The center of 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 CarCar-tan 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

Remark3.5(d), if condition (11) below is satisfied, then we may as well assume that a^!_i^t.u/Du. Recall that

R^ƒ.n;h/Dh˝_kR^ƒ.n/;

R^!t.n;h/Dh˝_k!t R^!t.n/:

When no confusion is possible, we will abbreviate V^ƒDM

n

R^ƒ.n/-grproj; V^!t DM

n

R^!t.n/-grproj:

Finally, we consider the condition

Qij.u; v/Drij.uv/^aij for somerij inksuch that

rij D.1/^aijrj i for alli¤j: (11)

THEOREM3.15

Letnbe a positive integer, and let˛2QC be of height n. If the condition (11) is satisfied, then the following hold:

(a) There is anh⁰-algebra isomorphism R^ƒ.n;h⁰/!M

.nt/

Mat_Sn=Q tS_nt

O

t2I_ƒ

R^!t.nt;h⁰/

;

where.nt/runs over the set ofIƒ-tuples of nonnegative integers with sumn.

(b) There is an isomorphismh⁰˝_kV^ƒ!h⁰˝_hN

t2I_ƒV^!t of (nongraded) cate-goricalg_h⁰-representations taking the functorF_˛toL

.˛t/

N

tF_˛t, where the sum runs over the set ofIƒ-tuples.˛t/of elements ofQCwith sum˛. More precisely, the canonical homomorphism

O

t2Iƒ

R.˛tIh⁰/!End O

t2Iƒ

F˛_t

is the composition of the inclusionN

t2I_ƒR.˛tIh⁰/R.˛Ih⁰/underlying (a) and of the canonical homomorphismR.˛Ih⁰/!End.F_˛/.

Proof

FixM 2R^ƒ.nIh⁰/-mod andg.u/2h⁰Œu. From [16, pp. 715–716], we get g.xa/e. /MD0

)Q_a;_aC1.xa; xaC1/g.xaC1/e sa. /

M D0; 8a2Œ1; n/;8 2Iⁿ:

SetQ.u; v/DQ

i¤jQij.u; v/. We deduce that g.xa/e. /MD0)Q.xa; xaC1/g.xaC1/e

sa. /

M D0; 8a2Œ1; n/: (12) Now, assume that the polynomialQ.u; v/2h⁰Œu; vhas the following form,

Q.u; v/Dr Y

2S

.uv/; (13)

for some finite familySof elements ofh⁰and some elementr2h⁰. Let sp_e./M.xa/ h⁰be the set of2h⁰such that the operatorxaid2End_h⁰.e. /M /is not invert-ible. Since xa and xaC1 commute with each other, from (12) and (13) we deduce that

sp_e.sa.//M.xaC1/ sp_e./M.xa/t

sp_e.sa.//M.xa/S : Thus, switching andsa. /, we deduce that

sp_e./M.xaC1/ sp_e.sa.//M.xa/t

sp_e./M.xa/S : Next, recall thatg.x1/.e. /M /D0ifg.u/Da^ƒ

1.u/. We deduce that sp_e./M.x1/ ¹y₁;pIpD1; : : : ; ƒ₁º:

Therefore, an easy induction implies that

sp_e./M.xa/ ¹y_b;pIpD1; : : : ; ƒ_b; bD1; : : : ; aº NS; 8a2Œ1; n:

Assume further that condition (11) holds. Then we haveSD ¹0º, and hence

sp_e./M.xa/ ¹y_b;pIpD1; : : : ; ƒ_b; bD1; : : : ; aº; 8a2Œ1; n: (14) In the rest of the proof we writeIQDIƒto simplify the notation. For eachn-tuple Q2 QIⁿ, set

M_QD®

m2MI.x_kCy_Qk/^DmD0;8k2Œ1; n;8D0¯ :

Considering the decomposition of the regular module, we deduce from (14) that there is a complete collection of orthogonal idempotents¹e. /I QQ 2 QIⁿºinR^ƒ.n;h⁰/such thate. /MQ DMQ. The groupSnacts onIQⁿin the obvious way. The following hold:

e. /e. /Q De. /e.Q /,

xle. /Q De. /xQ l,

'_ke. /Q De.s_k.Q //'_k,

_ke. /Q De. /Q _kif Q_kD Q_kC1,

where k, l, , andQ run over the sets Œ1; n/,Œ1; n,IQⁿ, and Iⁿ, respectively. The relations above are immediate, except the last one. To prove it, note that ifQ_kD Q_kC1, then the operatorsx_kCy_Qk andx_kC1Cy_Qk commute to each other and their sum and product commute withk and act nilpotently onke. /MQ . ThereforexkCyQ_k and xkC1CyQ_kalso act nilpotently onke. /MQ . Note that the first relation above implies that we may view¹e. /I QQ 2 QIⁿºas a complete collection of orthogonal idempotents inR^ƒ.˛;h⁰/for each˛2QCof heightn.

LEMMA3.16

For eachM 2R^ƒ.nIh⁰/-mod, the map'ke. /Q We. /MQ !e.sk.Q //M is invertible whenever Qk¤ QkC1.

Proof

The lemma is an immediate consequence of the following relations, for each 2Iⁿ, '_k²e. /D1 if kD _kC1;

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

because (11) implies thatQ_k;_kC1.xk; xkC1/is invertible if k¤ kC1 and Qk¤ QkC1.

SetQQCDNIQ. For any element˛Q DP

t2I_ƒntt inQQCwe set ht.˛/Q DP

tat. Assume that ht.˛/Q Dn; then we consider the setIQ^˛Q D ¹ Q 2 QIⁿIP

kQk D Q˛º. We will say that twon-tuples ;Q Q⁰2 QIⁿ are equivalent, and we write Q Q⁰ if we have Q

;Q⁰2 QI^˛Q for some˛Q 2 QQC. We define the idempotente.˛/Q inR^ƒ.n;h⁰/bye.˛/Q D P

Q

2 QI^˛Qe. /.Q

Next, fix a total order onIQ, and setIQ_Cⁿ D ¹ Q2 QIⁿIi < j ) Qi Qjº. For any tuple Q 2 QIⁿ there is a unique element Q^C2 QI_Cⁿ such that Q Q^C. Let ˛Q^C be the unique element inIQ_Cⁿ\ QI^˛Q. The theorem is an easy consequence of the following lemma.

LEMMA3.17

Let˛2QCand˛Q2 QQCbe of heightn, with˛QDP

t2 QIntt. The following hold:

(a) e. /RQ ^ƒ.˛;h⁰/e.Q⁰/D0unless Q Q⁰;

(b) e.˛/RQ ^ƒ.˛;h⁰/e.˛/Q 'MatIQ^˛Q.e.˛Q^C/R^ƒ.˛;h⁰/e.˛Q^C//ash⁰-algebras;

(c) e.˛Q^C/R^ƒ.˛;h⁰/e.˛Q^C/'L

.˛_t/

N

t2 QIR^!t.˛t;h⁰/ash⁰-algebras, where.˛t/ runs over the set of allIQ-tuples of elements inQCsuch thatP

t˛tD˛and ht.˛t/Dnt.

Proof

Part (a) of the lemma follows from the following.

CLAIM3.18

Theh⁰-algebraR^ƒ.n;h⁰/is generated by the subset

®e. /e.Q /; _he. /; 'Q _ke. /; xQ _le. /IQ h; k2Œ1; n/; l2Œ1; n;

2Iⁿ;Q2 QIⁿ;Q_hD Q_hC1;Q_k¤ Q_kC1¯ :

Let us concentrate on part (b). For each tupleQ2 QI^˛Q, we fix a sequence of simple reflectionssl₁; sl₂; : : : ; sl_j inSnsuch that

QDs_l1s_l2 s_lj.˛Q^C/;

thel_hth entry ofs_lhC1s_lhC2 s_lj.˛Q^C/inIQis smaller than the.l_hC1/th one for eachh2Œ1; j .

In particular,wDsl₁sl₂ sl_j is a reduced decomposition andwis minimal in its left coset inSnrelatively to the stabilizer of˛Q^C. Hence, we have

'_l1'_l2 '_lj D'w; (15) and the element_Qine. /RQ ^ƒ.˛;h⁰/e.˛Q^C/given by

QDe. /'Q we.˛Q^C/ (16) depends only onQand not on the choice ofl₁; l₂; : : : ; l_j. It is invertible by Lemma3.16.

We deduce that there is anh⁰-algebra isomorphism MatIQ^˛Q

e.˛Q^C/R^ƒ.˛;h⁰/e.˛Q^C/

!e.˛/RQ ^ƒ.˛;h⁰/e.˛/;Q

E_;QQ⁰.m/7!_Qm¹_Q0 : (17) HereE;QQ⁰.m/is the elementary matrix with anmat the spot.Q ;Q⁰/and0s elsewhere.

Part (b) is proved.

Now, we prove part (c). Set¹t2 QIInt¤0º D ¹t1; t2; : : : ; tmºwitht1< t2< <

t_m. Fix anIQ-tuple.˛_t/of elements inQ_Csuch thatP

t˛_tD˛and ht.˛_t/Dn_t for eacht. There is a canonical inclusion

O

t2 QI

R.˛t;h⁰/DR.˛t₁;h⁰/˝R.˛t₂;h⁰/˝ ˝R.˛t_m;h⁰/!R.˛;h⁰/:

Composing it with the multiplication by the idempotente.˛Q^C/, we get a map O

t2 QI

R.˛t;h⁰/!e.˛Q^C/R.˛;h⁰/e.˛Q^C/ (18)

such that, for eachIQ-tuple. t/with t2I^˛t, we have

e. t₁/˝e. t₂/˝ ˝e. t_m/7!e. t₁ t₂ t_m/e.˛Q^C/:

To prove (c) we must check that the direct sum of all maps (18), where.˛t/runs over the set of all tuples as above, gives anh⁰-algebra isomorphism

M

.˛t/

O

t2 QI

R^!t.˛t;h⁰/! e.˛Q^C/R^ƒ.˛;h⁰/e.˛Q^C/: (19) First, we will prove that theh⁰-algebra homomorphism (19) is well defined. To do that, it is enough to check that for each tuple.˛t/the map (18) factors to an algebra homomorphism

O

t2 QI

R^!t.˛t;h⁰/!e.˛Q^C/R^ƒ.˛;h⁰/e.˛Q^C/; (20)

because (20) maps into orthogonal direct summands of e.˛Q^C/R^ƒ.˛;h⁰/e.˛Q^C/ for different tuples.˛t/. To simplify the notation, we will check this in a particular case.

The proof of the general case is very similar. Assume that

ƒD2!iC!j; IQD ¹t1; t2; t3º; t1D.i; 1/; t2D.i; 2/; t3D.j; 1/;

withi¤j andt1< t2< t3. Thus, we have

a^ƒ_i .u/D.uCyt₁/.uCyt₂/; a^ƒ_j .u/DuCyt₃; a_k^ƒ.u/D1; 8k¤i; j;

!t₁D!i; !t₂D!i; !t₃D!j:

Next, fix a positive integernand an element˛Q 2 QQCof heightn. We have Q

˛Dnt₁t1Cnt₂t2Cnt₃t3; Q

˛^CD.˛Q₁^C;˛Q₂^C; : : : ;˛Q^C_n/D

.t1/^nt1.t2/^nt2.t3/^nt3

;

withn_t1Cn_t2Cn_t3Dn. To simplify we will assume that the integersn_t1; n_t2; n_t3 are positive.

CLAIM3.19

For any 2Iⁿ, the following relations hold ine.˛Q^C/R^ƒ.n;h⁰/e.˛Q^C/:

.x1Cyt₁/e.˛Q^C/e. /D0if 1Di, ande.˛Q^C/e. /D0otherwise;

.x1Cn_t1Cyt2/e.˛Q^C/e. /D0if 1Cn_t1 Di, ande.˛Q^C/e. /D0otherwise;

.x1Cn_t1Cn_t2Cyt₃/e.˛Q^C/e. /D0if 1Cn_t1Cn_t2Dj, ande.˛Q^C/e. /D0 oth-erwise.

The first relation is obvious, because

.x1Cyt₁/.x1Cyt₂/e. /D0if 1Di;

.x₁Cy_t3/e. /D0if 1Dj;

e. /D0if 1¤i; j;

.x1Cyt₂/e.˛Q^C/,.x1Cyt₃/e.˛Q^C/are invertible ine.˛Q^C/R^ƒ.n;h⁰/e.˛Q^C/.

To prove the second relation, note that

'1'2 'n_t1.x1Cn_t1Cyt₂/e.˛Q^C/e. /D.x1Cyt₂/e. /e. /'Q 1'2 'n_t1; where QDs1s2 sn_t1.˛Q^C/andDs1s2 sn_t1. /. SinceQ1Dt2and1D 1Cn_t1, we deduce that

'₁'₂ '_nt1.x_1Cnt1Cy_t2/e.˛Q^C/e. /D0if 1Cn_t1Di,

'1'2 'n_t1e.˛Q^C/e. /D0otherwise.

Further, by Lemma3.16, the operator

'1'2 'n_t1e.˛Q^C/We.˛Q^C/R^ƒ.˛;h⁰/e.˛Q^C/!e. /RQ ^ƒ.˛;h⁰/e.˛Q^C/ is invertible. Thus, we have

.x1Cn_t1Cyt₂/e.˛Q^C/e. /D0if 1Cn_t1 Di,

e.˛Q^C/e. /D0otherwise,

proving the second relation. The third relation is proved in a similar way, using the product of intertwiners'1'2 'n_t1Cn_t2 instead of'1'2 'n_t1. The claim is proved.

The claim implies that the homomorphism (20) is well defined, and so (19) is also well defined. We must check that it is invertible. To prove the surjectivity, we must check that theh⁰-algebra isomorphism (19) yields a surjective map

O

t2 QI

R^!t.nt;h⁰/! e.˛Q^C/R^ƒ.n;h⁰/e.˛Q^C/:

This is a consequence of the following fact.

CLAIM3.20

Theh⁰-algebrae.˛Q^C/R^ƒ.n;h⁰/e.˛Q^C/is generated by the subset

®e. /e.˛Q^C/; he.˛Q^C/; xle.˛Q^C/Ih2Œ1; n/; l2Œ1; n; 2Iⁿ;˛Q_h^CD Q˛_hC1^C ¯

: (21) Let us concentrate on the injectivity. We will construct a left inverse to the map (19). To do that, recall that parts (a) and (b) yield anh⁰-algebra isomorphism

M

Q

˛

MatIQ^˛Q

e.˛Q^C/R^ƒ.˛;h⁰/e.˛Q^C/

!R^ƒ.˛;h⁰/; (22) where the sum is over the set of all elements˛Q2 QQCof heightn.

CLAIM3.21

(a) FixQ2 QI^˛Qandw2Snas in (16). For eachh; k; l2Œ1; nsuch thath; k¤n, Q_hD Q_hC1, andQ_k¤ Q_kC1, the map (22) is such that

E;QQ

x_w1.l/e.˛Q^C/

7!xle. /;Q Es_k./;QQ

e.˛Q^C/

7!'ke. /;Q E;QQ

_w1.h/e.˛Q^C/

7!he. /:Q (b) Consider the assignment

xle. /Q 7!E;QQ.x_w1.l//;

'_ke. /Q 7!E_sk./;QQ .1/;

_he. /Q 7!E_;QQ._w1.h//;

for eachh; k; l2Œ1; nand Q2 QI^˛Q such thath; k¤n, Q_hD Q_hC1, and Q_k¤ QkC1, wherew2Snas in (16). It extends uniquely to anh⁰-algebra homo-morphism

R^ƒ.˛;h⁰/!M

Q

˛

M

.˛_t/

MatIQ^˛Q

O

t2 QI

R^!t.˛t;h⁰/

: (23)

Proof

Part (a) follows from (17) and from the following computations:

xle. /Q DxlQe.˛Q^C/¹_Q DQx_w1.l/e.˛Q^C/¹_Q ; '_ke. /Q D'_kQe.˛Q^C/¹_Q

D'_ke.Q /'we.˛Q^C/_Q¹ De

s_k.Q /

'_k'_we.˛Q^C/_Q¹ Ds_k.Q/e.˛Q^C/¹_Q ; he. /Q DhQe.˛Q^C/_Q¹

Dhe. /'Q we.˛Q^C/¹_Q De. /Q h'we.˛Q^C/¹_Q De. /'Q w_w1.h/e.˛Q^C/¹_Q D_Qw1.h/e.˛Q^C/¹_Q :

Here, we used the equalityw¹.hC1/Dw¹.h/C1, which follows from the defini-tion ofwin (15), and the equalitiess_h.Q /D Q ands_w1.h/.˛Q^C/D Q˛^C, which follow from the identities Q_hD Q_hC1and QDw.˛Q^C/.

Now, let us concentrate on (b). Since the elements e. /e.Q /, xle. /,Q 'ke.Q /, he. /Q generate R^ƒ.n;h⁰/ by Claim 3.18, it is enough to check that the defining relations ofR^ƒ.˛;h⁰/given in Section3.1.3are satisfied. This is obvious.

Note that the element _w1.h/ belongs indeed to N

t2 QIR^!t.˛t;h⁰/ because s_w1.h/.˛Q^C/D Q˛^C.

Composing (22) and (23), we get anh⁰-algebra homomorphism M

Q

˛

MatIQ^˛Q

e.˛Q^C/R^ƒ.˛;h⁰/e.˛Q^C/

!M

Q

˛

M

.˛t/

MatIQ^˛Q

O

t2 QI

R^!t.˛t;h⁰/ :

For each˛Q it restricts to anh⁰-algebra homomorphism e.˛Q^C/R^ƒ.˛;h⁰/e.˛Q^C/!M

.˛_t/

O

t2 QI

R^!t.˛t;h⁰/;

which is a left inverse to (19).

3.2.4. The isotypic filtration

Recall thateDsl2. A.P Z/-gradedk-categoryC is a direct sum of categories of the formCDL

2PC, where eachCis aZ-gradedk-category. We will callthe P-degreeofC.

Giveni2I there is ansl2-tripleeig. In this section, we study the restriction of a categoricalgk-representation to such ansl2-triple. To simplify the notation, in this section we fix an elementi2I and identifyeDei.

Recall that for each weight2P we set _iD h˛_i^_; i. By a .P Z/-graded categoricale_k-representation onC, we will mean a representation such that for each 2P thek-subcategoryChas the weightirelative to thee-action. In particular, if C is an integrable categorical representation ofgk, restricting theg-action onCtoei

yields an integrable.PZ/-graded categoricale-representation onCof degreedi. Now, fix an integerk 2N. Given a .P Z/-graded k-categoryM, such that M D0 whenever _i ¤k, and a Z-graded k-algebra homomorphism Z^k;Œdi! Z.M=Z/, we equip the tensor productV^k;Œdi˝_Zk;Œdi Mwith the.P Z/-graded categoricale-representation of degreedi such that

eacts on the left factor;

the summandVn^k;Œdi˝_Zk;Œdi Mhas theP-degreen˛ifor eachn2N.

The vanishing M D0 whenever i ¤k is imposed so that the tensor product V^k;Œdi˝_Zk;Œdi Mis a.PZ/-graded categoricale-representation as defined above.

PROPOSITION3.22

Fix an integrable categorical representation ofg_konC of degree1which is bounded above. For eachi2I there is a decreasing filtration C₁ C0DC ofC by

Fix an integrable categorical representation ofg_konC of degree1which is bounded above. For eachi2I there is a decreasing filtration C₁ C0DC ofC by

Dans le document ON THE CENTER OF QUIVER HECKE ALGEBRAS P. SHAN, M. VARAGNOLO, and E. VASSEROT (Page 11-48)