z.a; b/DHomC=Z.a; b/^Œ`; 8a; b:

Finally, for any categoryCwe denote byC^cthe idempotent completion.

2.1.2. Trace and center

LetC be ak-linear category, and letHH.C/be the Hochschild homology ofC(see [20, Section 3.1]). It is aZ-gradedk-module. We set tr.C/DHH0.C/and CF.C/D Hom_k.tr.C/;k/. We call tr.C/thecocenteror thetraceofC and CF.C/the set of central forms onC. Recall that

tr.C/D M

a2Ob.C/

EndC.a/

=X

f;g

kŒf; g for anyf Wa!b; gWb!a:

For any morphismf inC, let tr.f /denote its image in tr.C/.

Now, letAbe anyk-algebra. Unless specified otherwise, all algebras are assumed to be unital. Let Z.A/be the center ofA, and letHH.A/be its Hochschild homology.

Define tr.A/and CF.A/as above; that is, tr.A/DA=ŒA; A, where ŒA; AA is thek-submodule spanned by the commutators of elements of A. For any element a2A, let tr.a/denote its imageaCŒA; Ain tr.A/. LetA-mod andA-proj be the categories of finitely generated modules and finitely generated projective modules.

For any commutativek-algebraRand anyk-moduleM we abbreviateRMDR˝_k M. The following is well known.

PROPOSITION2.1

LetA; B bek-algebras, and letB;C bek-linear categories.

(a) IfBC is full and any object ofC is isomorphic to a direct summand of a direct sum of objects ofB, then the embeddingBCyields an isomorphism tr.B/!tr.C/.

(b) If C DA-mod or A-proj, then Z.A/DZ.C/. If C DA-proj, then tr.A/D tr.C/.

(c) For any commutativek-algebraR, we havetr.RA/DRtr.A/.

(d) We havetr.A˝_kB/Dtr.A/˝_ktr.B/andZ.A˝_kB/DZ.A/˝_kZ.B/.

(e) Z.C/acts ontr.C/via the mapZ.C/!End_k.tr.C//,a7!.tr.a⁰/7!tr.aa⁰//.

(f) A short exact sequence ofk-categories0!I!C!B!0yields an exact sequence ofk-linear mapstr.I/!tr.C/!tr.B/!0.

For a future use, let us give some details on part.f /. Assume thatCDC^c. For any object X, let add.X /C be the smallest k-subcategory containing X which is closed under taking direct summands. Then, the functor HomC.X;/ yields an equivalence add.X /!EndC.X /^op-proj. In particular, if C has a finite number of indecomposable objects X1; X2; : : : ; Xn (up to isomorphisms) and XDLd

iD0Xi, then we have an equivalenceC'EndC.X /^op-proj.

Now, assume thatCDA-proj, whereAis a finitely generatedk-algebra. Given a Serrek-subcategoryIC, there is an idempotent e2Asuch thatIDeAe-proj and the functorI!Cis given byM 7!Ae˝_eAeM. SetBDC=I. Then, we have B^cDB-proj, whereBDA=AeAand the composed functorC!B!B^c is given byM 7!B˝AM. We must prove that taking the trace we get an exact sequence ofk- modules tr.I/!tr.C/!tr.B/!0. Equivalently, we must check that the following complex is exact:

eAe=ŒeAe; eAe ⁱ A=ŒA; A

j

B=ŒB; B 0:

Note that kerj D.AeACŒA; A/=ŒA; Aand imiD.eAeCŒA; A/=ŒA; A. Since aebDebaeCŒae; eb for all a; b2A, we deduce that kerj Dimi, proving the claim.

2.1.3. Operators on the trace Definition 2.2

Given a functorF WC!C⁰ between twok-categories and a morphism of functors x2End.F /, thetraceofF onxis the linear map

trF.x/Wtr.C/!tr.C⁰/; tr.f /7!tr

x.a/ıF .f /

; wheref 2End.a/andx.a/ıF .f /2End.F .a//.

Note thatx.a/ıF .f /DF .f /ıx.a/ by functoriality. Below are some basic properties of the trace map, whose proofs are standard and are left to the reader.

LEMMA2.3

(a) For each F1, F2 WC !C⁰, x 2End.F1 ˚F2/, we have trF1˚F2.x/D trF₁.x11/CtrF₂.x22/, wherex112End.F1/,x222End.F2/are the diago- nal coordinates ofx.

(b) For two morphismsWF1!F2, WF2!F1, we havetrF₁. ı/DtrF₂.ı /. In particular, ifWF1!F2 is an isomorphism of functors, then for any x2End.F₁/we havetrF2.ıxı¹/DtrF1.x/.

(c) For eachF WC !C⁰,GWC⁰!C⁰⁰,x2End.F /andy2End.G/, we have trGF.yx/DtrG.y/ıtrF.x/.

2.1.4. Adjunction

Given twok-categoriesC1,C2, apair of adjoint functors.E; F /fromC1 toC2 is the datum .E; F; E; "E/of functors E WC1 !C2, F WC2 !C1 and morphisms of functors EW1_C1!FE and "E WEF !1_C2, calledunit andcounit, such that ."EE/ı.EE/DE and.F "E/ı.EF /DF, where we abbreviateED1E and F D1_F.

A pair of biadjoint functors C1!C2 is the datum .E; F; E; "E; F; "F/ of functors EWC1!C2, F WC2 !C1 and morphisms of functorsE W1C₁ !FE,

"EWEF !1C₂such that.E; F; E; "E/and.F; E; F; "F/are adjoint pairs.

Example 2.4

Given two pairs of adjoint functors.E; F /,.E⁰; F⁰/fromC1 toC2, the direct sum .E˚E⁰; F ˚F⁰/is an adjoint pair such that

E˚E⁰D.E; 0; 0; E⁰/W1C₁!FE˚FE⁰˚F⁰E˚F⁰E⁰;

"E˚E⁰D"EC"E⁰WEF ˚EF⁰˚E⁰F ˚E⁰F⁰!1C₂:

If E WC1!C2 and E⁰WC2 !C3, then .E⁰E; FF⁰/ is an adjoint pair such that E⁰ED.F E⁰E/ıE and"E⁰ED"E⁰ı.E⁰"EF⁰/.

Suppose that.E; F /,.E⁰; F⁰/are two pairs of adjoint functors fromC1 toC2. For any morphismxWE!E⁰, theleft transpose^_xWF⁰!F is the composition of the chain of morphisms

F⁰

_EF⁰

FEF^{0 F xF}

0

FE⁰F⁰

F "_E0

F:

For any morphismyWF⁰!F, theright transposey^_WE!E⁰is the composition E

E _E0

EF⁰E⁰

EyE⁰

EFE⁰

"_EE⁰

E⁰: 2.1.5. Operators on the center

Let C1,C2 be two k-categories, and let .E; F; E; "E; F; "F/ be a pair of biad- joint functors C1 !C2. The isomorphisms 1C₂EDEDE1C₁ yield a canonical .Z.C1/;Z.C2//-bimodule structure on End.E/. Let Z.C2/!End.E/,z7!zE and Z.C1/!End.E/,z7!Ezdenote the correspondingk-algebra homomorphisms.

Definition 2.5 (see [5])

For eachx2End.E/we define a map

ZE.x/WZ.C2/!Z.C1/ by sending an elementz2Z.C₂/to the composed morphism

1C_{1 E}

F 1C₂E ^{F zx} F 1C₂E

"_F

1C₁:

We defineZF.x/WZ.C1/!Z.C2/for eachx 2End.F /in the same manner but with the roles ofEandF exchanged.

The proof of the following proposition is standard and is left to the reader.

PROPOSITION2.6

Let.E; F; E; "E; F; "F/,.E⁰; F⁰; E⁰; "E⁰; F⁰; "F⁰/be two pairs of biadjoint func- tors. Letx2End.E/,x⁰2End.E⁰/. Then, we have the following:

(a) ZE.x/WZ.C2/!Z.C1/isk-linear.

(b) ZE⁰E.xı/DZE⁰.x⁰/ıZE.x/andZE˚E⁰.x˚x/DZE.x/CZE⁰.x⁰/.

(c) The mapZEWEnd.E/!Hom_k.Z.C2/;Z.C1//is.Z.C1/;Z.C2//-bilinear.

(d) LetWE!E⁰ be an isomorphism with^_D^_; thenZ_E⁰.ıxı¹/D ZE.x/.

2.2. Symmetric algebras LetA; B; C bek-algebras.

2.2.1. Kernels

There is an equivalence of categories between the category of.A; B/-bimodules and the categories of functors fromB-Mod toA-Mod. It associates an.A; B/-bimodule Kwith the functorˆKWB-Mod!A-Mod given byN 7!K˝BN. We say thatKis thekernelofˆK. SinceˆK.B/DK, the kernel is uniquely determined by the func- torˆ_K. For two.A; B/-bimodulesK,K⁰we have HomA;B.K; K⁰/'Hom.ˆ_K; ˆ_K⁰/ given byf 7!f ˝Bid.

2.2.2. Induction and restriction

We call aB-algebraak-algebraAwith ak-algebra homomorphismiWB!A. We consider the restriction and induction functors

Res^A_BWA-Mod!B-Mod; Ind^A_BDA˝_B WB-Mod!A-Mod:

The pair.Ind^A_B;Res^A_B/is adjoint with the counit"WInd^A_BRes^A_B!1represented by the.A; A/-bimodule homomorphismWA˝BA!Agiven by the multiplication, and the unitW1!Res^A_BInd^A_B is represented by the morphismi, which is.B; B/- bilinear. LetA^Bbe the centralizer ofBinA. For anyf 2A^Bwe set

f WA˝BA!A; a˝a⁰Daf a⁰: (1) 2.2.3. Frobenius and symmetrizing forms

We refer to [36] for more details on this section.

LetAbe aB-algebra that is projective and finite as aB-module. A morphism of .B; B/-bimodulest WA!B is called aFrobenius formif the morphism of.A; B/- bimodulestOWA!HomB.A; B/,a7!.a⁰7!t .a⁰a// is an isomorphism. If such a form exists, then we say thatAis aFrobeniusB-algebra. If we havet .aa⁰/Dt .a⁰a/

for eacha2A,a⁰2A^B, thent is called asymmetrizing formandAis asymmetric B-algebra.

Givent WA!B a Frobenius form, the composition of the isomorphismA˝_B A! HomB.A; B/˝Agiven bya˝a⁰7! Ot .a/˝a⁰and the canonical isomorphism HomB.A; B/˝BA! EndB.A/yields an isomorphismA˝BA! EndB.A/. The preimage of the identity under this map is theCasimir element2.A˝BA/^A. We have.t˝1/./D.1˝t /./D1.

There is a bijection between the set of Frobenius forms and the set of adjunctions .Res^A_B;Ind^A_B/given as follows. Given a Frobenius form t WA!B, the counit"OW Res^A_BInd^A_B!1B is represented by the .B; B/-linear mapt WA!B, and the unit

O

W1A!Ind^A_BRes^A_B is represented by the unique.A; A/-linear mapOWA!A˝BA such that.1O A/D. This yields an adjunction for.Res^A_B;Ind^A_B/. Conversely, if "O andO are counit and unit, respectively, for.Res^A_B;Ind^A_B/, then the.B; B/-linear map tWA!Bwhich represents"Ois a Frobenius form.

Recall that tr.A/ is a Z.A/-module. We equip CF.A/Dtr.A/ with the dual Z.A/-action. Let us recall a few basic facts.

PROPOSITION2.7

LetA; B; C bek-algebras which are projective and finite ask-modules.

(a) Ift WA!B andt⁰WB!C are symmetrizing forms, thent⁰ıt WA!C is again a symmetrizing form.

(b) A symmetrizing formtWA!kinduces aZ.A/-bilinear form tWZ.A/tr.A/!k; .a; b/7!t .ab/:

It is perfect onZ.A/; that is, it induces an isomorphism of Z.A/-modules tOWZ.A/! CF.A/which sendsztot .z/.

(c) IftWA!kis a symmetrizing form, thentr.A/is a faithfulZ.A/-module.

Proof

Part (a) is proved in [36, Lemma 2.10]. To prove part (b), note that the bilinear form is well defined, since multiplication by an element in Z.A/sendsŒA; Ato itself. The pairing is perfect on Z.A/, because the induced map tOWZ.A/!tr.A/DCF.A/

is given by taking.A; A/-invariants for the.A; A/-linear isomorphismtOWA!A; hence the result is again an isomorphism. The compatibility with the Z.A/-module structure follows from the definition. To prove part (c), we must show that the map Z.A/!End_k.tr.A// is injective. Indeed, if there exists z 2Z.A/ such that za2 ŒA; Afor alla2A, thent .zA/D0, and hencezD0.

3. The center of quiver Hecke algebras 3.1. Quiver Hecke algebras

Assume thatkDL

n2Nkⁿis Noetherian andN-graded and thatk⁰is a field. We may abbreviatekDk⁰, and we will identifykwith the quotientk=k^>0without mentioning it explicitly.

3.1.1. Cartan datum

A Cartan datum consists of a finite-rank free abelian group P called the weight lattice whose dual lattice, called the coweight lattice, is denoted P^_; a finite set of vectors ˆD ¹˛1; : : : ; ˛nº P called simple roots; and a finite set of vectors ˆ^_D ¹˛₁^_; : : : ; ˛_n^_º P^_ calledsimple coroots. Let Q_CDNˆP be the semi- group generated by the simple roots, and let PCP be the subset of dominant weights, that is, the set of weightsƒsuch thatƒiD h˛_i^_; ƒi 0for alli2I. We will call aBruhat orderthe partial order onP such thatwhenever2QC. Set I D ¹1; : : : ; nº, and let h;i be the canonical pairing on P^_P. The .I I /-matrix Awith entries aij D h˛_i^_; ˛jiis assumed to be a generalized Car- tan matrix. We will assume that the Cartan datum is nondegenerate (i.e., the simple roots are linearly independent) and symmetrizable (i.e., there exist nonzero integers di such thatdiaij Ddjaj i for alli; j). The integersdi are unique up to an overall common factor. They can be assumed positive and mutually prime.

Let.j/be the symmetric bilinear form onhDQ˝_ZP given by.˛ij˛j/D diaij. Letgbe the symmetrizable Kac–Moody algebra over kassociated with the generalized Cartan matrixAand the lattice of integral weightsP. Leth;n^Cgbe the Cartan subalgebra and the maximal nilpotent subalgebra spanned by the positive root vectors ofg. For any dominant weightƒ2PC, letV .ƒ/be the correspond- ing integrable simpleg-module. For each2P, let V .ƒ/ V .ƒ/be the weight subspace of weight.

3.1.2. Quiver Hecke algebras

Fix an element ci;j;p;q 2k for each i; j 2I, p; q 2N such that deg.ci;j;p;q/D 2d_i.a_ijCp/2d_jqandc_i;j;aij;0is invertible. Fix a matrixQD.Q_ij/_i;j2I with entries inkŒu; vsuch that

Qij.u; v/DQj i.v; u/;

Qi i.u; v/D0;

Q_ij.u; v/D X

p;q0

c_i;j;p;qu^pv^q ifi¤j:

Definition 3.1

Thequiver Hecke algebra(or QHA) of rankn0associated withAandQ is the k-algebraR.nIQ;k/generated bye. /,xk,l with 2Iⁿ,k; l2Œ1; n,l¤n, sat- isfying the following defining relations:

(a) e. /e. ⁰/Dı;⁰e. /,P

e. /D1, (b) x_kx_lDx_lx_k,x_ke. /De. /x_k,

(c) le. /De.sl. //l,klDlkifjklj> 1, (d) _l²e. /DQ_l;_lC1.xl; xlC1/e. /,

(e) .kxlxs_k.l/k/e. /Dı_k;_kC1.ıl;kC1ıl;k/e. /,

(f) ._kC1kkC1kkC1k/e. /Dı_k;_kC2@_k;kC2Q_k;_kC1.x_k; x_kC1/e. /, where@_k;l is the Demazure operator onkŒx1; x2; : : : ; xnwhich is defined by

@k;l.f /D.f .k; l/.f /=.xkxl/:

The algebraR.nIQ;k/is free as ak-module. It admits aZ-grading given by deg

e. /

D0; deg xke. /

D2d_k; deg ke. /

D d_ka_k;_kC1: For˛2QCsuch that ht.˛/Dn, we set

I^˛D®

D. 1; : : : ; n/2IⁿI˛₁C C˛_nD˛¯ : The idempotent e.˛/DP

2I^˛e. / is central in R.nIQ;k/. Given 2Iⁿ, ⁰2 I^m we write ⁰2I^nCm for their concatenation. Sete.˛; ⁰/DP

2I^˛e. ⁰/and e.n; ⁰/DP

2Iⁿe. ⁰/. The quiver Hecke algebra of rank˛is the algebra R.˛IQ;k/De.˛/R.nIQ;k/e.˛/:

3.1.3. Cyclotomic quiver Hecke algebras Given a dominant weightƒ2PCwe set

IƒD®

.i; p/Ii2I; pD1; : : : ; ƒi

¯:

For a future use, let

Iƒ!I; t7!it (2)

denote the canonical map such that .i; p/7!i. Fix a family of commuting formal variables¹c_tIt2I_ƒº. Letk^ƒbe theN-graded ring given by

k^ƒDkŒctIt 2Iƒ; deg.cip/D2pdi: We will abbreviatekDk^ƒ, and we will writeci 0D1.

Now, fix aN-gradedk-algebrak. Letctdenote both the element inkabove and its image in kby the canonical map k!k(which is homogeneous of degree 0).

Then, set

a_i^ƒ.u/D

ƒ_i

X

pD0

cipu^ƒip2kŒu: (3)

The monic polynomial a^ƒ_i .u/ is called the ith cyclotomic polynomial associated withk.

For each˛2QCand1kht.˛/, we set a_˛^ƒ.xk/D X

2I^˛

a^ƒ

k.xk/e. /: (4)

Note thata^ƒ_˛.x_k/e. /is a homogeneous element ofR.˛IQ;k/with degree2d_kƒ_k. Definition 3.2

The cyclotomic quiver Hecke algebra of rank ˛ and level ƒ is the quotient R^ƒ.˛IQ;k/ofR.˛IQ;k/by the two-sided ideal generated bya_˛^ƒ.x1/.

To simplify notation, we write R.˛/DR.˛Ik/DR.˛IQ;k/ and R^ƒ.˛/D R^ƒ.˛Ik/DR^ƒ.˛IQ;k/. We may also write RDL

˛R.˛/,R.k/DL

˛R.˛Ik/, R^ƒDL

˛R^ƒ.˛/, and so on. The following is proved in [16, Corollary 4.4, Theo- rem 4.5].

PROPOSITION3.3

Thek-algebraR^ƒ.˛Ik/is free of finite type as ak-module.

Remark 3.4

A morphism ofN-gradedk-algebras k!hyields canonical gradedh-algebra iso- morphismsh˝_kR.˛Ik/!R.˛Ih/andh˝_kR^ƒ.˛Ik/!R^ƒ.˛Ih/.

Example 3.5

(a) SetR^ƒ.˛/DR^ƒ.˛Ik/. We callR^ƒ.˛/theglobal(oruniversal) cyclotomic quiver Hecke algebra.

(b) If kDk, then a^ƒ_i .u/ Du^ƒi for each i. We call R^ƒ.˛Ik/ the local (or restricted) cyclotomic quiver Hecke algebra.

(c) For eachi2Iwe fix an elementci2kof degree2di. We define the following:

k⁰Dkas ak-algebra with the newk-algebra structure associated with the elementsc⁰_ip given byc_ip⁰ Dep.yi1ci; yi 2ci; : : : /, where the y_ip’s are commuting formal variables such thatc_ipDe_p.y_i1; y_{i 2}; : : : / and ep is the pth elementary symmetric polynomial. Then, the ith cyclotomic polynomial associated withk⁰isa^ƒ_i .uci/for eachi2I.

Q⁰_ij.u; v/DQij.uci; vcj/. In particular, we haveQ_ij⁰ .u; v/D Q⁰_{j i}.v; u/.

Then, the assignment e. /; x_ke. /; _le. / 7!e. /; .x_k Cc_k/e. /; _le. / extends uniquely to ak-algebra isomorphismR^ƒ.˛IQ;k/! R^ƒ.˛IQ⁰;k⁰/.

(d) Under the hypothesis above, assume that i2I and that ƒD!i is the ith fundamental weight. Assume also that the polynomialQij.u; v/satisfies con- dition (11) below. Ifa^ƒ_i .u/DuCci, then we have ak-algebra isomorphism

R^!i.˛IQ;k/'k˝_kR^!i.˛IQ;k/: (5) Definition 3.6

For each k2Œ1; n1, the kth intertwiner operator is the element 'k 2R^ƒ.n/

defined by 'ke. /Dke. / if k¤ kC1 and by the following formulas if kD

kC1:

'ke. /D.xkkkxk/e. /D.kxkC1xkC1k/e. / D

.x_kx_kC1/_kC1

e. /D

_k.x_kC1x_k/1 e. /:

We have the following facts (see [18, Section 5.1] for details):

x_{sk.`/</sub>'<sub>k</sub>e. /D'<sub>k</sub>x<sub>`}e. /.

¹'kºsatisfies the braid relations.

Ifw2Snsatisfiesw.kC1/Dw.k/C1, then'wkDw.k/'w.

'_k²e. /De. /if kD kC1 and'_k²e. /DQ_k;_kC1.xk; xkC1/e. /if k¤

kC1.

3.1.4. Induction and restriction

Leti2I and˛2QCbe of heightn. SetDƒ˛andiD h˛_i^_; i.

We have a Z-graded k-algebra embedding i WR.˛/ ,!R.˛ C˛i/ given by e. /; x_k; _l7!e. ; i /; x_k; _l for each 2I^˛ with1knand 1ln1. It induces aZ-gradedk-algebra homomorphism_iWR^ƒ.˛/!R^ƒ.˛C˛_i/.

The restriction and induction functors form an adjoint pair.F_i⁰; E_i⁰/with E_i⁰WR^ƒ.˛C˛i/-grmod!R^ƒ.˛/-grmod; N7!e.˛; i /N;

F_i⁰WR^ƒ.˛/-grmod!R^ƒ.˛C˛i/-grmod;

M7!R^ƒ.˛C˛i/e.˛; i /˝_Rƒ.˛/M:

The counit "⁰_i;WF_i⁰E_i⁰1!1 and the unit ⁰_i;W1!E_i⁰F_i⁰1 are represented, respectively, by the multiplication mapand the map_i:

"⁰_i;WR^ƒ.˛/e.˛˛i; i /˝_Rƒ.˛˛_i/e.˛˛i; i /R^ƒ.˛/!R^ƒ.˛/;

⁰_i;WR^ƒ.˛/!e.˛; i /R^ƒ.˛C˛i/e.˛; i /:

Finally, let_ij;⁰ WF_i⁰E_j⁰1!E_j⁰F_i⁰1be the morphism represented by the linear map

R^ƒ.˛˛j C˛i/e.˛˛j; i /˝_Rƒ.˛˛_j/e.˛˛j; j /R^ƒ.˛/

!e.˛˛j C˛i; j /R^ƒ.˛C˛i/e.˛; i /;

x˝y7!xny:

Forj Di, the elementn2R^ƒ.˛C˛i/centralizes the subalgebrae.˛˛i; i²/ R^ƒ.˛C˛i/e.˛˛i; i²/, so we have_{i i;}⁰ D_n(see (1)).

THEOREM3.7 (see [16])

For each˛2QCof heightn, we have the following.

(a) Ifi0, then the following morphism of endofunctors onR^ƒ.˛/-Modis an isomorphism:

_i;⁰ D_{i i;}⁰ C

_i1

X

kD0

.E_i⁰x^k/ı⁰_i;WF_i⁰E_i⁰1˚

_i1

M

kD0

kx^k˝1!E_i⁰F_i⁰1: (b) Ifi0, then the following morphism of endofunctors onR^ƒ.˛/-Modis an

isomorphism:

⁰_i;D

_{i i;}⁰ ; "⁰_i;ı.F_i⁰x⁰/; : : : ; "⁰_i;ı.F_i⁰xⁱ¹/ W

F_i⁰E_i⁰1!E_i⁰F_i⁰1˚

_i1

M

kD0

k.x¹/^k˝1:

The theorem can be rephrased as follows.

Assume thati0: for anyz2e.˛; i /R^ƒ.˛C˛i/e.˛; i /there are unique ele- ments.z/2R^ƒ.˛/e.˛˛_i; i /˝_Rƒ.˛˛_i/e.˛˛_i; i /R^ƒ.˛/andp_k.z/2 R^ƒ.˛/such that

zDn

.z/

C

_i1

X

kD0

pk.z/x_nC1^k : (6)

Assume thati 0: for any z2e.˛; i /R^ƒ.˛C˛i/e.˛; i / and any z0; : : : ; z_i12R^ƒ.˛/, there is a unique elementy2R^ƒ.˛/e.˛˛i; i /˝_Rƒ.˛˛_i/

e.˛˛_i; i /R^ƒ.˛/such that _n.y/Dz; _xk

n.y/Dzk; 8k2Œ0;i1: (7)

For a future use, let us introduce the following notation. Assume thati0and thatz2e.˛; i /R^ƒ.˛C˛i/e.˛; i /. For each`2Œ0;i1, let

Q

z;Q_`2R^ƒ.˛/e.˛˛i; i /˝_Rƒ.˛˛_i/e.˛˛i; i /R^ƒ.˛/ (8) be the unique elements such that

n.z/Q Dz; _xk

n.Qz/D0; n.Q`/D0; _xk

n.Q`/Dık;`:

THEOREM3.8 (see [18])

The pair.E_i⁰; F_i⁰/ is adjoint with the counit"O⁰_i;WE_i⁰F_i⁰1!1 and the unitO⁰_i;W 1!F_i⁰E_i⁰1represented by the morphisms

O

"⁰_i;We.˛; i /R^ƒ.˛C˛i/e.˛; i /!R^ƒ.˛/;

O

⁰_i;WR^ƒ.˛/!R^ƒ.˛/e.˛˛i; i /˝_Rƒ.˛˛_i/e.˛˛i; i /R^ƒ.˛/

such that

"O⁰_i;.z/Dp_i1.z/ifi> 0and

x_nⁱ.Qz/ifi0,

O⁰_i;.1/D .x_nC1ⁱ /ifi0andQ_i1ifi< 0.

We abbreviate"⁰_iD"⁰_i;,⁰_i D⁰_i;,"O⁰_i D O"⁰_i;,O⁰_i D O⁰_i;, and so on, when is clear from the context.

COROLLARY3.9

The linear maps "⁰_i, ⁰_i are homogeneous of degree 0. The linear maps "O⁰_i, O⁰_i are homogeneous of degrees2di.1i/,2di.1Ci/, respectively. The linear map_ij⁰ is homogeneous of degreediaij.

3.1.5. The symmetrizing form For each˛2QCwe set

dƒ;˛D.ƒjƒ/.ƒ˛jƒ˛/:

We will need the following result from [40].

PROPOSITION3.10 ([40, Remark 3.19])

Thek-algebra R^ƒ.˛/ is symmetric and admits a symmetrizing formtƒ;˛ which is homogeneous of degreedƒ;˛.

The definition oftƒ;˛is given in DefinitionA.6. We will abbreviatet˛Dtƒ;˛and t_ƒDP

˛t_˛. Since we have not found any proof of the proposition in the literature, we have given one in AppendixA.

3.2. Categorical representations

Letkbe anN-graded commutative ring as in Section3.1. WritegkDk˝_kg. Fix an integer`.

3.2.1. Definition

For each2P, letCbe an.`Z/-gradedk-category. SetCDL

Cand denote by 1the obvious functor1WC!C. For eachi; j 2I,2P, we fix

aZ-gradedk-algebra homomorphismk<sup>Œ`</sup>!Z.C=`Z/;

a functor 1_˛iFi D Fi1 with a right adjoint 1EiŒ`di.1 i/ D E<sub>i</sub>1<sub>˛i</sub>Œ`d_i.1_i/;

morphisms of functors xi1W Fi1 !Fi1Œ2`di and ij1 WFiFj1 ! FjFi1Œ`diaij.

ThusC=`Zis ak-category, and the functorsFi1,Ei1arek-linear. Let

"_i1WF_iE_i1!1

`d_i.1C_i/

; _i1W1!E_iF_i1

`d<sub>i</sub>.1<sub>i</sub>/ be the counit and the unit of the adjoint pair.1Fi; Ei1Œ`di.1Ci//. We will abbreviate

EiDM

Ei1; FiDM

Fi1; F˛DM

2I^˛

F; and so on;

where FDF₁F₂ F_n for D. 1; 2; : : : ; n/. Next, we define the following morphisms:

ij D.EjFi"j/ı.Ejj iEi/ı.jFiEj/WFiEj !EjFi;

i1 D i i1 C P_i1

lD0 ."i1/ ı .x_i^lEi1/ W FiEi1 ! EiFi1 ˚ L_i1

lD0 1Œ`di.1C2lCi/ifi0;

i1Di i1CP_i1

lD0 .Eix_i^l1/ı.i1/WFiEi1˚L_i1

lD0 1Œ`di.1C2l i/!EiFi1ifi0.

Definition 3.11

Acategorical representationofg_kof degree`inC is a tupleC,Ei,Fi,"i,i,xi, ij as above such that the following hold:

the assignmente. /7!1F,x_ke. /7!x_k1F,_le. /7!_l;_lC11Ffor each 2 I^˛ defines a Z-graded k^{Œ`</sup>-algebra homomorphism R.˛Ik/<sup>Œ`} ! EndC=`Z.F˛/;

the morphismsi1,ij,i¤j, are isomorphisms.

Morphisms of categorical representations are defined in the obvious way.

We will call the mapR.˛Ik/<sup>Œ`</sup>!EndC=`Z.F˛/thecanonical homomorphism associated with the categorical representation ofgkinC.

Unless specified otherwise, a categorical representation will be of degree 1.

Degrees`¤1are used only in the nonsymmetric case and the reader interested only in symmetric ones may set`D1everywhere. Note that, given a categorical represen- tation ofgkinC, there is a canonical categorical representation ofgkof degree`in C<sup>Œ`</sup>called its`-twistsuch that theZ-gradedk<sup>Œ`</sup>-algebra homomorphism

R.˛Ik/^{Œ`</sup>!End<sub>C</sub>Œ`=`Z.F<sub>˛</sub>/DEndC=Z.F<sub>˛</sub>/<sup>Œ`}

is equal to the homomorphismR.˛Ik/!EndC=Z.F˛/associated with thegk-action onC.

We will also use the following definitions:

CisintegrableifEi,Fiare locally nilpotent for alli.

Cisbounded aboveif the set of weights ofC is contained in a finite union of cones of typeQCwith2P.

Thehighest weight subcategoryC^hwC is the full subcategory given by C^hwD®

M 2CIEi.M /D0;8i2I¯ : Remark 3.12

(a) Taking the left transpose of the morphisms of functors

xi1WFi1!Fi1Œ2`di; ij1WFiFj1!FjFi1Œ`diaij we get the morphisms of functors

1_

xiW1Ei!1EiŒ2`di; 1_

ij1W1EiEj !1EjEiŒ`diaij: We will abbreviatexiD^_xi andij D^_ij.

(b) Forgetting the grading at each place we define as above a categorical repre- sentation ofgkin a (not graded)k-categoryC.

(c) For each short exact sequence of Z-graded k-categories 0!I !C ! B!0such thatI!C is a morphism of categorical representations ofg_k, there is a unique categorical representation ofg_konBsuch thatC!Bis a morphism of categorical representations.

(d) Given a categorical representation ofgkon C, there is a unique categorical representation ofgkonC^c such that the canonical fully faithful functorC !C^c is a morphism of categorical representations. Recall that the objects of the idempotent completionC^c are the pairs.M; e/, whereM is an object of C ande is an idem- potent of EndC.M /, and that HomC^c..M; e/; .N; f //DfHomC.M; N /e. Then, we haveFi.M; e/D.Fi.M /; Fi.e//,Ei.M; e/D.Ei.M /; Ei.e//, andxi1,ij1 are defined in a similar way.

3.2.2. The minimal categorical representation

Fix a dominant weightƒ2PCand anN-gradedk-algebrak. Given˛2QCwe write Dƒ˛. Recall that we abbreviateR^ƒ.˛/DR^ƒ.˛Ik/. LetkA^ƒ DR^ƒ.˛/-grmod be the Z-graded abelian k-category consisting of the finitely generated Z-graded R^ƒ.˛/-modules, and let kV^ƒDR^ƒ.˛/-grproj be the full subcategory formed by the projective Z-graded modules. When there is no confusion, we will abbreviate A^ƒ DkA^ƒ andV^ƒDkA^ƒ. LetA^ƒ,V^ƒbe the categories

A^ƒDM

A^ƒ; V^ƒDM

V^ƒ:

Fix an integer`. LetV<sup>ƒ;Œ`</sup>D.V^ƒ/<sup>Œ`</sup> be the`-twist of V^ƒ, and letR^ƒ.˛/<sup>Œ`</sup> be the`-twist ofR^ƒ.˛/. Thus,R^ƒ.˛/<sup>Œ`</sup>is a.`Z/-gradedk^{Œ`</sup>-algebra andV<sup>ƒ;Œ`}is the category of finitely generated projective.`Z/-graded modules; that is,

V^{ƒ;Œ`</sup>DR<sup>ƒ</sup>.˛/<sup>Œ`}-grproj:

Definition 3.13

Theminimal categorical representationofgkof highest weightƒand degree`is the representation onV<sup>ƒ;Œ`</sup>given by the following:

E_i1DE_i⁰Œ`d_i.1C_i/;

Fi1DF_i⁰;

"i1D"⁰_i;andi1D⁰_i;;

xi12Hom.Fi1; Fi1Œ2`di/is represented by the right multiplication by xnC1onR^ƒ.˛C˛i/e.˛; i /;

ij12Hom.FiFj1; FjFi1Œ`diaij/is represented by the right multipli- cation bynC1onR^ƒ.˛C˛iC˛j/e.˛; j i /.

The categoryA^{ƒ;Œ`</sup> is Krull–Schmidt with a finite number of indecomposable projective objects. The categoryV<sup>ƒ;Œ`}=.`Z/is the category of.`Z/-graded finitely generated projectiveR<sup>ƒ;Œ`</sup>-modules with morphisms which are not necessarily homo- geneous. We will call it the category of all.`Z/-gradableprojective modules.

Example 3.14

We will abbreviateeDsl2. Assume thatgDe,ƒDk!1, and˛Dn˛1withk; n2N. In this case, we write V^kDV^ƒ and V_k2n^k DV^ƒ. Consider the polynomial ring Z^kDkŒc1; : : : ; c_k with deg.cp/D2p for all p. Let H^k_n be the global cyclotomic affine nil Hecke algebra of rank n and level k, that is, the Z-graded Z^k-algebra denoted by Hn;k in [35, Section 4.3.2]. Note that we have H^k₀ DZ^k and kDZ^k. Given anN-graded Z^k-algebrak, the cyclotomic quiver Hecke algebraR^ƒ.˛Ik/is isomorphic to k˝_Zk H^k_n as aZ-gradedk-algebra by [35, Lemma 4.27]. In partic- ular, we haveR^ƒ.˛/DH^k_n. For each integer`, we abbreviate Z<sup>k;Œ`</sup>D.Z^k/^{Œ`</sup> and H<sup>k;Œ`}_n D.H^k_n/^Œ`. We have

V^k;Œ`DM

n0

.k<sup>Œ`</sup>˝<sub>Z</sub>k;Œ`H^{k;Œ`</sup><sub>n</sub> /-grproj; (9) and the trace tr.V<sup>k;Œ`}/ is given in Proposition 3.30 below. We will also identify R^ƒ.˛Ik/with thelocal cyclotomic affine nil Hecke algebraof rank nand level k, which is the quotient H^knDk˝_ZkH^k_nof H^k_nby the ideal.c1; : : : ; c_k/.

3.2.3. Factorization

Fix a dominant weightƒ2PC. Recall the mapIƒ!I,t7!it introduced in (2).

We will abbreviate !t D!i_t and dt Ddi_t. Consider the N-graded k-algebra hD kŒytIt2Iƒ, whereyt is a formal variable of degree deg.yt/D2dt. It has a natural structure of anN-gradedk-algebra such that the elementc_ip2his given byc_ipD ep.yi1; : : : ; yi ƒ_i/. The corresponding cyclotomic polynomials are

a_i^ƒ.u/D

ƒ_i

Y

pD1

.uCyip/; 8i2I: (10)

Leth⁰ be the fraction field ofh. Then we have the algebrasR^ƒ.n;h/andR^ƒ.n;h⁰/ overhandh⁰such thatR^ƒ.n;h⁰/Dh⁰˝_hR^ƒ.n;h/. Next, for eacht2Iƒ, we have

hDO

t2I_ƒ

kŒytDO

t2I_ƒ

k^!t:

Therefore, we can viewhas ak^!t-algebra, and hence theh-algebraR^!t.n;h/asso- ciated with the cyclotomic polynomialsa^!_i^t.u/DuCytis well defined. Note that by