Algebraic approach to q,t-characters

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Advances in Mathematics 187 (2004) 1–52

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Algebraic approach to q; t-characters David Hernandez
E´cole Normale Supe´rieure - DMA, 45, Rue d’Ulm F-75230 Paris Cedex 05, France Received 1 April 2003; accepted 28 July 2003 Communicated by P. Etingof

Abstract Frenkel and Reshetikhin (in: Recent Developments in Quantum Affine Algebras and Related Topics, Contemporary Mathematics, Vol. 248, 1999, pp. 163–205) introduced qcharacters to study finite dimensional representations of the quantum affine algebra Uq ð#gÞ: In the simply laced case Nakajima (in: Physics and Combinatorics, Proceedings of the Nagoya 2000 International Workshop, World Scientific, Singapore, 2001, pp. 181–212; Preprint arXiv:math.QA/0105173) defined deformations of q-characters called q; t-characters. The definition is combinatorial but the proof of the existence uses the geometric theory of quiver varieties which holds only in the simply laced case. In this article we propose an algebraic general (non-necessarily simply laced) new approach to q; t-characters motivated by the deformed screening operators (Internat. Math. Res. Not. 2003 (8) (2003) 451). The tdeformations are naturally deduced from the structure of Uq ðg# Þ: the parameter t is analog to the central charge cAUq ð#gÞ: The q; t-characters lead to the construction of a quantization of the Grothendieck ring and to general analogues of Kazhdan–Lusztig polynomials in the same spirit as Nakajima did for the simply laced case. r 2003 Elsevier Inc. All rights reserved.

1. Introduction We suppose qAC
is not a root of unity. In the case of a semi-simple Lie algebra g; the structure of the Grothendieck ring RepðUq ðgÞÞ of finite dimensional representations of the quantum algebra Uq ðgÞ is well understood. It is analogous to the


Corresponding author. E-mail address: [email protected]. URL: http://www.dma.ens.fr/~dhernand.

0001-8708/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2003.07.016

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classical case q ¼ 1: In particular we have ring isomorphisms: RepðUq ðgÞÞCRepðgÞCZ½LW CZ½T1 ; y; Tn  deduced from the injective homomorphism of characters w: X wðV Þ ¼ dimðVl Þl; lAL

where Vl are weight spaces of a representation V and L is the weight lattice. For the general case of Kac–Moody algebras the picture is less clear. In the affine case Uq ðg# Þ; Frenkel and Reshetikhin [5] introduced an injective ring homomorphism of q-characters: 7 wq : RepðUq ðg# ÞÞ-Z½Yi;a 1pipn;aAC
¼ Y:

The homomorphism wq allows to describe the ring RepðUq ðg# ÞÞCZ½Xi;a iAI;aAC
; where the Xi;a are fundamental representations. It particular RepðUq ðg# ÞÞ is commutative. The morphism of q-characters hasTa symmetry property analogous to the classical action of the Weyl group Imðwq Þ ¼ iAI KerðSi Þ: Frenkel and Reshetikhin defined n S screening operators Si such that Imðwq Þ ¼ iAI KerðSi Þ (the result was proved by Frenkel and Mukhin for the general case in [6]). In the simply laced case Nakajima introduced t-analogues of q-characters [11,12]: it is a Z½t7 -linear map 7 7 wq;t : RepðUq ðg# ÞÞ#Z Z½t7 -Yt ¼ Z½Yi;a ; t iAI;aAC
;

which is a deformation of wq and multiplicative in a certain sense. A combinatorial axiomatic definition of q; t-characters is given. But the existence is non-trivial and is proved with the geometric theory of quiver varieties which holds only in the simply laced case. In [8] we introduced t-analogues of screening operators Si;t such that in the simply laced case: \ KerðSi;t Þ ¼ Imðwq;t Þ: iAI

It is a first step in the algebraic approach to q; t-characters proposed in this article: we define and construct q; t-characters in the general (non-necessarily simply laced) case. The motivation of the construction appears in the non-commutative structure # # Þ; the study of screening currents and of of the Cartan subalgebra Uq ðhÞCU q ðg deformed screening operators. In particular, this article proves a conjecture that

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Nakajima made for the simply laced case (Remark 3.10 in [12]): there exists a purely combinatorial proof of the existence of q; t-characters. As an application we construct a deformed algebra structure, an involution of the Grothendieck ring, and analogues of Kazhdan–Lusztig polynomials in the general case in the same spirit as Nakajima did for the simply laced case. This article is organized as follows: after some backgrounds in Section 2, we define 7 7 a deformed non-commutative algebra structure on Yt ¼ Z½Yi;a ; t iAI;aAC
(Section # 3): it is naturally deduced from the relations of Uq ðhÞCUq ðg# Þ (Theorem 3.11) by using the quantization in the direction of the central element c: In particular in the simply laced case it can be used to construct the deformed multiplication of Nakajima [12] (Proposition 3.16) and of Varagnolo–Vasserot [15] (Section 3.5.3). This picture allows us to introduce the deformed screening operators of [8] as commutators of Frenkel–Reshetikhin’s screening currents of [4] (Section 4). In [8] we gave explicitly the kernel of each deformed screening operator (Theorem 4.10). T In analogy to the classical case where Imðwq Þ ¼ iAI KerðSi Þ; we have to describe the intersection of the kernels of deformed screening operators. We introduce a completion of this intersection (Section 5.2) and give its structure in Proposition 5.17. It is easy to see that it is not too big (Lemma 5.7); but the point is to prove that it contains enough elements: it is the main result of our construction in Theorem 5.11 which is crucial for us. It is proved by induction on the rank n of g: We define a t-deformed algorithm (Section 5.7.1) analog to the Frenkel–Mukhin’s algorithm [6] to construct q; t-characters in the completion of Yt : An algorithm was also used by Nakajima in the simply laced case in order to compute the q; tcharacters for some examples [11] assuming they exist (which was geometrically proved). Our aim is different: we do not know a priori the existence in the general case. That is why we have to show that the algorithm is well-defined, never fails (Lemma 5.22) and gives a convenient element (Lemma 5.23). This construction gives q; t-characters for fundamental representations; we deduce from them the injective morphism of q; t-characters wq;t (Definition 6.1). We study the properties of wq;t (Theorem 6.2). Some of them are generalization of the axioms that Nakajima defined in the simply laced case [12]; in particular we have constructed the morphism of [12]. We have some applications: the morphism gives a deformation of the Grothendieck ring because the image of wq;t is a subalgebra for the deformed multiplication (Section 6.2). Moreover, we define an antimultiplicative involution of the deformed Grothendieck ring (Section 6.3); the construction of this involution is motivated by the new point view adopted in this paper: it is just replacing c by c in Uq ðg# Þ: In particular, we define constructively analogues of Kazhdan–Lusztig polynomials and a canonical basis (Theorem 6.9) motivated by the introduction of [12]. We compute explicitly the polynomials for some examples. In Section 7 we raise some questions: we conjecture that the coefficients of q; tcharacters are in N½t7 CZ½t7 : In the ADE-case it a result of Nakajima; we give an alternative elementary proof for the A-cases in Section 7.1. The cases G2 ; B2 ; C2 are

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also checked in the appendix. The cases F4 ; Bn ; Cn ðnp10Þ have been checked on a computer. We also conjecture that the generalized analogues to Kazhdan–Lusztig polynomials give at t ¼ 1 the multiplicity of simple modules in standard modules. We propose some generalizations and further applications which will be studied elsewhere. In the appendix we give explicit computations of q; t-characters for Lie algebras of rank 2. They are used in the proof of Theorem 5.11. For convenience of the reader we give at the end of this paper an index of the main notations introduced in the text.

2. Background 2.1. Cartan matrix A generalized Cartan matrix of rank n is a matrix C ¼ ðCi;j Þ1pi;jpn such that Ci;j AZ and ðiajÞ: Ci;i ¼ 2;

Ci;j p0;

Ci;j ¼ 0 3 Cj;i ¼ 0:

Let I ¼ f1; y; ng: We say that C is symmetrizable if there is a matrix D ¼ diagðr1 ; y; rn Þ ðri AN
Þ such that B ¼ DC is symmetric. Let qAC
be the parameter of quantization. In the following we suppose it is not a root of unity. z is an indeterminate. If C is symmetrizable, let qi ¼ qri ; zi ¼ zri and CðzÞ ¼ ðCðzÞi;j Þ1pi;jpn the matrix with coefficients in Z½z7  such that ðiajÞ: CðzÞi;j ¼ ½Ci;j z ;

CðzÞi;i ¼ ½Ci;i zi ¼ zi þ z1 i ; l

l

z where for lAZ we use the notation ½lz ¼ zzz 1 : In particular, the coefficients of CðzÞ are symmetric Laurent polynomials (invariant under z/z1 Þ: We define the diagonal matrix Di;j ðzÞ ¼ di;j ½ri z and the matrix BðzÞ ¼ DðzÞCðzÞ: In the following we suppose that C is of finite type, in particular detðCÞa0: In this case C is symmetrizable; if C is indecomposable there is a unique choice of ri AN
such that r1 4?4rn ¼ 1: We have Bi;j ðzÞ ¼ ½Bi;j z and BðzÞ is symmetric. See [1] or [9] for a classification of those finite Cartan matrices. C is said to be simply laced if r1 ¼ ? ¼ rn ¼ 1: In this case C and CðzÞ ¼ BðzÞ are symmetric. In the classification those matrices are of type ADE: PðzÞ Denote by UCQðzÞ the subgroup Z-linearly spanned by the Qðz 1 Þ such that

PðzÞAZ½z7 ; QðzÞAZ½z; the zeros of QðzÞ are roots of unity and Qð0Þ ¼ 1: It is a subring of QðzÞ; and for Rðzm ÞAU; mAZ we have Rðqm ÞAU and Rðqm ÞAC makes ˜ sense. It follows from Lemma 1.1 of [6] that CðzÞ has inverse CðzÞ with coefficients of the form RðzÞAU:

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2.2. Finite quantum algebras We refer to [14] for the definition of the finite quantum algebra Uq ðgÞ associated to a finite Cartan matrix, the definition and properties of the type 1-representations of Uq ðgÞ; the Grothendieck ring RepðUq ðgÞÞ and the injective ring morphism of characters w : RepðUq ðgÞÞ-Z½y7 i : 2.3. Quantum affine algebras The quantum affine algebra associated to a finite Cartan matrix C is a C-algebra 7
Uq ðg# Þ with generators x7 i;m ðiAI; mAZÞ; ki ðiAIÞ; hi;m ðiAI; mAZ Þ; central elements 1

#Þ c72 : We refer to [5] for the relations and the definition of the f7 i;m AUq ðg ðiAI; mAZÞ: In particular we will use the relations: ½hi;m ; hj;m0  ¼ dm;m0

1 cm  cm ½mBij q : m q  q1

One has an embedding Uq ðgÞCUq ðg# Þ and a Hopf algebra structure on Uq ðg# Þ (see [5] # # Þ is the C-subalgebra of Uq ðg# Þ for example). The Cartan subalgebra Uq ðhÞCU q ðg 7 generated by the hi;m ; c ðiAI; mAZ  f0gÞ: 2.4. Finite dimensional representations of Uq ðg# Þ A finite dimensional representation V of Uq ðg# Þ is called of type 1 if c acts as Id and V is of type 1 as a representation of Uq ðgÞ: Denote by RepðUq ðg# ÞÞ the Grothendieck ring of finite dimensional representations of type 1. The operators ff7 i;7m ; iAI; mAZg commute on a such V : So, we have a pseudo-weight space decomposition: V¼

M

Vg ;

gACI Z CI Z

where for g ¼ ðgþ ; g Þ; Vg is a simultaneous generalized eigenspace: 7 p Vg ¼ fxAV =(pAN; 8iAf1; y; ng; 8mAZ; ðf7 i;m  gi;m Þ :x ¼ 0g:

The g7 i;m are called pseudo-eigen values of V : Theorem 2.1 (Chari and Pressley [2,3]). Every simple representation V ARepðUq ðg# ÞÞ is a highest weight representation, that is to say there is v0 AV (highest weight vector)

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g7 i;m AC (highest weight) such that: V ¼ Uq ðg# Þ:v0 ;

1

c2 :v0 ¼ v0 7 f7 i;m :v0 ¼ gi;m v0 :

8iAI; mAZ; xþ i;m :v0 ¼ 0;

Moreover we have an I-uplet ðPi ðuÞÞiAI of (Drinfel’d-)polynomials such that Pi ð0Þ ¼ 1 and: g7 i ðuÞ ¼

X

7 g7 i;7m u ¼ qi

degðPi Þ

mAN

Pi ðuq1 i Þ AC½½u7  Pi ðuqi Þ

and ðPi ÞiAI parameterizes simple modules in RepðUq ðg# ÞÞ: Theorem 2.2 (Frenkel and Reshetikhin [5]). The eigenvalues gi ðuÞ7 AC½½u7  of a representation V ARepðUq ðg# ÞÞ have the form: g7 i ðuÞ ¼ qi

degðQi ÞdegðRi Þ

Qi ðuq1 i ÞRi ðuqi Þ ; Qi ðuqi ÞRi ðuq1 i Þ

where Qi ðuÞ; Ri ðuÞAC½u and Qi ð0Þ ¼ Ri ð0Þ ¼ 1: Note that the polynomials Qi ; Ri are uniquely defined by g: Denote by Qg;i ; Rg;i the polynomials associated to g: 2.5. q-characters 7 Let Y be the commutative ring Y ¼ Z½Yi;a iAI;aAC
:

Definition 2.3. For V ARepðUq ðg# ÞÞ a representation, the q-character wq ðV Þ of V is: wq ðV Þ ¼

X

dimðVg Þ

g

Y

lg;i;a mg;i;a

Yi;a

AY;

iAI;aAC


where for gACI Z CI Z ; iAI; aAC
the lg;i;a ; mg;i;a AZ are defined by: Qg;i ðzÞ ¼

Y


aAC

ð1  zaÞlg;i;a ;

Rg;i ðzÞ ¼

Y

ð1  zaÞmg;i;a :




aAC

Theorem 2.4 (Frenkel and Reshetikhin [5]). The map wq : RepðUq ðg# ÞÞ-Y is an injective ring homomorphism and the following diagram is

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commutative: wq

7 RepðUq ðg# ÞÞ ! Z½Yi;a iAI;aAC


kres

kb w

RepðUq ðgÞÞ !

Z½y7 i iAI

where b is the ring homomorphism such that bðYi;a Þ ¼ yi ðiAI; aAC
Þ: For

mAY

of

the

form

m¼

Q

iAI;aAC


u ðmÞ

Yi;ai;a

ðui;a ðmÞX0Þ;

denote Q

by

Vm ARepðUq ðg# ÞÞ the simple module with Drinfel’d polynomials Pi ðuÞ ¼ aAC
ð1  uaÞui;a ðmÞ : In particular for iAI; aAC
let Vi;a ¼ VYi;a and Xi;a ¼ wq ðVi;a Þ: The simple modules Vi;a are called fundamental representations. #ui;a ðmÞ

Q

Let Mm ¼ #iAI;aAC
Vi;a iAI;aAC


: It is called a standard module and his q-character is

u ðmÞ Xi;ai;a :

Corollary 2.5 (Frenkel and Reshetikhin [5]). The ring RepðUq ðg# ÞÞ is commutative and isomorphic to Z½Xi;a iAI;aAC
: Proposition 2.6 (Frenkel and Mukhin [6]). For iAI; aAC
; we have Xi;a A 7 Z½Yj;aq l jAI;lX0 : In particular for aAC
we have an injective ring homomorphism: 7 waq : Repa ¼ Z½Xi;aql iAI;lAZ -Ya ¼ Z½Yi;aq l iAI;lAZ :

For a; bAC
let ab;a : Repa -Repb and bb;a : Ya -Yb be the canonical ring homomorphism. As a consequence of Theorem 4.2 (or see [5,6]), we have bb;a 3waq ¼ wbq 3ab;a : In particular it suffices to study w1q : In the following denote Rep ¼ Rep1 ; Xi;l ¼ Xi;ql ; Y ¼ Y1 and wq ¼ w1q : Rep-Y:

3. Twisted polynomial algebras related to quantum affine algebras The aim of this section is to define the t-deformed algebra Yt and to describe its structure (Theorem 3.11). We define the Heisenberg algebra H; the subalgebra Yu CH½½h and eventually Yt as a quotient of Yu :

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3.1. Heisenberg algebras related to quantum affine algebras Definition 3.1. H is the C-algebra defined by generators ai ½m ðiAI; mAZ  f0gÞ; central elements cr ðr40Þ and relations ði; jAI; m; rAZ  f0gÞ: ½ai ½m; aj ½r ¼ dm;r ðqm  qm ÞBi;j ðqm Þcjmj : This definition is motivated by the structure of Uq ðg# Þ: in H the cr are algebraically # such that independent, but we have a surjective homomorphism from H to Uq ðhÞ 1 cr cr ai ½m/ðq  q Þhi;m and cr / r : P For jAI; mAZ we set yj ½m ¼ iAI C˜ i;j ðqm Þai ½mAH: Lemma 3.2. We have the Lie brackets in H ði; jAI; m; rAZÞ: ½ai ½m; yj ½r ¼ ðqmri  qri m Þdm;r di;j cjmj ; ½yi ½m; yj ½r ¼ dm;r C˜ j;i ðqm Þðqmrj  qmrj Þcjmj : Let pþ and p be the C-algebra endomorphisms of H such that ðiAI; m40; ro0Þ: pþ ðai ½mÞ ¼ ai ½m; p ðai ½mÞ ¼ 0;

pþ ðai ½rÞ ¼ 0;

p ðai ½rÞ ¼ ai ½r;

pþ ðcm Þ ¼ 0; p ðcm Þ ¼ 0:

They are well-defined because the relations are preserved. We set Hþ ¼ Imðpþ ÞCH and H ¼ Imðp ÞCH: Note that Hþ (resp. H Þ is the subalgebra of H generated by the ai ½m; iAI; m40 (resp. mo0Þ: So Hþ and H are commutative algebras, and Hþ CH CC½ai ½miAI;m40 : A product of the ai ½m; cr is called a H-monomial. Lemma 3.3. There is a unique C-linear endomorphism :: of H such that for all m Hmonomials : m :¼ pþ ðmÞp ðmÞ: In particular, there is a triangular decomposition HCHþ #C½cr r40 #H : Proof. The H-monomials span the C-vector space H; so the map is unique. But there are non-trivial linear combinations between them because of the relations of H: it suffices to show that for m1 ; m2 H-monomials the definition of :: is compatible with the relations ði; jAI; l; kAZ  f0gÞ: m1 ai ½kaj ½lm2  m1 aj ½lai ½km2 ¼ dk;l ðqk  qk ÞBi;j ðqk Þm1 cjkj m2 :

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We can conclude because Hþ and H are commutative and pþ ðm1 cjkj m2 Þ ¼ p ðm1 cjkj m2 Þ ¼ 0: &

3.2. The deformed algebra Yu 3.2.1. Construction of Yu Consider the C-algebra Hh ¼ H½½h and the application exp : hHh -Hh : For lAZ; iAI; introduce A˜ i;l ; Y˜ i;l AHh such that: ! ! X X m lm m lm h ai ½mq h ai ½mq A˜ i;l ¼ exp exp ; m40

Y˜ i;l ¼ exp

X

m40

! hm yi ½mqlm exp

m40

X

! hm yi ½mqlm :

m40

Note that A˜ i;l and Y˜ i;l are invertible in Hh : Recall the definition UCQðzÞ of Section 2.1. For RAU; introduce tR AHh : ! X 2m m tR ¼ exp h Rðq Þcm : m40

˜7 Definition 3.4. Yu is the Z-subalgebra of Hh generated by the Y˜ 7 i;l ; Ai;l ; tR ðiAI; lAZ; RAUÞ: In this section, we give properties of Yu and subalgebras of Yu which will be useful in Section 3.3.

3.2.2. Relations in Yu Lemma 3.5. We have the following relations in Yu ði; jAI l; kAZÞ: ˜ 1 A˜ i;l Y˜ j;k A˜ 1 i;l Yj;k ¼ tdi;j ðzri zri ÞðzðlkÞ þzðklÞ Þ ;

ð1Þ

˜ 1 Y˜ i;l Y˜ j;k Y˜ 1 i;l Yj;k ¼ tC˜ j;i ðzÞðzrj zrj ÞðzðlkÞ þzðklÞ Þ ;

ð2Þ

˜ 1 A˜ i;l A˜ j;k A˜ 1 i;l Aj;k ¼ tBi;j ðzÞðz1 zÞðzðlkÞ þzðklÞ Þ :

ð3Þ

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Proof. For A; BAhHh such that ½A; BAhC½cr r40 ; we have: expðAÞexpðBÞ ¼ expðBÞexpðAÞexpð½A; BÞ: So a straightforward computation and Lemma 3.2 give the result.

&

3.2.3. Commutative subalgebras of Hh The C-algebra endomorphisms pþ ; p of H are naturally extended to C-algebra endomorphisms of Hh : As Yu CHh ; we have by restriction the Z-algebra morphisms p7 : Yu -Hh : Introduce Y ¼ pþ ðYu ÞCHþ ½½h: In this Section 3.2.3 we study Y: In particular, we will see in Proposition 3.8 that the notation Y is consistent with Section 2.5. For iAI; lAZ; denote ! X Y 7 ¼ pþ ðY˜ 7 Þ ¼ exp 7 hm yi ½mqlm ; i;l

i;l

m40

A7 i;l

¼ pþ ðA˜ 7 i;l Þ ¼ exp 7

X

! m

h ai ½mq

lm

:

m40

Lemma 3.6. For iAI; lAZ; we have 0 10 Y Y Ai;l ¼ Yi;lri Yi;lþri @ Yj;l1 A@ j=Cj;i ¼1

10 1 1 A@ Yj;lþ1 Yj;l1

j=Cj;i ¼2

Y

1 1 1 A : Yj;lþ2 Yj;l1 Yj;l2

j=Cj;i ¼3

In particular, Y is generated by the Yi;l7 ðiAI; lAZÞ: The formula already appeared in [5]. We have the last point because Yu is ˜7 generated by the Y˜ 7 i;l ; Ai;l ; tR : We need a technical lemma to describe Y: Lemma 3.7. Let J ¼ f1; y; rg and let L be the polynomial commutative algebra L ¼ C½lj;m jAJ;mX0 : For R ¼ ðR1 ; y; Rr ÞAUr ; consider: ! X m m LR ¼ exp h Rj ðq Þlj;m AL½½h: jAJ;m40

Then the ðLR ÞRAUr are C-linearly independent. In particular, the Lj;l ¼ Lð0;y;0;zl ;0;y;0Þ ðjAJ; lAZÞ are C-algebraically independent. Proof. Suppose we have a linear combination ðmR AC; only a finite number of mR a0Þ: X mR LR ¼ 0: RAUr

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The coefficients of hL in LR are of the form Rj1 ðql1 ÞL1 Rj2 ðql2 ÞL2 yRjN ðqlN ÞLN lLj11;l1 lLj22;l2 ylLjNN;lN where l1 L1 þ ? þ lN LN ¼ L: So for NX0; j1 ; y; jN AJ; l1 ; y; lN 40; L1 ; y; LN X0 we have: X mR Rj1 ðql1 ÞL1 Rj2 ðql2 ÞL2 yRjN ðqlN ÞLN ¼ 0: RAUr

If we fix L2 ; y; LN ; we have for all L1 ¼ lX0: X X al1 mR Rj2 ðql2 ÞL2 yRjN ðqlN ÞLN ¼ 0: RAUr =Rj1 ðql1 Þ¼a1

a1 AC

We get a Van der Monde system which is invertible, so for all a1 AC: X mR Rj2 ðql2 ÞL2 yRjN ðqlN ÞlN ¼ 0: RAUr =Rj1 ðql1 Þ¼a1

By induction we get for r0 pN and all a1 ; y; ar0 AC: X mR Rjr0 þ1 ðqlr0 þ1 ÞLr0 þ1 yRjN ðqlN ÞLN ¼ 0: RAUr =Rj1 ðql1 Þ¼a1 ;y;Rj 0 ðqlr0 Þ¼ar0 r

0

And so for r ¼ N: X

mR ¼ 0:

RAUr =Rj1 ðql1 Þ¼a1 ;y;RjN ðqlN Þ¼aN

Let be SX0 such that for all mR ; mR0 a0; jAJ we have Rj  R0j ¼ 0 or Rj  R0j has at most S  1 roots. We set N ¼ Sr and ððj1 ; l1 Þ; y; ðjS ; lS ÞÞ ¼ ðð1; 1Þ; ð1; 2Þ; y; ð1; SÞ; ð2; 1Þ; y; ð2; SÞ; ð3; 1Þ; y; ðr; SÞÞ: We get for all aj;l AC ðjAJ; 1plpSÞ: X mR ¼ 0: RAUr =8jAJ;1plpS;Rj ðql Þ¼aj;l

It suffices to show that there is at most one term in this sum. But consider P; QAU such that for all 1plpS; Pðql Þ ¼ P0 ðql Þ: As q is not a root of unity the ql are different and P  P0 has S roots, so is 0. Q uj;l For the last assertion, we can write a monomial jAJ;lAZ Lj;l ¼ P LP l l : In particular, there is no trivial linear combination between u z lAZ 1;l

;y;

u z lAZ r;l

those monomials.

&

It follows from Lemmas 3.6 and 3.7:

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Proposition 3.8. The Yi;l AY are Z-algebraically independent and generate the Zalgebra Y: In particular, Y is the commutative polynomial algebra Z½Yi;l7 iAI;lAZ : The A1 i;l AY are Z-algebraically independent. In particular, the subalgebra of Y 1 generated by the A1 i;l is the commutative polynomial algebra Z½Ai;l iAI;lAZ : 3.2.4. Generators of Yu The C-linear endomorphism :: of H is naturally extended to a C-linear endomorphism of Hh : As Yu CHh ; we have by restriction a Z-linear morphism :: from Yu to Hh : We say that mAYu is a Yu -monomial if it is a product of generators A˜ 7 ; Y˜ 7 ; tR : In the following, for a product of non-commuting terms, denote Qi;l- i;l s¼1::S Us ¼ U1 U2 yUS : Lemma 3.9. The algebra Yu is generated by the Y˜ 7 i;l ; tR ðiAI; lAZ; RAUÞ: Proof. Let be iAI; lAZ: It follows from Proposition 3.8 that pþ ðA˜ i;l Þ is of the form Q Q Q u ˜ ui;l pþ ðA˜ i;l Þ ¼ iAI;lAZ Yi;li;l and that : m :¼: iAI Yi;l : : So it suffices to show lAZ that for m a Yu -monomial, there is a unique Rm AU such that m ¼ tRm : m : : Let us Q ˜ 7 ˜7 write m ¼ tR s¼1::S Us where Us AfAi;l ; Yi;l giAI;lAZ are generators. Then : m :¼

Y

!

Y

pþ ðUs Þ

s¼1::S

! p ðUs Þ :

s¼1::S

And we can conclude because it follows from the proof of Lemma 3.5 that for 1ps; s0 pS; there is Rs;s0 AU such that pþ ðUs Þp ðUs0 Þ ¼ tRs;s0 p ðUs0 Þpþ ðUs Þ:

&

In particular, it follows from this proof that : Yu : CYu :

3.3. The deformed algebra Yt P Denote by Zððz1 ÞÞ the ring of series of the form P ¼ rpRP Pr zr where RP AZ and the coefficients Pr AZ: Recall the definition U of Section 2.1. We have an embedding UCZððz1 ÞÞ by expanding Qðz11 Þ in Z½½z1  for QðzÞAZ½z such that Qð0Þ ¼ 1: So we can introduce maps

pr : U-Z;

P¼

X kpRP

Pk zk /Pr :

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Definition 3.10. We define Yt (resp. Ht Þ as the algebra quotient of Yu (resp. Hh Þ by relations tR ¼ tR0

if p0 ðRÞ ¼ p0 ðR0 Þ:

˜7 We keep the notations Y˜ 7 i;l ; Ai;l for their image in Yt : Denote by t the image of P t1 ¼ expð m40 h2m cm Þ in Yt : As p0 is additive, the image of tR in Yt is tpo ðRÞ : In ˜7 7 particular Yt is generated by the Y˜ 7 i;l ; Ai;l ; t : As the defining relations of Ht involve only the cl and pþ ðcl Þ ¼ p ðcl Þ ¼ 0; the algebra endomorphisms pþ ; p of Ht are well-defined. So we can define  þ  Hþ in the same way as in Section 3.2.3 and :: a C-linear t ; Ht ; Yt ; Yt endomorphism of Ht as in Section 3.2.4. The Z½t7 -subalgebra Yt CHt verifies : Yt : CYt (proof of Lemma 3.9). We have Yþ t CY: We say that mAYt (resp. mAYÞ 7 is a Yt -monomial (resp. a Y-monomial) if it is a product of the generators Y˜ 7 i;m ; t 7 (resp. Yi;m Þ: Theorem 3.11. The algebra Yt is defined by generators Y˜ 7 i;l ðiAI; lAZÞ; central elements t7 and relations ði; jAI; k; lAZÞ: Y˜ i;l Y˜ j;k ¼ tgði;l;j;kÞ Y˜ j;k Y˜ i;l ; where g : ðI ZÞ2 -Z is given by: X gði; l; j; kÞ ¼ pr ðC˜ j;i ðzÞÞðdlk;rj r  dlk;rrj þ dlk;rj r þ dlk;rj þr Þ: rAZ

Proof. As the image of tR in Yt is tpo ðRÞ ; we can deduce the relations from Lemma 3.5. For example we use formula 2: p0 ðC˜ j;i ðzÞðzrj  zrj ÞðzðlkÞ þ zðklÞ ÞÞ ¼ gði; l; j; kÞ: 7 It follows from Lemma 3.6 that Yt is generated by the Y˜ 7 i;l ; t : It follows from Lemma 3.7 that the tR AYu ðRAUÞ are Z-linearly independent. So the Z-algebra Z½tR RAU is defined by generators ðtR ÞRAU and relations tRþR0 ¼ tR tR0 for R; R0 AU: In particular the image of Z½tR RAU in Yt is Z½t7 : Let A be the classes of Yt -monomials modulo tZ : So we have: X Z½t7 :m ¼ Yt : mAA

P We prove the sum is direct: suppose we have a linear combination mAA lm ðtÞm ¼ 0 where lm ðtÞAZ½t7 : We saw in Proposition 3.8 that YCZ½Yi;l7 iAI;lAZ : So lm ð1Þ ¼ 0 P ð1Þ ð1Þ 7 and lm ðtÞ ¼ ðt  1Þlð1Þ m ðtÞ where lm ðtÞAZ½t : In particular mAA lm ðtÞ ðtÞm ¼ 0

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and we get by induction lm ðtÞAðt  1Þr Z½t7  for all rX0: This is possible if and only if all lm ðtÞ ¼ 0: & In the same way using the last assertion of Proposition 3.8, we have: Proposition 3.12. The sub Z½t7 -algebra of Yt generated by the A˜ 1 i;l is defined by generators A˜ 1 ; t7 ðiAI; lAZÞ and relations: i;l

aði;l;j;kÞ ˜ 1 ˜ 1 ˜ 1 Aj;k Ai;l ; A˜ 1 i;l Aj;k ¼ t

where a : ðI ZÞ2 -Z is given by aði; l; i; kÞ ¼ 2ðdlk;2ri þ dlk;2ri Þ; X ðdlk;ri þr þ dlk;ri þr Þ aði; l; j; kÞ ¼ 2

ðif iajÞ:

r¼Ci;j þ1;Ci;j þ3;y;Ci;j 1

Moreover, we have the following relations in Yt : A˜ i;l Y˜ j;k ¼ tbði;l;j;kÞ Y˜ j;k A˜ i;l ; where b : ðI ZÞ2 -Z is given by bði; l; j; kÞ ¼ 2di;j ðdlk;ri þ dlk;ri Þ: 3.4. Notations and technical complements In this section, we study some technical properties of the Y-monomials and the Yt -monomials which will be used in the following. Denote by A the set of Ymonomials. It is a Z-basis of Y (Proposition 3.8). Let us define an analog Z½t7 -basis of Yt : denote A0 the set of Yt -monomials of the form m ¼: m : : It follows from Theorem 3.11 that: Yt ¼

M

Z½t7 m:

mAA0

The map p : A0 -A defined by pðmÞ ¼ pþ ðmÞ is a bijection. In the following we identify A and A0 : In particular, we have an embedding YCYt and an isomorphism of Z½t7 -modules Y#Z Z½t7 CYt : We P say that w1 AYt has P the same monomials as w2 AY if in the decompositions w1 ¼ mAA lm ðtÞm; w2 ¼ mAA mm m we have lm ðtÞ ¼ 03mm ¼ 0: Q u ðmÞ For m a Y-monomial we set ui;l ðmÞAZ such that m ¼ iAI;lAZ Yi;li;l and ui ðmÞ ¼ P lAZ ui;l ðmÞ: For m a Yt -monomial, we set ui;l ðmÞ ¼ ui;l ðpþ ðmÞÞ and ui ðmÞ ¼ ui ðpþ ðmÞÞ: Note that ui;l is invariant by multiplication by t and compatible with the

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identification of A and A0 : Section 3.3 implies that for iAI; lAZ and m a Yt monomial we have: A˜ i;l m ¼ t2ui;lri ðmÞþ2ui;lþri ðmÞ mA˜ i;l : Denote by Bi CA the set of i-dominant T Y-monomials, that is to say mABi if 8lAZ; ui;l ðmÞX0: For JCI denote by BJ ¼ iAJ Bi the set of J-dominant Y-monomials. In particular, B ¼ BI is the set of dominant Y-monomials. We recall that we can define a partial ordering on A by putting mpm0 if there is a 0 Y-monomial M which is a product of A7 i;l ðiAI; lAZÞ such that m ¼ Mm (see for example [8]). A maximal (resp. lowest, higher, etc.) weight Y-monomial is a maximal (resp. minimal, higher, etc.) element of A for this ordering. We deduce from pþ a partial ordering on the Yt -monomials. Following [6], a Y-monomial m is said to be right negative if the factors Yj;l appearing in m; for which l is maximal, have negative powers. A product of right negative Y-monomials is right negative. It follows from Lemma 3.6 that the A1 i;l are right negative. A Yt -monomial is said to be right negative if pþ ðmÞ is right negative. Lemma 3.13. Let ði1 ; l1 Þ; y; ðiK ; lK Þ be in ðI ZÞK : For UX0; the set of the m ¼ Q vik ;lk ðmÞ ðvik ;lk ðmÞX0Þ such that miniAI;kAZ ui;k ðmÞX  U is finite. k¼1yK Aik ;lk Proof. Suppose it is not the case: let be ðmp ÞpX0 such that miniAI;kAZ ui;k ðmp ÞX  U P but ! þN: So there is at least one k such that k¼1yK vik ;lk ðmp Þ p-N vik ;lk ðmp Þ p-N ! þN: Denote by R the set of such k: Among those kAR; such that lk is maximal suppose that rik is maximal (recall the definition of ri in Section 2.1). In particular, we have uik ;lk þrik ðmp Þ ¼ vik ;lk ðmp Þ þ f ðpÞ where f ðpÞ depends only of the vik0 ;lk0 ðmp Þ; k0 eR: In particular, f ðpÞ is bounded and uik ;lk þrik ðmp Þ p-N ! N: & Lemma 3.14. For MAB; KX0 the following set is finite: 1 fMA1 i1 ;l1 yAiR ;lR =RX0; l1 ; y; lR XKg-B:

Proof. Let us write M ¼ Yi1 ;l1 yYiR ;lR such that l1 ¼ minr¼1yR lr ; lR ¼ maxr¼1yR lr Q v and consider m in the set. It is of the form m ¼ MM 0 where M 0 ¼ iAI;lXK Ai;l i;l ðvi;l X0Þ: Let L ¼ maxflAZ=(iAI; ui;l ðM 0 Þo0g: M 0 is right negative so for all iAI; Q v l4L ) vi;l ¼ 0: But m is dominant, so LplR : In particular M 0 ¼ iAI;KplplR Ai;l i;l : It suffices to prove that the vi;l ðmr Þ are bounded under the condition m dominant. This follows from Lemma 3.13. &

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3.5. Presentations of deformed algebras Our construction of Yt using Hh (Section 3.3) is a ‘‘concrete’’ presentation of the deformed structure. Let us look at another approach: in this section we define two bicharacters N; Nt related to basis of Yt : All the information of the multiplication of Yt is contained is those bicharacters because we can construct a deformed
multiplication on the ‘‘abstract’’ Z½t7 -module Y#Z Z½t7  by putting for m1 ; m2 AA Y-monomials (for example with NÞ: m1
m2 ¼ tNðm1 ;m2 ÞNðm2 ;m1 Þ m2
m1 : Those presentations appeared earlier in the literature [12,15] for the simply laced case. In particular, this section identifies our approach with those articles and gives an algebraic motivation of the deformed structures of [12,15] related to the structure of Uq ðg# Þ: 3.5.1. The bicharacters N and Nt It follows from the proof of Lemma 3.9 that for m a Yt -monomial, there is NðmÞAZ such that m ¼ tNðmÞ : m : : For m1 ; m2 Yt -monomials we define Nðm1 ; m2 Þ ¼ Nðm1 m2 Þ  Nðm1 Þ  Nðm2 Þ: We have NðYi;l Þ ¼ NðAi;l Þ ¼ 0: Note that for a; bAZ we have Nðta mÞ ¼ a þ NðmÞ and Nðta m1 ; tb m2 Þ ¼ Nðm1 ; m2 Þ: In particular the map N : A A-Z is welldefined and independent of the choice of a representant in p1 þ ðAÞ: Lemma 3.15. The map N : A A-Z is a bicharacter, that is to say for m1 ; m2 ; m3 AA; we have: Nðm1 m2 ; m3 Þ ¼ Nðm1 ; m3 Þ þ Nðm2 ; m3 Þ; Nðm1 ; m2 m3 Þ ¼ Nðm1 ; m2 Þ þ Nðm1 ; m3 Þ: Moreover for m1 ; y; mk Yt -monomials, we have X

Nðm1 m2 ymk Þ ¼ Nðm1 Þ þ Nðm2 Þ þ ? þ Nðmk Þ þ

Nðmi ; mj Þ:

1piojpk

Proof. It is a consequence of the relations ðm1 ; m2 Yt -monomials): p ðm1 Þpþ ðm2 Þ ¼ tNðm1 ;m2 Þ pþ ðm2 Þp ðm1 Þ:

&

Q u ðmÞ For m a Yt -monomial and lAZ; denote pl ðmÞ ¼ jAI Y˜ j;lj;l : It is well-defined because for i; jAI and lAZ we have Y˜ i;l Y˜ j;l ¼ Y˜ j;l Y˜ i;l (Theorem 3.11). For m a Yt Q ˜ monomial denote m ˜ ¼ lAZ pl ðmÞ; and At the set of Yt -monomials of the form m:

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From Theorem 3.11 there is a unique Nt ðmÞAZ such that m ¼ tNt ðmÞ m; ˜ and: Yt ¼

M

Z½t7 m:

mAAt

We define Nt : A A-Z as N and we have results analogues result of Lemma 3.15 for Nt because ðm1 ; m2 Yt -monomials): X Nt ðm1 ; m2 Þ ¼ ðNðpl ðm1 Þ; pl 0 ðm2 ÞÞ  Nðpl 0 ðm2 Þ; pl ðm1 ÞÞÞ: l4l 0

3.5.2. Presentation related to the basis At and identification with [12] We suppose we are in the ADE-case. Q Q y0 v0 y v Let m1 ¼: iAI;lAZ Y˜ i;li;l A˜ i;l i;l :; m2 ¼: iAI;lAZ Y˜ i;li;l A˜ i;l i;l : AYt : We set m1y ¼: 0 Q Q y ˜ yi;l ˜ yi;l iAI;lAZ Yi;l : and m2 ¼: iAI;lAZ Yi;l : : Proposition 3.16. We have Nt ðm1 ; m2 Þ ¼ Nt ðm1y ; m2y Þ þ 2dðm1 ; m2 Þ; where X X dðm1 ; m2 Þ ¼ vi;lþ1 u0i;l þ yi;lþ1 v0i;l ¼ ui;lþ1 v0i;l þ vi;lþ1 y0i;l ; iAI;lAZ

iAI;lAZ

ui;l ¼ yi;l  vi;l1  vi;lþ1 þ 0 j=Ci;j ¼1 vj;l :

where P

P

j=Ci;j ¼1 vj;l

u0i;l ¼ y0i;l  v0i;l1  v0i;lþ1 þ

and

Proof. First note that we have ðiAI; lAZÞ: 1 1 Nt ðYi;l ; A1 i;l1 Þ ¼ Nt ðAi;lþ1 ; Yi;l Þ ¼ Nt ðAi;lþ1 ; Yi;l Þ ¼ 2; 1 1 1 Nt ðYi;lþ1 ; Yi;l1 Þ ¼ Nt ðA1 i;lþ1 ; Ai;l1 Þ ¼ 2:

We have Nt ðm1 ; m2 Þ ¼ A þ B þ C þ D where A ¼ Nt ðm1y ; m2y Þ; X X yi;l v0j;k Nt ðYi;l ; A1 yi;l v0i;l1 ; B¼ j;k Þ ¼ 2 i;jAI;l;kAZ

C¼

X

iAI;lAZ

vi;l y0j;k Nt ðA1 i;l ; Yj;k Þ

i;jAI;l;kAZ

D¼

X

X

¼2

iAI;lAZ 1 vi;l v0j;k Nt ðA1 i;l ; Aj;k Þ

i;jAI;l;kAZ

¼ 2

vi;lþ1 y0i;l ;

X

iAI;lAZ

ðvi;lþ1 v0i;l1 þ vi;l vi;l 0 Þ þ 2

X Cj;i ¼1;lAZ

And we have the announced formula for B þ C þ D: &

vi;lþ1 v0j;l :

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The bicharacter d was introduced for the ADE-case by Nakajima in [12] motivated by geometry. It particular this Proposition 3.16 gives a new motivation for this deformed structure. 3.5.3. Presentation related to the basis A and identification with [15] Lemma 3.17. For m1 ; m2 AA; we have X Nðm1 ; m2 Þ ¼ ui;l ðm1 Þuj;k ðm2 ÞððC˜ j;i ðzÞÞrj þlk  ðC˜ j;i ðzÞÞrj þlk Þ: i;jAI;l;kAZ

Proof. Indeed we have Y˜ i;l Y˜ j;k ¼ tC˜ j;i ðzÞzkl ðzrj zrj Þ : Y˜ i;l Y˜ j;k : : ˜ In sl2 -case we have CðzÞ ¼ z þ z1 and CðzÞ ¼ zþz1 1 ¼ Y˜ l Y˜ k ¼ ts : Y˜ l Y˜ k : where

& P

rX0

ð1Þr z2r1 : So

s ¼ 0 if l  k ¼ 1 þ 2r; rAZ; s ¼ 0 if l  k ¼ 2r; r40; s ¼ 2ð1Þrþ1 s ¼ 1

if l  k ¼ 2r; ro0;

if l ¼ k:

It is analogous to the multiplication introduced for the ADE-case by Varagnolo– Vasserot (see the notations in [15]): Yi;l Yj;m ¼ t

2ezl o ;zm o 2ezm o ;zl o i

j

j

i

Yj;m Yi;l ;

el;m ¼ p0 ððz1 O1 ðlÞjmÞÞ:

But we have ezl oi ;zm oj ¼ ðC˜ i;j ðzÞÞlþ1m and if we set e0l;m ¼ p0 ððzO1 ðlÞjmÞÞ then we and have e0 l m ¼ ðC˜ i;j ðzÞÞ z oi ;z oj

l1m

ezl oi ;zm oj  e0zl oi ;zm oj ¼ NðYi;l ; Yj;m Þ:

4. Deformed screening operators Motivated by the screening currents of [4] we give in this section a ‘‘concrete’’ approach to deformations of screening operators. In particular the t-analogues of screening operators defined in [8] will appear as commutators in Hh : Let us begin with some background about classical screening operators.

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4.1. Reminder: classical screening operators [5,6] 4.1.1. Classical screening operators and symmetry property of q-characters Definition 4.1. The ith-screening operator is the Z-linear map defined by Si : Y-Yi ¼ P 8mAA; Si ðmÞ ¼

L

lAZ Y:Si;l ; Y:ðS i;lþ2ri  Ai;lþri :Si;l Þ lAZ

X lAZ

ui;l ðmÞSi;l :

Note that the ith-screening operator can also be defined as the derivation such that Si ð1Þ ¼ 0 and Si ðYj;l Þ ¼ di;j Yi;l :Si;l ðjAI; lAZÞ: Theorem 4.2 (Frenkel and Reshetikhin [5]; Frenkel and Mukhin [6]). The image of wq : Z½Xi;l iAI;lAZ -Y is: \ Imðwq Þ ¼ KerðSi Þ: iAI W It is analogous to the classical symmetry property of w: ImðwÞ ¼ Z½y7 i iAI :

4.1.2. Structure of the kernel of Si Let Ki ¼ KerðSi Þ: It is a Z-subalgebra of Y: Theorem 4.3 (Frenkel and Reshetikhin [5]; Frenkel and Mukhin [6]). The Z7 subalgebra Ki of Y is generated by the Yi;l ð1 þ A1 i;lþri Þ; Yj;l ðjai; lAZÞ: For mABi ; let Ei ðmÞ ¼ m

Q

lAZ ð1

ui;l ðmÞ þ A1 AKi : i;lþri Þ

Corollary 4.4. The Z-module Ki is freely generated by the Ei ðmÞ ðmABi Þ:

4.1.3. Examples in the sl2 -case We suppose in this section that g ¼ sl2 : For mAB; let LðmÞ ¼ wq ðVm Þ be the q-character of the Uq ðsl#2 Þ-irreducible representation of highest weight m: In particular LðmÞAK and K ¼ "mAB ZLðmÞ: In [5] an explicit formula for LðmÞ is given: a sCZ is called a 2-segment if s is of the form s ¼ fl; l þ 2; y; l þ 2kg: Two 2-segment are said to be in special position if their union is a 2-segment that properly contains each of them. All finite subset of Z with multiplicity ðl; ul ÞlAZ ðul X0Þ can be broken in a unique way into a union of 2-segments which are not in pairwise special position.

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Q Q For mAB we decompose m ¼ j lAsj Yl AB where the ðsj Þj is the decomposition Q Q of the ðl; ul ðmÞÞlAZ : We have LðmÞ ¼ j Lð lAsj Yl Þ: So it suffices to give the formula for a 2-segment: 1 LðYl Ylþ2 Ylþ4 yYlþ2k Þ ¼ Yl Ylþ2 Ylþ4 yYlþ2k þ Yl Ylþ2 yYlþ2ðk1Þ Ylþ2ðkþ1Þ 1 1 1 1 1 þYl Ylþ2 yYlþ2ðk2Þ Ylþ2k Ylþ2ðkþ1Þ þ ? þ Ylþ2 Ylþ2 yYlþ2ðkþ1Þ :

We say that m is irregular if there are j1 aj2 such that: sj1 Csj2 and sj1 þ 2Csj2 : Lemma 4.5 (Frenkel and Reshetikhin [5]). There is a dominant Y-monomial other than m in LðmÞ if and only if m is irregular. 4.1.4. Complements: another basis of Ki Let us go back to the general case. Let Ysl2 ¼ Z½Yl7 lAZ the ring Y for the sl2 -case. Let i be in I and for 0pkpri  1; let ok : A-Ysl2 be the map defined by ok ðmÞ ¼ Q ui;kþlri ðmÞ and nk : Z½ðYl1 Ylþ1 Þ1 lAZ -Y be the ring homomorphism such that lAZ Yl nk ððYl1 Ylþ1 Þ1 Þ ¼ A1 i;kþlri : For mABi ; ok ðmÞ is dominant in Ysl2 and so we can define Lðok ðmÞÞ (see Section 4.1.3). We have Lðok ðmÞÞok ðmÞ1 AZ½ðYl1 Ylþ1 Þ1 lAZ : We introduce Y Li ðmÞ ¼ m nk ðLðok ðmÞÞok ðmÞ1 ÞAKi : 0pkpri 1

In analogy with Corollary 4.4 we have Ki ¼ "mABi ZLi ðmÞCZðBi Þ : 4.2. Screening currents Following [4], for iAI; lAZ; introduce S˜ i;l AHh : ! ! X X a ½m a ½m i i S˜ i;l ¼ exp hm m qlm exp hm m qlm : qi  qm qi  qm i i m40 m40 Lemma 4.6. We have the following relations in Hh : A˜ i;l S˜ i;lri ¼ tz2ri 1 S˜ i;lþri ; S˜ i;l A˜ j;k ¼ tCi;j ðzÞðzðklÞ þzðlkÞ Þ A˜ j;k S˜ i;l ; S˜ i;l Y˜ j;k ¼ tdi;j ðzðklÞ þzðlkÞ Þ Y˜ j;k S˜ i;l : The proof is a straightforward computation as for Lemma 3.5.

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4.3. Deformed bimodules In this section we define and study a t-analogue Yi;t of the module Yi : For iAI; let Yi;u be the sub Yu -left-module of Hh generated by the S˜ i;l ðlAZÞ: It follows from Lemma 4.6 that ðS˜ i;l Þri plori generate Yi;u and that it is also a subbimodule of Hh : Denote by S˜ i;l AHt the image of S˜ i;l AHh in Ht : Definition 4.7. Yi;t is the sub left-module of Ht generated by the S˜ i;l ðlAZÞ: In particular, it is the image of Yi;u in Ht : It follows from Lemma 4.6 that for lAZ; we have in Yi;t : A˜ i;l S˜ i;lri ¼ t1 S˜ i;lþri : So Yi;t is generated by the ðS˜ i;l Þri plori : It follows from Lemma 4.6 that for m a Yt -monomial S˜ i;l :m ¼ t2ui;l ðmÞ m:S˜ i;l and so Yi;t a sub Yt -bimodule of Ht : Proposition 4.8. The Yt -left-module Yi;t is freely generated by ðS˜ i;l Þri plori : Proof. We saw that ðS˜ i;l Þri plori generate Yi;t : We prove they are Yt -linearly independent: for ðR1 ; y; Rn ÞAUn ; introduce: X

YR1 ;y;Rn ¼ exp

! h yj ½mRj ðq Þ AHþ t : m

m

m40;jAI

It follows from Lemma 3.7 that the ðYR ÞRAUn are Z-linearly independent. Note that we have pþ ðYi;t ÞC"RAUn ZYR and that Y ¼ "RAZ½z7 n ZYR : Suppose we have a linear combination ðlr AYt Þ lri S˜ i;ri þ ? þ lri 1 S˜ i;ri 1 ¼ 0: Introduce mk;R AZ such P that pþ ðlk Þ ¼ RAZ½z7 n mk;R YR and Ri;k ¼ ðR1i;k ðzÞ; y; Rni;k ðzÞÞAUn such that pþ ðS˜ i;k Þ ¼ YR : If we apply pþ to the linear combination, we get i;k

X RAZ½z7 n ;ri pkpri 1

mk;R YR YRi;k ¼ 0

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and we have for all R0 AU: X

mk;R0 Ri;k ¼ 0:

ri pkpri 1=R0 Ri;k AZ½z7 n

Suppose we have ri pk1 ak2 pri  1 such that R0  Ri;k1 ; R0  Ri;k2 AZ½z7 n : So P Ri;k1  Ri;k2 AZ½z7 n : But ai ½m ¼ jAI Cj;i ðqm Þyj ½m; so for jAI: Cj;i ðzÞ

zk1  zk2 ¼ ðRji;k1 ðzÞ  Rji;k2 ðzÞÞAZ½z7 : zri  zri ri

ri

k1

k2

ðz þz Þðz z Þ z 2 In particular for j ¼ i we have Ci;i ðzÞzzri1z AZ½z7 : This is ri ¼ zri zri impossible because jk1  k2 jo2ri : So we have only one term in the sum and all mk;R ¼ 0: So pþ ðlk Þ ¼ 0; and lk Aðt  1ÞYt : We have by induction for all m40; lk Aðt  1Þm Yt : It is possible if and only if lk ¼ 0: & k

k

Denote by Yi the Y-bimodule pþ ðYi;t Þ: It is consistent with Section 4.1. 4.4. t-analogues of screening operators We introduced t-analogues of screening operators in [8]. The picture of the last section enables us to define them from a new point of view. For m a Yt -monomial, we have ½S˜ i;l ; m ¼ S˜ i;l m  mS˜ i;l ¼ ðt2ui;l ðmÞ  1ÞmS˜ i;l ¼ tui;l ðmÞ ðt  t1 Þ½ui;l ðmÞt mS˜ i;l : So for lAYt we have ½S˜ i;l ; lAðt2  1ÞYi;t ; and ½S˜ i;l ; la0 only for a finite number of lAZ: So we can define: Definition 4.9. The ith t-screening operator is the map Si;t : Yt -Yi;t such that ðlAYt Þ: Si;t ðlÞ ¼

t2

1 X ˜ ½Si;l ; lAYi;t :  1 lAZ

In particular, Si;t is Z½t7 -linear and a derivation. It is the map of [8]. For m a Yt monomial, we have: pþ ðSi;t ðmÞÞ ¼ pþ ðtui;l ðmÞ1 ½ui;l ðmÞt Þpþ ðmS˜ i;l Þ ¼ ui;l ðmÞpþ ðmS˜ i;l Þ: and so pþ 3Si;t ¼ Si 3pþ :

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4.5. Kernel of deformed screening operators 4.5.1. Structure of the kernel We proved in [8] a t-analogue of Theorem 4.3: Theorem 4.10 (Hernandez [8]). The kernel of the ith t-screening operator Si;t is the ˜7 Z½t7 -subalgebra of Yt generated by the Y˜ i;l ð1 þ tA˜ 1 i;lþri Þ; Yj;l ðjai; lAZÞ: Proof. For the first inclusion we compute: 2 ˜ ˜ ˜ ˜ ˜ 1 Si;t ðY˜ i;l ð1 þ tA˜ 1 i;lþri ÞÞ ¼ Yi;l Si;l þ tYi;l Ai;lþri ðt ÞSi;lþ2ri ¼ 0:

For the other inclusion we refer to [8].

&

Let Ki;t ¼ KerðSi;t Þ: It is a Z½t7 -subalgebra of Yt : In particular, we have pþ ðKi;t Þ ¼ Ki (consequence of Theorems 4.3 and 4.10). Q For mABi introduce: (recall that lAZ Ul means yU1 U0 U1 U2 yÞ: ! Y u ðmÞ Y u ðmÞ j;l 1 i;l ˜ Y˜ Ei;t ðmÞ ¼ ðY˜ i;l ð1 þ tAi;lþr ÞÞ : j;l

i

jai

lAZ

It is well-defined because it follows from Theorem 3.11 that for jai; lAZ; ðY˜ i;l ð1 þ tA˜ 1 ÞÞ and Y˜ j;l commute. For mABi ; the formula shows that the Yt -monomials of i;lþri

Ei;t ðmÞ are the Y-monomials of Ei ðmÞ (with identification by pþ Þ: Such elements were used in [12] for the ADE case. The Theorem 4.10 allows us to describe Ki;t : Corollary 4.11. For all mABi ; "mABi Z½t7 Ei;t ðmÞCZ½t7 ðBi Þ :

we

have

Ei;t ðmÞAKi;t :

Moreover,

Ki;t ¼

Proof. First Ei;t ðmÞAKi;t as product of elements of Ki;t : We show easily that the Ei;t ðmÞ are Z½t7 -linearly independent by looking at a maximal Yt -monomial in a linear combination. Let us prove that the Ei;t ðmÞ are Z½t7 -generators of Ki;t : for a product w of the algebra-generators of Theorem 4.10, let us look at the highest weight Yt monomial m: Then Ei;t ðmÞ is this product up to the order in the multiplication. But for p ¼ 1 or pX3; Yi;l Yi;lþpri is the unique dominant Y-monomial of Ei ðYi;l ÞEi ðYi;lþpri Þ; so: Z ˜ ˜ 1 ˜ 1 ˜ 1 ˜ ˜ Y˜ i;l ð1 þ tA˜ 1 i;lþri ÞYi;lþpri ð1 þ tAi;lþpri þri ÞAt Yi;lþpri ð1 þ tAi;lþpri þri ÞYi;l ð1 þ tAi;lþri Þ:

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24

And for p ¼ 2: ˜ 1 ˜ 1 ˜ 1 ˜ ˜ ˜ Y˜ i;l ð1 þ tA˜ 1 i;lþri ÞYi;lþ2ri ð1 þ tAi;lþ3ri Þ  Yi;lþ2ri ð1 þ tAi;lþ3ri ÞYi;l ð1 þ tAi;lþri Þ ˜ 1 ˜ AZ½t7  þ tZ Y˜ i;l ð1 þ tA˜ 1 i;lþri ÞYi;lþ2ri ð1 þ tAi;lþ3ri Þ:

&

4.5.2. Elements of Ki;t with a unique i-dominant Yt -monomial Proposition 4.12. For mABi ; there is a unique Fi;t ðmÞAKi;t such that m is the unique i-dominant Yt -monomial of Fi;t ðmÞ: Moreover: Ki;t ¼

M

Fi;t ðmÞ

mABi

Proof. It follows from Corollary 4.11 that an element of Ki;t has at least one i-dominant Yt -monomial. In particular, we have the uniqueness of Fi;t ðmÞ: For the existence, let us look at the sl2 -case. Let m be in B: It follows from Lemma 1 3.14 that fMA1 i1 ;l1 yAiR ;lR =RX0; l1 ; y; lR XlðMÞg-B is finite (where lðMÞ ¼ minflAZ=(iAI; ui;l ðMÞa0gÞ: We define on this set a total ordering compatible with the partial ordering: mL ¼ m4mL1 4?4m1 : Let us prove by induction on l the existence of Ft ðml Þ: The unique dominant Yt -monomial of Et ðm1 Þ is m1 so Ft ðm1 Þ ¼ Et ðm1 Þ: In general let l1 ðtÞ; y; ll1 ðtÞAZ½t7  be the coefficient of the dominant Yt -monomials m1 ; y; ml1 in Et ðml Þ: We put X Ft ðml Þ ¼ Et ðml Þ  lr ðtÞFt ðmr Þ: r¼1yl1 7 Notice that this construction gives Ft ðmÞAmZ½A˜ 1 l ; t lAZ : For the general case, let i be in I and m be in Bi : Consider ok ðmÞ as in Section 7 4.1.4. The study of the sl2 -case allows us to set wk ¼ ok ðmÞ1 Ft ðok ðmÞÞAZ½A˜ 1 l ; t l : 7 ˜ 1 7 And using the Z½t7 -algebra homomorphism nk;t : Z½A˜ 1 l ; t lAZ -Z½Ai;l ; t iAI;lAZ defined by nk;t ðA˜ 1 Þ ¼ A˜ 1 ; we set (the terms of the product commute): l

i;kþlri

Fi;t ðmÞ ¼ m

Y

nk;t ðwk ÞAKi;t :

0pkpri 1

P For the last assertion, we have Ei;t ðmÞ ¼ l¼1yL ll ðtÞFi;t ðml Þ where m1 ; y; mL are the i-dominant Yt -monomials of Ei;t ðmÞ with coefficients l1 ðtÞ; y; lL ðtÞAZ½t7 : & In the same way there is a unique Fi ðmÞAKi such that m is the unique i-dominant Y-monomial of Fi ðmÞ: Moreover Fi ðmÞ ¼ pþ ðFi;t ðmÞÞ: 4.5.3. Examples in the sl2 -case In this section, we suppose that g ¼ sl2 and we compute Ft ðmÞ ¼ F1;t ðmÞ in some examples with the help of Section 4.1.3.

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Lemma 4.13. Let s ¼ fl; l þ 2; y; l þ 2kg Y˜ l Y˜ lþ2 yY˜ lþ2k AB: Then Ft ðms Þ is equal to:

be

a

2-segment

25

and

ms ¼

2 ˜ 1 k ˜ 1 ˜ 1 ˜ 1 ˜ 1 ms ð1 þ tA˜ 1 lþ2kþ1 þ t Alþð2kþ1Þ Alþð2k1Þ þ ? þ t Alþð2kþ1Þ Alþð2k1Þ yAlþ1 Þ:

If s1 ; s2 are 2-segments not in special position, we have Ft ðms1 ÞFt ðms2 Þ ¼ tNðms1 ;ms2 ÞNðms2 ;ms1 Þ Ft ðms2 ÞFt ðms1 Þ: If s1 ; y; sR are 2-segments such that ms1 ymsr is regular, we have Ft ðms1 ymsR Þ ¼ Ft ðms1 ÞyFt ðmsR Þ: In particular if mAB verifies 8lAZ; ul ðmÞp1 then it is of the form m ¼ ms1 ymsR where the sr are 2-segments such that maxðsr Þ þ 2ominðsrþ1 Þ: So the Lemma 4.13 gives an explicit formula Ft ðmÞ ¼ Ft ðms1 ÞyFt ðmsR Þ: Proof. First we need some relations in Y1;t : we know that for lAZ we have tS˜ l1 ¼ 2 ˜ 1 ˜ ˜ ˜ 1 ˜ ˜ 1 A˜ 1 l Slþ1 ¼ t Slþ1 Al ; so t Sl1 ¼ Slþ1 Al : So we get by induction that for rX0 ˜ 1 ˜ 1 ˜ 1 ˜ 1 ˜ 1 ˜ 1 tr S˜ lþ12r ¼ S˜ lþ1 A˜ 1 l Al2 yAl2ðr1Þ : As ui;lþ1 ðAl Al2 yAl2ðr1Þ Þ ¼ ui;lþ1 ðAl Þ ¼ 1; we get ˜ 1 ˜ ˜ 1 tr S˜ lþ12r ¼ t2 A˜ 1 l Al2 yAl2ðr1Þ Slþ1 : 0 ˜ 1 ˜ 1 For r0 X0; by multiplying on the left by A˜ 1 lþ2r0 Alþ2ðr0 1Þ yAlþ2 and putting r ¼ 1 þ R0 ; r ¼ R  R0 ; l ¼ L  1  2R0 ; we get for 0pR0 pR: 0 R2 ˜ 1 ˜ 1 ˜ 1 ˜ ˜ ˜ 1 ALþ1 AL1 yA˜ 1 tR A˜ 1 Lþ1 AL1 yALþ12R0 SL2R ¼ t Lþ12R SL2R0 :

Now let be m ¼ Y˜ 0 Y˜ 2 yY˜ l and wAYt given by the formula in the lemma. The last remark enables us to compute S˜ t ðwÞ ¼ 0: So wAKt : But we see on the formula that m is the unique dominant monomial of w: So w ¼ Ft ðmÞ: For the second point, we have two cases: if ms1 ms2 is regular, it follows from Lemma 4.5 that Lðms1 ÞLðms2 Þ has no dominant monomial other than ms1 ms2 : But our formula shows that Ft ðms1 Þ (resp. Ft ðms2 Þ) has the same monomials than Lðms1 Þ (resp. Lðms2 ÞÞ: So Ft ðms1 ÞFt ðms2 Þ  tNðms1 ;ms2 ÞNðms2 ;ms1 Þ Ft ðms2 ÞFt ðms1 Þ has no dominant Yt -monomial. If ms1 ms2 is irregular, we have for example sj1 Csj2 and sj1 þ 2Csj2 : Let us write sj1 ¼ fl1 ; l1 þ 2; y; p1 g and sj2 ¼ fl2 ; l2 þ 2; y; p2 g: So we have l2 pl1 and p1 pp2  2: Let m ¼ m1 m2 be a dominant Y-monomial of Lðms1 ms2 Þ ¼ Lðms1 ÞLðms2 Þ where m1 (resp. m2 Þ is a Y-monomial of Lðms1 Þ (resp. Lðms2 ÞÞ: If m2 is not ms2 ; we have

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Yp1 in m2 which cannot be canceled by m1 : So m ¼ m1 ms2 : Let us write m1 ¼ 2 1 ˜ 1 ˜ 1 ms1 Ap1 þ1 yA1 p1 þ12r : So we just have to prove ½Ap1 þ1 yAp1 þ12r ; ms2  ¼ 0: This ˜ ˜ ˜ ˜ 1 ˜ follows from ðlAZÞ: A˜ 1 l Yl1 Ylþ1 ¼ Yl1 Ylþ1 Al : For the last assertion it suffices to show that Ft ðms1 ÞyFt ðmsR Þ has a unique dominant Yt -monomial than ms1 ymsR : But Ft ðms1 ÞyFt ðmsR Þ has the same monomials than Lðms1 ÞyLðmsR Þ ¼ Lðms1 ymsR Þ and ms1 ymsR is regular. & 4.5.4. Technical complements Let us go back to the general case. We give some technical results which will be used in the following to compute Fi;t ðmÞ in some cases (see Proposition 5.15 and the appendix). Lemma 4.14. Let i be in I; lAZ; MAA such that ui;l ðMÞ ¼ 1 and ui;lþ2ri ¼ 0: Then we 1 1 ˜ 1 have NðM; A˜ 1 i;lþri Þ ¼ 1: In particular p ðMAi;lþri Þ ¼ tM Ai;lþri : Proof. We can suppose M ¼: M : and we compute in Yu : ˜ 1 M A˜ 1 i;lþri ¼ tR : M Ai;lþri : P where RðzÞ ¼ ðz2ri  z2ri Þ þ rAZ ui;r ðMÞzðlþri rÞ ðzri  zri Þ: So X ui;r ðMÞðz2ri þlr  zlr Þ0 ¼ ui;l ðMÞ þ ui;lþ2ri ðMÞ ¼ 1: NðM; A˜ 1 i;lþri Þ ¼

&

rAZ

Lemma 4.15. Let m be in Bi such that 8l; rAZ; ui;l ðmÞp1 and the set fkAZ=ui;rþ2kri ðmÞ ¼ 1g is a 1-segment. Then we have Fi;t ðmÞ ¼ p1 ðFi ðmÞÞ: Proof. Let us look at the sl2 -case: m ¼ m1 m2 ¼ ms1 ms2 where s1 ; s2 are 2-segment. So Lemma 4.13 gives an explicit formula for Ft ðmÞ and it follows from Lemma 4.14 that Ft ðmÞ ¼ p1 ðF ðmÞÞ: We go back to the general case: let us write m ¼ m0 m1 ym2ri where m0 ¼ Q ui;rþ2lr ðmÞ uj;l ðmÞ and mr ¼ lAZ Yi;rþ2lrii : We have mr of the form mr ¼ jai;lAZ Yj;l Yi;lr Yi;lr þ2ri yYi;lr þ2ni ri and Q

0

Fi;t ðmÞ ¼ tNðm m1 ymr Þ m0 Fi;t ðm1 ÞyFi;t ðm2ri Þ: The study of the sl2 -case gives Fi;t ðmr Þ ¼ p1 ðFi ðmr ÞÞ: It follows from Lemma 4.14 that: 0

tNðm m1 ymr Þ m0 p1 ðFi ðm1 ÞÞyp1 ðFi ðmr ÞÞ ¼ p1 ðm0 Fi ðm1 ÞyFi ðmr ÞÞ ¼ p1 ðFi ðmÞÞ:

&

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5. Intersection of kernels of deformed screening operators T Motivated by Theorem 4.2 we study the structure of a completion of Kt ¼ iAI KerðSi;t Þ in order to construct wq;t in Section 6. Note that in the sl2 -case we have Kt ¼ KerðS1;t Þ that was studied in Section 4. 5.1. Reminder: classical case [5,6] For JCI; denote the Z-subalgebra KJ ¼

T

iAJ

Ki CY and K ¼ KI :

Lemma 5.1 (Frenkel and Reshetikhin [5]; Frenkel and Mukhin [6]). A non-zero element of KJ has at least one J-dominant Y-monomial. Proof. It suffices to look at a maximal weight Y-monomial m of wAKJ : for iAJ we have mABi because wAKi : & Theorem 5.2 (Frenkel and Reshetikhin [5]; Frenkel and Mukhin [6]). For iAI there is a unique EðYi;0 ÞAK such that Yi;0 is the unique dominant Y-monomial in EðYi;0 Þ: The uniqueness follows from Lemma 5.1. For the existence we have EðYi;0 Þ ¼ wq ðVoi ð1ÞÞ (Theorem 4.2). Note that the existence of EðYi;0 ÞAK suffices to characterize wq : Rep-K: It is the ring homomorphism such that wq ðXi;l Þ ¼ sl ðEðYi;0 ÞÞ where sl : Y-Y is given by sl ðYj;k Þ ¼ Yj;kþl : For mAB; we defined Mm and Vm in Section 2. We set Y sl ðEðYi;0 ÞÞui;l ðmÞ ¼ wq ðMm ÞAK; LðmÞ ¼ wq ðVm ÞAK: EðmÞ ¼ mAB

We have K¼

M

ZEðmÞ ¼

mAB

M

ZLðmÞCZðBÞ :

mAB

For mAB; we can also define a unique F ðmÞAK such that m is the unique dominant Y-monomial which appears in F ðmÞ (see for example the proof of Proposition 4.12). For JCI; let gJ be the semi-simple Lie algebra of Cartan Matrix ðCi;j Þi;jAJ and Uq ðg# ÞJ the associated quantum affine algebra with coefficient ðri ÞiAJ : In analogy with the definition of Ei ðmÞ; Li ðmÞ using the sl2 -case (Section 4.1.4), we define for mABJ : EJ ðmÞ; LJ ðmÞ; FJ ðmÞAKJ using Uq ðg# ÞJ : We have KJ ¼

M mABJ

ZEJ ðmÞ ¼

M mABJ

ZLJ ðmÞ ¼

M mABJ

As a direct consequence of Proposition 2.6 we have:

ZFJ ðmÞCZðBJ Þ :

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Lemma 5.3. For mAB; minflAZ=(iAI; ui;l ðmÞa0g:

we

have

EðmÞAZ½Yi;l iAI;lXlðmÞ

where

lðmÞ ¼

5.2. Completion of the deformed algebras T In this section, we introduce completions of Yt and of KJ;t ¼ iAJ Ki;t CYt ðJCIÞ: We have the following motivation: we have seen pþ ðKJ;t ÞCKJ (Section 4). In order to prove an analogue of the other inclusion (Theorem 5.11) we have to introduce completions where infinite sums are allowed. 5.2.1. The completion YN t of Yt N N Q Let A t ¼ mAA Z½t7 :mCZ½t7 A : We have "mAA Z½t7 :m ¼ Yt C A t : The N

algebra structure of Yt gives a structure of Yt -bimodule on A t but cannot be N

N

N naturally extended to A t : We define a Z½t7 -submodule YN t with Yt CYt C A t ; for

which it is the case: 7 ˜ 1 Let YA t be the Z½t -subalgebra of Yt generated by the ðAi;l ÞiAI;lAZ : It follows A;K where for KX0: from Proposition 3.12 that YA t ¼ "KX0 Yt

M

YA;K ¼ t

Z½t7 :mCYA t :

m¼:A˜ 1 yA˜ 1 : i ;l i ;l 1 1

K K

1 2 1 þK2 YA;K CYA;K for the multiplication of Yt : So YA Note that for K1 ; K2 X0; YA;K t t t t is A;K A;N a graded algebra if we set degðxÞ ¼ K for xAYt : Denote by Yt the completion

N

7 of YA t for this gradation. It is a sub-Z½t -module of A t : N

A;N Definition 5.4. YN : t is the sub Yt -leftmodule of A t generated by Yt

In particular, we have YN t ¼

P

MAA

N

M:YA;N C At: t

Lemma 5.5. There is a unique algebra structure on YN t compatible with the structure of Yt CYN : t are infinite sums of Proof. The structure is unique because the elements of YN t A;N N elements of Yt : For MAA; we have YA;N :MCM:Y ; so Y is a sub Yt -bimodule t t t N

of A t : For MAA and lAYA;N denote lM AYA;N such that l:M ¼ M:lM : We define t t

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A;N 0 0 the Z½t7 -algebra structure on YN Þ: t by ðM; M AA; l; l AYt 0

ðM:lÞðM 0 :l0 Þ ¼ MM 0 :ðlM l0 Þ: It is well-defined because for M1 ; M2 ; MAA; l; l2 AYA t M M M1 l1 ¼ M2 l2 ) M1 Ml1 ¼ M2 Ml2 : &

we

have

5.2.2. The completion KN i;t of Ki;t We define a completion of Ki;t analog to the completed algebra YN t : A;N 7 M For MAA; we define a Z½t -linear endomorphism Ei;t : MYt -MYA;N such t that ðm YA -monomial): t M ðMmÞ ¼ 0 if : Mm : eBi ; Ei;t M Ei;t ðMmÞ ¼ Ei;t ðMmÞ if : Mm : ABi :

and : Mm : ABi we have Ei;t ðMmÞA It is well-defined because if mAYA;K t A;K 0 M"K 0 XK Y : Definition 5.6. We define KN i;t ¼ For JCI; we set KN J;t ¼

T

iAJ

P

MAA

M ImðEi;t ÞCYN t :

N N KN i;t and Kt ¼ KI;t :

Lemma 5.7. A non-zero element of KN J;t has at least one J-dominant Yt -monomial. For N 7 -Y ¼ K : Moreover KN JCI; we have KN t J;t J;t J;t is a Z½t -subalgebra of Yt : Proof. For the first point the proof is analog to the proof of Lemma 5.1. Next it suffices to prove the results for J ¼ fig: First for mABi we have Ei;t ðmÞ ¼ N N m 7 ðmÞAKN Ei;t i;t and so Ki;t ¼ "mABi Z½t Ei;t ðmÞCKi;t -Yt : Now let w be in Ki;t such that w has only a finite number of Yt -monomials. In particular it has only a finite number of i-dominant Yt -monomials m1 ; y; mr with coefficients l1 ðtÞ; y; lr ðtÞ: In particular it follows from the first point that w ¼ l1 ðtÞFi;t ðm1 Þ þ ? þ lr ðtÞFi;t ðmr ÞAKi;t (see Proposition 4.12 for the definition of Fi;t ðmÞÞ: For the last assertion, consider M1 ; M2 AA and m1 ; m2 YA t -monomials such that : M1 m1 :; : M2 m2 : ABi : Then Ei;t ðM1 m1 ÞEi;t ðM2 m2 Þ is in the sub-algebra Ki;t CYt M1 M2 and in ImðEi;t Þ: & N In the same way for t ¼ 1 we define the Z-algebras YN and KN J CY : The N surjective map pþ : Yt -Y is naturally extended to a surjective map pþ : Yt -YN : N N N For iAI; we have pþ ðKN i;t Þ ¼ Ki and for JCI; pþ ðKJ;t ÞCKJ : The other inclusion is equivalent to Theorem 5.11.

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5.2.3. Special submodules of YN t ˜ 1 For mAA; KX0 we construct a subset Dm;K CmfA˜ 1 i1 ;l1 yAiK ;lK g stable by the maps S m 0 Ei;t such that KX0 Dm;K is countable: we say that m ADm;K if and only if there is a finite sequence ðm0 ¼ m; m1 ; y; mR ¼ m0 Þ of length RpK; such that for all 1prpR; there is r0 or; JCI such that mr0 ABJ and for r0 or00 pr; mr00 is a Y-monomial of 1 EJ ðmr0 Þ and mr00 m1 r00 1 AfAj;l =lAZ; jAJg: The definition means that ‘‘there is chain of monomials of some EJ ðm00 Þ from m to m0 ’’. Lemma 5.8. The set Dm;K is finite. In particular, the set Dm is countable. Moreover For m; m0 AA such that m0 ADm we have Dm0 DDm : For MAA; the set B-DM is finite. Proof. Let us prove by induction on KX0 that Dm;K is finite: we have Dm;0 ¼ fmg and: [ fY-monomials of EJ ðm0 Þg Dm;Kþ1 C JCI;m0 ADm;K -BJ

Consider ðm0 ¼ m; m1 ; y; mR ¼ m0 Þ a sequence adapted to the definition of Dm : Let m00 be in Dm0 and ðmR ¼ m0 ; mRþ1 ; y; mR0 ¼ m00 Þ a sequence adapted to the definition of Dm0 : So ðm0 ; m1 ; y; mR0 Þ is adapted to the definition of Dm ; and m00 ADm : Let us look at mAB-DM : we can see by induction on the length of a sequence ðm0 ¼ M; m1 ; y; mR ¼ mÞ adapted to the definition of DM that m is of the form Q v m ¼ MM 0 where M 0 ¼ iAI;lXl1 Ai;l i;l ðvi;l X0Þ: So the last assertion follows from Lemma 3.14. & Definition 5.9. D˜ m is the Z½t7 -submodule of YN t whose elements are of the form ðlm0 ðtÞm0 Þm0 ADm : For mAA introduce m0 ¼ m4m1 4m2 4? the countable set Dm with a total ordering compatible with the partial ordering. For kX0 consider an element Fk AD˜ mk : Note that some infinite sums make sense in D˜ m : for kX0; we have D Cfmk ; mkþ1 ; yg: So mk appears only in the Fk0 with k0 pk and the infinite sum Pmk ˜ kX0 Fk makes sense in Dm : 5.3. Crucial result for our construction Definition 5.10. For nX1 denote PðnÞ the property ‘‘for all semi-simple Lie-algebras ˜ g of rank rkðgÞ ¼ n; for all mAB there is a unique Ft ðmÞAKN t -Dm such that m is the unique dominant Yt -monomial of Ft ðmÞ:’’. Our construction of q; t-characters is based on Theorem 5.11. Theorem 5.11. For all nX1; the property PðnÞ is true.

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Note that for n ¼ 1; that is to say g ¼ sl2 ; the result follows from Section 4. In general the uniqueness follows from Lemma 5.7: if w1 ; w2 AKN t are solutions, then w1  w2 has no dominant Yt -monomial, so w1 ¼ w2 : Remark: in the simply laced case the existence is a consequence of the geometric theory of quivers [11,12], and in An ; Dn -cases of algebraic explicit constructions [13]. In the rest of this Section 5 we give an algebraic proof of this theorem in the general case. Let us outline the proof: first we give some preliminary technical results (Section 5.4) in which we construct t-analogues of the EðmÞ: Next we prove PðnÞ by induction on n: Our proof has 3 steps: Step 1 (Section 5.5): we prove Pð1Þ and Pð2Þ using a more precise property QðnÞ such that QðnÞ ) PðnÞ: The property QðnÞ has the following advantage: it can be verified by computation in elementary cases n ¼ 1; 2: Step 2 (Section 5.6): we give some consequences of PðnÞ which will be used in the proof of PðrÞ ðr4nÞ: we give the structure of KN t (Proposition 5.17) for rkðgÞ ¼ n and the structure of KN where JCI; jJj ¼ n and jIj4n (Corollary 5.18). J;t Step 3 (Section 5.7): we prove PðnÞ ðnX3Þ assuming PðrÞ; rpn are true. We give an algorithm (Section 5.7.1) to construct explicitly Ft ðmÞ: It is called t-algorithm and is a t-analogue of Frenkel–Mukhin algorithm [6] (a deformed algorithm was also used by Nakajima in the ADE-case [11]). As we do not know a priori that the algorithm is well-defined the general case, we have to show that it never fails (Lemma 5.22) and gives a convenient element (Lemma 5.23). 5.4. Preliminary: construction of the Et ðmÞ ˜ ˜ Lemma 5.12. We suppose that for iAI; there is Ft ðY˜ i;0 ÞAKN t -DY˜ i;0 such that Yi;0 is the unique dominant Yt -monomial of Ft ðY˜ i;0 Þ: Then: (i) All Yt -monomials of Ft ðY˜ i;0 Þ; except the highest weight Yt -monomial, are right negative. (ii) All Yt -monomials of Ft ðY˜ i;0 Þ are products of Y˜ 7 j;l with lX0: (iii) The only Yt -monomial of Ft ðY˜ i;0 Þ which contains a Y˜ 7 j;0 ðjAIÞ is the highest weight monomial Y˜ i;0 : (iv) The Ft ðY˜ i;0 Þ ðiAIÞ commute.

Note that (i),(ii) and (iii) appeared in [6] for t ¼ 1: Proof. (i) It suffices to prove that all Yt -monomials m0 ¼ Yi;0 ; m1 ; y of DYi;0 except Yi;0 are right negative. But m1 is the monomial Yi;0 A1 i;1 of Ei ðYi;0 Þ and it is right negative. We can now prove the statement by induction: suppose that mr is a monomial of EJ ðmr0 Þ; where mr0 is right negative. So mr is a product of mr0 by some

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A1 j;l ðlAZÞ:Those monomials are right negative because a product of right negative monomial is right negative. 7 (ii) Suppose that mAA is product of Yk;l with lX0: It follows from Lemma 5.3 that 7 all monomials of Dm are product of Yk;l with lX0: ˜ (iii) All Y-monomials of DYi;0 except Yi;0 are in DY

1 i;0 Ai;r i

: But lðYi;0 A1 i;ri ÞX1 and we

can conclude with the help of Lemma 5.3. (iv) Let iaj be in I and look at Ft ðY˜ i;0 ÞFt ðY˜ j;0 Þ: Suppose we have a dominant Yt monomial m0 ¼ m1 m2 in Ft ðY˜ i;0 ÞFt ðY˜ j;0 Þ different from the highest weight Yt monomial Y˜ i;0 Y˜ j;0 : We have for example m1 aY˜ i;0 ; so m1 is right negative. Let l1 be the maximal l such that a Y˜ k;l appears in m1 : We have uk;l ðm1 Þo0 and l40: As uk;l ðm0 ÞX0 we have uk;l ðm2 Þ40 and m2 aYj;0 : So m2 is right negative and there is k0 AI and l 0 4l such that uk0 ;l 0 ðm2 Þo0: So uk0 ;l 0 ðm1 Þ40; contradiction. So the highest weight Yt -monomial of Ft ðY˜ i;0 ÞFt ðY˜ j;0 Þ is the unique dominant Yt -monomial. In the same way the highest weight Yt -monomial of Ft ðY˜ j;0 ÞFt ðY˜ i;0 Þ is the unique dominant Yt -monomial. But we have Y˜ i;0 Y˜ j;0 ¼ Y˜ j;0 Y˜ i;0 ; so Ft ðY˜ i;0 ÞFt ðY˜ j;0 Þ  Ft ðY˜ j;0 Þ Ft ðY˜ i;0 ÞAKN t has no dominant Yt -monomial, so is equal to 0. & N 7 For lAZ denote by sl : YN t -Yt the endomorphism of Z½t -algebra such that sl ðY˜ j;k Þ ¼ Y˜ j;kþl (it is well-defined because the defining relations of Yt are invariant for k/k þ lÞ: If the hypothesis of Lemma 5.12 are verified, we can define for mAtZ B:

Et ðmÞ ¼ m

- Y Y

!1 u ðmÞ Y˜ i;li;l

lAZ iAI

- Y Y

sl ðFt ðY˜ i;0 ÞÞui;l ðmÞ AKN t

lAZ iAI

because for iaj ½sl ðFt ðY˜ i;0 ÞÞ; sl ðFt ðY˜ j;0 ÞÞ ¼ 0 (Lemma 5.12). 5.5. Step 1: Proof of Pð1Þ and Pð2Þ The aim of this section is to prove Pð1Þ and Pð2Þ: First we define a more precise property QðnÞ such that QðnÞ ) PðnÞ: Definition 5.13. For nX1 denote QðnÞ the property ‘‘for all semi-simple Lie-algebras g of rank rkðgÞ ¼ n; for all iAI there is a unique Ft ðY˜ i;0 ÞAKt -D˜ Y˜ i;0 such that Y˜ i;0 is the unique dominant Yt -monomial of Ft ðY˜ i;0 Þ: Moreover, Ft ðY˜ i;0 Þ has the same monomials as EðYi;0 Þ’’. The property QðnÞ is more precise than PðnÞ because it implies that Ft ðY˜ i;0 Þ has only a finite number of monomials. Lemma 5.14. For nX1; the property QðnÞ implies the property PðnÞ:

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Proof. We suppose QðnÞ is true. In particular Section 5.4 enables us to construct Et ðmÞAKN t for mAB: The defining formula of Et ðmÞ shows that it has the same monomials as EðmÞ: So Et ðmÞAD˜ m and Et ðmÞAKt : Let us prove PðnÞ: let m be in B: The uniqueness of Ft ðmÞ follows from Lemma 5.7. Let mL ¼ m4mL1 4?4m1 be the dominant monomials of Dm with a total ordering compatible with the partial ordering (it follows from Lemma 3.14 that Dm -B is finite). Let us prove by induction on l the existence of Ft ðml Þ: The unique dominant of Dm1 is m1 so Ft ðm1 Þ ¼ Et ðm1 ÞAD˜ m1 : In general, let l1 ðtÞ; y; ll1 ðtÞAZ½t7  be the coefficient of the dominant Yt -monomials m1 ; y; ml1 in Et ðml Þ: We put X lr ðtÞFt ðmr Þ: Ft ðml Þ ¼ Et ðml Þ  r¼1yl1

We see in the construction that Ft ðmÞAD˜ m because for m0 ADm we have Et ðm0 ÞAD˜ m0 DD˜ m (Lemma 5.8). & We need the following general technical result: Proposition 5.15. Let m be in B such that all monomial m0 of F ðmÞ verifies: 8iAI; m0 ABi implies 8lAZ; ui;l ðm0 Þp1 and for 1prp2ri the set flAZ=ui;rþ2lri ðm0 Þ ¼ 1g is a 1-segment. Then p1 ðF ðmÞÞAYt is in Kt and has a unique dominant monomial m: P 0 Proof. Let us write F ðmÞ ¼ m0 AA mðmP Þm0 ðmðm0 ÞAZÞ: Let i be in I and consider the decomposition of F ðmÞ in Ki F ðmÞ ¼ m0 ABi mðm0 ÞFi ðm0 Þ: But mðm0 Þa0 implies the hypothesis of Lemma 4.15 is verified for m0 ABi : So p1 ðFi ðm0 ÞÞ ¼ Fi;t ðm0 Þ: And P p1 ðF ðmÞÞ ¼ m0 ABi mðm0 ÞFi;t ðm0 ÞAKi;t : & For n ¼ 1 (Section 4.5), n ¼ 2 (appendix), we can give explicit formula for the EðYi;0 Þ ¼ F ðYi;0 Þ: In particular, we see that the hypothesis of Proposition 5.15 are verified, so: Corollary 5.16. The properties Qð1Þ; Qð2Þ and so Pð1Þ; Pð2Þ are true. This allow us to start our induction in the proof of Theorem 5.11. In Section 7.1 we will see other applications of Proposition 5.15. Note that the hypothesis of Proposition 5.15 are not verified for fundamental monomials m ¼ Yi;0 in general: for example for the D5 -case we have in F ðY2;0 Þ the 2 1 1 1 Y5;4 Y2;4 Y4;4 : monomial Y3;3 5.6. Step 2: consequences of the property PðnÞ Let nX1: We suppose in this section that PðnÞ is proved. We give some consequences of PðnÞ which will be used in the proof of PðrÞ ðr4nÞ:

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Let KN;f be the Z½t7 -submodule of KN generated by elements with a finite t t number of dominant Yt -monomials. Proposition 5.17. We suppose rkðgÞ ¼ n: We have ¼ KN;f t

M

Z½t7 Ft ðmÞCZ½t7 ðBÞ :

mAB

Moreover for MAA; we have ˜ KN t -DM ¼

M

Z½t7 Ft ðmÞCZ½t7 B-DM :

mAB-DM

Proof. Let w be in KN;f and m1 ; y; mL AB the dominant Yt -monomials of w and t 7 ðtÞ; y; l ðtÞAZ½t  their coefficients. It follows from Lemma 5.7 that w ¼ l 1 L P l ðtÞF ðm Þ: t l l¼1yL l Let us look at the second point: Lemma 5.8 shows that mAB-DM ) Ft ðmÞAD˜ M : In particular the inclusion + is clear. For the other inclusion we prove as in the first P 7 ˜ point that KN t -DM ¼ mAB-DM Z½t Ft ðmÞ: We can conclude because it follows from Lemma 3.14 that DM -B is finite. & We recall that some infinite sum make sense in D˜ M (Section 5.2.3). Corollary 5.18. We suppose rkðgÞ4n and let J be a subset of I such that jJj ¼ n: For mABJ ; there is a unique FJ;t ðmÞAKN J;t such that m is the unique J-dominant Yt monomial of FJ;t ðmÞ: Moreover FJ;t ðmÞAD˜ m : P ˜ For MAA; the elements of KN J;t -DM are infinite sums mABJ -DM lm ðtÞFJ;t ðmÞ: In particular: 7 BJ -DM ˜ : KN J;t -DM CZ½t 

Proof. The uniqueness of FJ;t ðmÞ follows from Lemma 5.7. Let us write m ¼ mJ m0 Q u ðmÞ where mJ ¼ iAJ;lAZ Yi;li;l : So mJ is a dominant Yt -monomial of Z½Yi;l7 iAJ;lAZ : In particular Proposition 5.17 with the algebra Uq ðg# ÞJ of rank n gives mJ w where U ðg# Þ ;1 U ðg# Þ ;7 (where for iAI; lAZ; A˜ q J ¼ bI;J ðA˜ 7 Þ where wAZ½A˜ q J ; t7  i;l

iAJ;lAZ

i;l

i;l

˜7 bI;J ðY˜ 7 So we can put Ft ðmÞ ¼ mnJ;t ðwÞ where nJ;t : Z i;l Þ ¼ diAJ Yi;l Þ: # U ð g Þ ;1 U ðg# Þ ;1 ½A˜ i;lq J ; t7 iAJ;lAZ -Yt is the ring homomorphism such that nJ;t ðA˜ i;lq J Þ ¼ A˜ 1 : The last assertion is proved as in Proposition 5.17. & i;l

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5.7. Step 3: t-algorithm and end of the proof of Theorem 5.11 In this section we prove the property PðnÞ by induction on nX1: In particular we define the t-algorithm which constructs explicitly the Ft ðmÞ: It follows from Section 5.5 that Pð1Þ and Pð2Þ are true. Let be nX3 and suppose that PðrÞ is proved for ron: Let mþ be in B and m0 ¼ mþ 4m1 4m2 4? the countable set Dmþ with a total ordering compatible with the partial ordering. For JiI and mABJ ; it follows from PðrÞ and Corollary 5.18 that there is a unique FJ;t ðmÞAD˜ m -KN J;t such that m is the unique J-dominant monomial of FJ;t ðmÞ: Moreover the elements of D˜ mþ -KN are the infinite sums of YN J;t t : P 7 mADm -BJ lm ðtÞFJ;t ðmÞ where lm ðtÞAZ½t : If mAA  BJ ; denote FJ;t ðmÞ ¼ 0: þ

! I denote by ½FJ;t ðmr0 Þmr AZ½t7  the coefficient of mr in FJ;t ðmr0 Þ: For r; r0 X0 and JD 5.7.1. Definition of the t-algorithm Definition 5.19. We call t-algorithm the following inductive definition of the sequences ðsðmr ÞðtÞÞrX0 AZ½t7 N ; ðsJ ðmr ÞðtÞÞrX0 AZ½t7 N ðJiIÞ: sðm0 ÞðtÞ ¼ 1;

sJ ðm0 ÞðtÞ ¼ 0

! I: and for rX1; JD sJ ðmr ÞðtÞ ¼

X r0 or

ðsðmr0 ÞðtÞ  sJ ðmr0 ÞðtÞÞ½FJ;t ðmr0 Þmr

if mr eBJ ; sðmr ÞðtÞ ¼ sJ ðmr ÞðtÞ if mr AB; sðmr ÞðtÞ ¼ 0: We have to prove that the t-algorithm defines the sequences in a unique way. We ! I: The sJ ðmR Þ see that if sðmr Þ; sJ ðmr Þ are defined for rpR so are sJ ðmRþ1 Þ for JD impose the value of sðmRþ1 Þ and by induction the uniqueness is clear. We say that the t-algorithm is well-defined to step R if there exist sðmr Þ; sJ ðmr Þ such that the formulas of the t-algorithm are verified for rpR: Lemma 5.20. The t-algorithm is well-defined to step r if and only if: 8J1 ; J2 iI; 8r0 pr; mr0 eBJ1 and mr0 eBJ2 ) sJ1 ðmr0 ÞðtÞ ¼ sJ2 ðmr0 ÞðtÞ: Proof. If for r0 or the sðmr0 ÞðtÞ; sJ ðmr0 ÞðtÞ are well-defined, so is sJ ðmr ÞðtÞ: If mr AB; sðmr ÞðtÞ ¼ 0 is well-defined. If mr eB; it is well-defined if and only if fsJ ðmr ÞðtÞ=mr eBJ g has one unique element. &

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5.7.2. The t-algorithm never fails If the t-algorithm is well-defined to all steps, we say that the t-algorithm never fails. If the t-algorithm is well-defined to step r; for JiI we set mJ ðmr ÞðtÞ ¼ sðmr ÞðtÞ  sJ ðmr ÞðtÞ; X wrJ ¼ mJ ðmr0 ðtÞÞFJ;t ðmr0 ÞAKN J;t : r0 pr

Lemma 5.21. If the t-algorithm is well-defined to step r; for JCI we have wrJ A

X

! sðmr0 ÞðtÞmr0

X

þ sJ ðmrþ1 ÞðtÞmrþ1 þ

r0 pr

Z½t7 mr0 :

r0 4rþ1

! I; we have For J1 CJ2D wrJ2 ¼ wrJ1 þ

X

lr0 ðtÞFJ1 ;t ðmr0 Þ;

r0 4r

where lr0 ðtÞAZ½t7 : In particular, if mrþ1 eBJ1 ; we have sJ1 ;t ðmrþ1 Þ ¼ sJ2 ;t ðmrþ1 Þ: Proof. For r0 pr let us compute the coefficient ðwrJ Þmr0 AZ½t7  of mr0 in wrJ : ðwrJ Þmr0 ¼

X r00 pr0

ðsðmr00 ÞðtÞ  sJ ðmr00 ÞðtÞÞ½FJ;t ðmr00 Þmr0

¼ ðsðmr0 ÞðtÞ  sJ ðmr0 ÞðtÞÞ½FJ;t ðmr0 Þmr0 X þ ðsðmr00 ÞðtÞ  sJ ðmr00 ÞðtÞÞ½FJ;t ðmr00 Þmr0 r00 or0

¼ ðsðmr0 ÞðtÞ  sJ ðmr0 ÞðtÞÞ þ sJ ðmr0 ÞðtÞ ¼ sðmr0 ÞðtÞ: Let us compute the coefficient ðwrJ Þmrþ1 AZ½t7  of mrþ1 in wrJ : ðwrJ Þmrþ1 ¼

X r00 orþ1

ðsðmr00 ÞðtÞ  sJ ðmr00 ÞðtÞÞ½FJ;t ðmr00 Þmrþ1 ¼ sJ ðmrþ1 Þ:

˜ ! I: We have wrJ2 AKN For the second point let J1 CJ2D J1 ;t -Dmþ and it follows from PðjJ1 jÞ and Corollary 5.18 (or Section 5.5 if jJ1 jp2Þ that we can introduce lmr0 ðtÞAZ½t7  such that wrJ2 ¼

X r0 X0

lmr0 ðtÞFJ1 ;t ðmr0 Þ:

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We show by induction on r0 that for r0 pr; mr0 ABJ1 ) lmr0 ðtÞ ¼ mJ1 ðmr0 ÞðtÞ: First we have lm0 ðtÞ ¼ ðwrJ2 Þm0 ¼ sðm0 ÞðtÞ ¼ 1 ¼ mJ1 ðm0 Þ: For r0 pr: sðmr0 ÞðtÞ ¼ lmr0 ðtÞ þ

X r00 or0

lmr0 ðtÞ ¼ sðmr0 ÞðtÞ 

X r00 or0

lmr00 ðtÞ½FJ1 ;t ðmr00 Þmr0 ;

mJ1 ðmr00 ÞðtÞ½FJ1 ;t ðmr00 Þmr0

¼ sðmr0 ÞðtÞ  sJ1 ðmr0 ÞðtÞ ¼ mJ1 ðmr0 ÞðtÞ: P For the last assertion if mrþ1 eBJ1 ; the coefficient of mrþ1 in r0 4r Z½t7 FJ1 ;t ðmr0 Þ is 0, and ðwrJ2 Þmrþ1 ¼ ðwrJ1 Þmrþ1 : It follows from the first point that sJ1 ;t ðmrþ1 Þ ¼ sJ2 ;t ðmrþ1 Þ: & Lemma 5.22. The t-algorithm never fails. Proof. Suppose that the sequence is well-defined until the step r  1 and let J1 ; J2 iI such that mr eBJ1 and mr eBJ2 : Let i be in J1 ; j in J2 such that mr eBi and mr eBj : r1 r1 Consider J ¼ fi; jgiI: The wr1 J ; wi ; wj AYt have the same coefficient sðmr0 ÞðtÞ on mr0 for r0 pr  1: Moreover si ðmr ÞðtÞ ¼ ðwr1 i Þmr ;

sj ðmr ÞðtÞ ¼ ðwr1 j Þmr ;

sJ ðmr ÞðtÞ ¼ ðwr1 J Þmr :

But mr eBJ ; so: ¼ wr1 J

X

mi ðmr0 ÞðtÞFi;t ðmr0 Þ þ

r0 pr1

X

lmr0 ðtÞFi;t ðmr0 Þ:

r0 Xrþ1

r1 So ðwr1 J Þmr ¼ ðwi Þmr and we have si ðmr ÞðtÞ ¼ sJ ðmr ÞðtÞ: In the same way we have si ðmr ÞðtÞ ¼ sJ1 ðmr ÞðtÞ; sj ðmr ÞðtÞ ¼ sJ ðmr ÞðtÞ and sj ðmr ÞðtÞ ¼ sJ2 ðmr ÞðtÞ: So we can conclude sJ1 ðmr ÞðtÞ ¼ sJ2 ðmr ÞðtÞ: &

5.7.3. Proof of PðnÞ P It follows from Lemma 5.22 that w ¼ rX0 sðmr ÞðtÞmr AYN t is well-defined. Lemma 5.23. We have wAKN t is m0 ¼ mþ :

T

D˜ mþ : Moreover, the only dominant Yt -monomial in w

Proof. The defining formula of w gives wAD˜ mþ : Let i be in I and wi ¼

X rX0

mi ðmr ÞðtÞFi;t ðmr ÞAKN i;t :

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Let us compute for rX0 the coefficient of mr in w  wi : X ðw  wi Þmr ¼ sðmr ÞðtÞ  mi ðmr0 ÞðtÞ½Fi;t ðmr0 Þmr r0 pr

¼ sðmr ÞðtÞ  si ðmr ÞðtÞ  mi ðmr ÞðtÞ½Fi;t ðmr Þmr ¼ ðsðmr ÞðtÞ  si ðmr ÞðtÞÞð1  ½Fi;t ðmr Þmr Þ: We have two cases: if mr ABi ; we have 1  ½Fi;t ðmr Þmr ¼ 0; if mr eBi ; we have sðmr ÞðtÞ  si ðmr ÞðtÞ ¼ 0: N So w ¼ wi AKN i;t ; and wAKt : The last assertion follows from the definition of the algorithm: for r40; mr AB ) sðmr ÞðtÞ ¼ 0: & Corollary 5.24. For nX3; if the PðrÞ ðronÞ are true, then PðnÞ is true. In particular Theorem 5.11 is proved by induction on n:

6. Morphism of q; t-characters and applications 6.1. Morphism of q; t-characters We set Rept ¼ Rep#Z Z½t7  ¼ Z½Xi;l ; t7 iAI;lAZ : We say that MARept is a Rept Q x monomial if it is of the form M ¼ iAI;lAZ Xi;li;l ðxi;l X0Þ: In this case denote xi;l ðMÞ ¼ xi;l : Recall the definition of the Et ðmÞ (Section 5.4). Definition 6.1. The morphism of wq;t : Rept -YN t such that ðui;l X0Þ: wq;t

q; t-characters !

Y

u Xi;li;l

Y

¼ Et

iAI;lAZ

is

the

Z½t7 -linear

map

! u Yi;li;l

iAI;lAZ

Theorem 6.2. We have pþ ðImðwq;t ÞÞCY and the following diagram is commutative: Rep

wq;t

! Imðwq;t Þ

id k

Rep

k pþ wq

!

Y

In particular the map wq;t is injective. The Z½t7 -linear map wq;t : Rept -YN is t characterized by the three following properties:

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Q x ðMÞ (1) For a Rept -monomial M define m ¼ p1 ð iAI;lAZ Yi;li;l ÞAA and mAA ˜ t as in Section 3.5.1. Then we have X ˜þ am0 ðtÞm0 ðwhere am0 ðtÞAZ½t7 Þ: wq;t ðMÞ ¼ m m0 om

(2) The image of Imðwq;t Þ is contained in KN t : (3) Let M1 ; M2 be Rept -monomials such that (, ) (, ) X X max l xi;l ðM1 Þ40 pmin l xi;l ðM2 Þ40 : iAI

iAI

Then we have wq;t ðM1 M2 Þ ¼ wq;t ðM1 Þwq;t ðM2 Þ: Note that properties (1)–(3) are generalizations of the defining axioms introduced by Nakajima in [12] for the ADE-case; in particular in the ADE-case wq;t is the morphism of q; t-characters constructed in [12]. Proof. pþ ðImðwq;t ÞÞCY means that only a finite number of Yt -monomials of Et ðmÞ have coefficient lðtÞeðt  1ÞZ½t7 : As Ft ðY˜ i;0 Þ has no dominant Yt -monomial other than Y˜ i;0 ; we have the same property for pþ ðFt ðY˜ i;0 ÞÞAKN and pþ ðFt ðY˜ i;0 ÞÞ ¼ EðYi;0 ÞAY: As Y is a subalgebra of YN we get pþ ðEt ðmÞÞAY with the help of the defining formula. The diagram is commutative because pþ 3sl ¼ sl 3pþ and pþ ðFt ðY˜ i;0 ÞÞ ¼ EðYi;0 Þ: It is proved by Frenkel, Reshetikhin in [5] that wq is injective, so wq;t is injective. Let us show that wq;t verifies the three properties: ˜ Y˜ i;l Þ: In (1) By definition we have wq;t ðMÞ ¼ Et ðmÞ: But sl ðFt ðY˜ i;0 ÞÞ ¼ Ft ðY˜ i;l ÞADð P 0 ˜ ˜ particular sl ðFt ðYi;0 ÞÞ is of the form Yi;l þ m0 oYi;l lm0 ðtÞm and we get the property for Et ðmÞ by multiplication. (2) We have sl ðFt ðY˜ i;0 ÞÞ ¼ Et ðY˜ i;l ÞAKN and KN is a subalgebra of YN t ; so t t N Imðwq;t ÞCKt : Q P x ðM Þ (3) If we set L ¼ maxfl= iAI xi;l ðM1 Þ40g; m1 ¼ iAI;lAZ Yi;li;l 1 and m2 ¼ Q xi;l ðM2 Þ we have iAI;lAZ Yi;l Et ðm1 Þ ¼

- Y Y lpL iAI

sl ðFt ðY˜ i;0 ÞÞxi;l ðM1 Þ ;

Et ðm2 Þ ¼

- Y Y

sl ðFt ðY˜ i;0 ÞÞxi;l ðM2 Þ

lXL iAI

and in particular: Et ðm1 m2 Þ ¼ Et ðm1 ÞEt ðm2 Þ: Finally, let f : Rept -YN be a Z½t7 -linear homomorphism which verifies t properties 1–3. We saw that the only element of KN t with highest weight monomial

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Y˜ i;l is sl ðFt ðY˜ i;0 ÞÞ: In particular we have f ðXi;l Þ ¼ Et ðYi;l Þ: Using property 3, we get for MARept a monomial f ðMÞ ¼

- Y Y

f ðXi;l Þui;l ðmÞ ¼

lAZ iAI

- Y Y

sl ðFt ðY˜ i;0 ÞÞui;l ðmÞ ¼ wq;t ðMÞ:

&

lAZ iAI

6.2. Quantization of the Grothendieck Ring In this section we see that wq;t allows us to define a deformed algebra structure on Rept generalizing the quantization of [12]. The point is to show that Imðwq;t Þ is a subalgebra of KN t : 6.2.1. Construction of the quantization Recall the definition of KN;f in Section 5.6. For mAB; all monomials of Et ðmÞ are t 1 yA =kX0; l XLg where L ¼ minflAZ; (iAI; ui;l ðmÞ40g: So it follows in fmA1 k i1 ;l1 iK ;lK has only a finite number of dominant Yt from Lemma 3.14 that Et ðmÞAKN t N;f monomials, that is to say Et ðmÞAKt : Proposition 6.3. The Z½t7 -module KN;f is freely generated by the Et ðmÞ: t ¼ KN;f t

M

Z½t7 Et ðmÞCZ½t7 ðBÞ :

mAB

is a subalgebra of KN Moreover KN;f t t : Proof. The Et ðmÞ are Z½t7 -linearly independent and we saw Et ðmÞAKN;f : It suffices t to prove that the Et ðmÞ generate the Ft ðmÞ: let us look at m0 AB and consider L ¼ minflAZ; (iAI; ui;l ðm0 Þ40g: In the proof of Lemma 3.14 we saw that there is vi ;l vi ;l only a finite dominant monomials in fm0 Ai1 ;l11 1 yAiR ;lRR R =RX0; ir AI; lr XLg: Let m0 4m1 4?4mD AB be those monomials with a total ordering compatible with the partial ordering. In particular, for 0pdpD the dominant monomials of Et ðmd Þ are in fmd ; mdþ1 ; y; mD g: So there are elements ðld;d 0 ðtÞÞ0pd;d 0 pD of Z½t7  such that P Et ðmd Þ ¼ dpd 0 pD ld;d 0 ðtÞFt ðmd 0 Þ: We have ld;d 0 ðtÞ ¼ 0 if d 0 od and ld;d ðtÞ ¼ 1: We have a triangular system with 1 on the diagonal, so it is invertible in Z½t7 : For the last point it suffices to prove that for m1 ; m2 AB; Et ðm1 ÞEt ðm2 Þ has only a finite number of dominant Yt -monomials. But Et ðm1 ÞEt ðm2 Þ has the same monomials as Et ðm1 m2 Þ: & It particular wq;t is a Z½t7 -linear isomorphism between Rept and KN;f : So we can t 7 define an associative deformed Z½t -algebra structure on Rept by 8l1 ; l2 ARept ; l1
l2 ¼ w1 q;t ðwq;t ðl1 Þwq;t ðl2 ÞÞ:

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6.2.2. Examples: sl2 -case We make explicit computation of the deformed multiplication in the sl2 -case: Proposition 6.4. In the sl2 -case, the deformed algebra structure on Rept ¼ Z½Xl ; t7 lAZ is given by Xl1
Xl2
?
Xlm ¼ Xl1 Xl2 yXlm Xl
Xl 0 ¼ tg Xl Xl 0 ¼ tg Xl 0
Xl

if l1 pl2 p?plm ;

if l4l 0 and lal 0 þ 2;

Xl
Xl2 ¼ t2 Xl Xl2 þ tg ð1  t2 Þ ¼ t2 Xl2
Xl þ ð1  t2 Þ; where gAZ is defined by Y˜ l Y˜ l 0 ¼ tg Y˜ l 0 Y˜ l : The proof is a straightforward computation which uses for lAZ the q; t-character ˜ 1 ˜ of Xl : wq;t ðXl Þ ¼ Y˜ l þ Y˜ 1 lþ2 ¼ Yl ð1 þ tAlþ1 Þ: Note that g were computed in Section 3.5.3. We see that the new Z½t7 -algebra structure is not commutative and not even twisted polynomial. 6.3. An involution of the Grothendieck ring In this section, we construct an antimultiplicative involution of the Grothendieck ring Rept : The construction is motivated by the point view adopted in this article: it is just replacing cjlj by cjlj : In the ADE-case such an involution were introduced Nakajima [12] with different motivations. Lemma 6.5. There is a unique C-linear isomorphism of Hh which is antimultiplicative and such that ðm40; iAI; rAZ  f0gÞ: cm ¼ cm ;

ai ½r ¼ ai ½r:

Moreover it is an involution. The Z-subalgebra Yu CHh verifies Yu CYu : Proof. It suffices to show that it is compatible with the defining relations of H ði; jAI; m; rAZ  f0gÞ: ½ai ½m; aj ½r ¼ ½ai ½m; aj ½r: It is an involution because cm ¼ cm and ai ½r ¼ ai ½r: For the last assertion it suffices to check on the generators of Yu ðRAU; iAI; lAZÞ: tR ¼ tR ;

Y˜ i;l ¼ tC˜ i;i ðqÞðqi q1 Þ Y˜ i;l ; i

Y˜ i;l

1

¼ tC˜ i;i ðqÞðqi q1 Þ Y˜ 1 i;l :

&

i

As for R; R0 AU; we have p0 ðRÞ ¼ p0 ðR0 Þ3p0 ðRÞ ¼ p0 ðR0 Þ; the involution of Yu (resp. of Hh Þ is compatible with the defining relations of Yt (resp. Ht Þ: We get a

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Z-linear involution of Yt (resp. of Ht Þ: For l; l0 AYt ; aAZ; we have: l:l0 ¼ l%0 :l% ;

ta l ¼ ta l% :

˜ ˜ ˜ Note that in Yu for iAI; lAZ; A˜ i;l ¼ tðq2i þq2 Þ Ai;l : So in Yt we have Ai;l ¼ Ai;l and i ˜ 1 A˜ 1 i;l ¼ Ai;l : Lemma 6.6. For iAI; the Yi;u CHh verifies Yi;u CYi;u : S˜ i;l AYi;u : Now for lAYu ; we Proof. First we compute for iAI; lAZ: S˜ i;l ¼ tqi þq1 i qi q1 i

have l:S˜ i;l ¼ tqi þq1 S˜ i;l l% : But it is in Yi;u because l% AYu (Lemma 6.5) and Yi;u is a Yu i qi q1 i

subbimodule of Hh (Lemma 4.6).

& 1

q þq In Ht we have S˜ i;l ¼ tS˜ i;l because p0 ðqii qi1 Þ ¼ 1: As said before we get a Z-linear i

involution of Yi;t such that lS˜ i;l ¼ tS˜ i;l l% : We introduced such an involution in [8]. With this new point of view, the compatibility with the relation A˜ i;lri S˜ i;l ¼ t1 S˜ i;lþri is a direct consequence of Lemma 4.6. Lemma 6.7. For iAI; the subalgebra Ki;t CYt verifies Ki;t CKi;t : Proof. Suppose lAKi;t ; that is to say Si;t ðlÞ ¼ 0: So ðt2  1ÞSi;t ðlÞ ¼ 0 and: X X ðS˜ i;l l  lS˜ i;l Þ ¼ 0 ) t ðl% S˜ i;l  S˜ i;l l% Þ ¼ 0: lAZ

lAZ

So tð1  t2 ÞSi;t ðl% Þ ¼ 0 and l% AKi;t : & P Note that P wAYt has the same monomials as w% ; that is to say if w ¼ mAA lðtÞm and w% ¼ mAA mðtÞm; we have lðtÞa03mðtÞa0: In particular, we can naturally extend our involution to an antimultiplicative involution on YN t : Moreover, we have N N;f KN ¼ Imðwq;t ÞCImðwq;t Þ: So we can define a Z-linear involution of t CKt and Kt Rept by

8lARept ;

l% ¼ w1 q;t ðwq;t ðlÞÞ:

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6.4. Analogues of Kazhdan–Lusztig polynomials In this section we define analogues of Kazhdan–Lusztig polynomials (see [10]) with the help of the antimultiplicative involution of Section 6.3 in the same spirit Nakajima did for the ADE-case [12]. Let us begin we some technical properties of the action of the involution on monomials. 6.4.1. Construction ˜ 1 We recall that the YA t -monomials are products of the Ai;l ðiAI; lAZÞ: Lemma 6.8. For M a Yt -monomial and m a YA t -monomial there is a unique aðM; mÞAZ such that taðM;mÞ Mm ¼ taðM;mÞ Mm: Proof. Let bAZ such that m ¼ tb m: We have Mm ¼ mM ¼ tbþg Mm where gA2Z (Section 3.4). So it suffices to prove that bA2Z: Q v Let us compute b: Let pþ ðmÞ ¼ iAI;lAZ Ai;l i;l : In Yu we have pþ ðmÞp ðmÞ ¼ tR p ðmÞpþ ðmÞ where p0 ðRÞ ¼ b and X

RðqÞ ¼

vi;r vj;r0

X

i;jAI;r;r0 AZ ½a ½l;a ½l

where for l40 we set i clj For i ¼ j; we have the term: X

vi;r vi;r0

r;r0 AZ

0 X @ ¼ l40

X

qlrlr

l40

X

0

qlrlr

l40

0

½ai ½l; aj ½l ; cl

¼ Bi;j ðql Þðql  ql ÞAZ½q7  which is antisymmetric.

½ai ½l; ai ½l cl 0

0

vi;r ðmÞvi;r0 ðmÞðqlðrr Þ þ qlðr rÞ Þ þ

fr;r0 gCZ;rar0

X

1 vi;r ðmÞ2 A

rAZ

½ai ½l; ai ½l : cl

It is antisymmetric, so it has no term in q0 : So p0 ðRÞ ¼ p0 ðR0 Þ where R0 is the sum of the contributions for iaj:   X X ½aj ½l; ai ½l lrlr0 ½ai ½l; aj ½l vi;r ðmÞvj;r0 ðmÞ q þ cl cl r;r0 AZ l40 ¼2

X

vi;r ðmÞvj;r0 ðmÞ

r;r0 AZ

In particular p0 ðR0 ÞA2Z:

X l40

qlrlr

0

½ai ½l; aj ½l : cl

&

aðm;MÞ Mm=m YA For M a Yt -monomial denote Ainv t -monomialg: In partiM ¼ ft 0 inv 01 1 inv Z 0 cular for m AAM we have m m ¼ MM : Let BM ¼ ðt BÞ-Ainv M :

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Theorem 6.9. For mAtZ B there is a unique Lt ðmÞAKN t such that: Lt ðmÞ ¼ ðmm1 ÞLt ðmÞ;

Et ðmÞ ¼ Lt ðmÞ þ

X

Pm0 ;m ðtÞLt ðm0 Þ;

m0 om;m0 ABinv m

where Pm0 ;m ðtÞAt1 Z½t1 : Those polynomials Pm0 ;m ðtÞ are called analogues to Kazhdan–Lusztig polynomials and the Lt ðmÞ ðmABÞ for a canonical basis of Kft ;N : Such polynomials were introduced by Nakajima [12] for the ADE-case. Proof. First consider Ft ðmÞ: it is in KN t and has only one dominant Yt -monomial m; so Ft ðmÞ ¼ mm1 Ft ðmÞ: Let be m ¼ mL 4mL1 4?4m0 the finite set ðtZ DðmÞÞ-Binv m (see Lemma 5.8) with a total ordering compatible with the partial ordering. Note that it follows from 1 Section 6.4.1 that for LXlX0; we have ml m1 l ¼ mm : inv We have Et ðm0 Þ ¼ Ft ðm0 Þ and so Et ðm0 Þ ¼ m0 m1 0 Et ðm0 Þ: As Bm0 ¼ fm0 g; we have Lt ðm0 Þ ¼ Et ðm0 Þ: We suppose by induction that the Lt ðml Þ ðL  1XlX0Þ are uniquely and well-defined. In particular, ml is of highest weight in Lt ðml Þ; Lt ðml Þ ¼ 1 ml m1 l Lt ðml Þ ¼ mm Lt ðml Þ; and we can write M

7 D˜ t ðmL Þ-KN t ¼ Z½t Ft ðmL Þ"

Z½t7 Lt ðml Þ:

0plpL1

In particular, consider al;L ðtÞAZ½t7  such that Et ðmÞ ¼ Ft ðmÞ þ

X

al;L ðtÞLt ðml Þ:

loL

We want Lt ðmÞ of the form Lt ðmÞ ¼ Ft ðmÞ þ

P

loL

bl;L ðtÞLt ðml Þ: The condition

Lt ðmÞ ¼ mm1 mLt ðmÞ means that the bl;L ðtÞ are symmetric. The condition Pm0 ;m ðtÞAt1 Z½t1  means al;L ðtÞ  bl;L ðtÞAt1 Z½t1 : So it suffices to prove that those two conditions uniquely define the bl;L ðtÞ: let us write al;L ðtÞ ¼ aþ l;L ðtÞ þ þ 0  7 7 7 a0l;L ðtÞ þ a l;L ðtÞ (resp. bl;L ðtÞ ¼ bl;L ðtÞ þ bl;L ðtÞ þ bl;L ðtÞÞ where al;L ðtÞAt Z½t  and

a0l;L ðtÞAZ (resp. for bÞ: The condition al;L ðtÞ  bl;L ðtÞAt1 Z½t1  means b0l;L ðtÞ ¼  The symmetry of bl;L ðtÞ means bþ a0l;L ðtÞ and b l;L ðtÞ ¼ al;L ðtÞ: l;L ðtÞ ¼  1  1 bl;L ðt Þ ¼ al;L ðt Þ: &

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6.4.2. Examples Proposition 6.10. We suppose that g ¼ sl2 : Let mAtZ B such that 8lAZ; ul ðmÞp1: Then Lt ðmÞ ¼ Ft ðmÞ: Moreover, X

Et ðmÞ ¼ Lt ðmÞ þ

0

tRðm Þ Lt ðm0 Þ;

m0 om=m0 ABinv m

1 0 inv where Rðm0 ÞX1 is given by pþ ðm0 m1 Þ ¼ A1 i1 ;l1 yAiR ;lR : In particular for m ABm such 0

that m0 om we have Pm0 ;m ðtÞ ¼ tRðm Þ : Proof. Note that a dominant monomial m0 om verifies 8lAZ; ul ðm0 Þp1 and appears in Et ðmÞ: We know that D˜ m -Kt ¼ "m0 AtZ Dm -Binv Z½t7 Ft ðm0 Þ: We can m introduce Pm0 ;m ðtÞAZ½t7  such that: X

Et ðmÞ ¼ Ft ðmÞ þ m0 AtZ D

m

Pm0 ;m ðtÞFt ðm0 Þ:

-Binv fmg

So by induction it suffices to show that Pm0 ;m ðtÞAt1 Z½t1 : Pm0 ;m ðtÞ is the coefficient of m0 in Et ðmÞ: A dominant Yt -monomial M which appears in Et ðmÞ is of the form: ˜ 1 ˜ 1 M ¼ mðm1 ymRþ1 Þ1 m1 tA˜ 1 l1 m2 tAl2 m3 ytAlR mRþ1 where l1 o?olR AZ verify flr þ 2; lr  2g-fl1 ; y; lr1 ; lrþ1 ; y; lR g is empty, Q ˜ ul ðmÞ : Such a monomial ulr 1 ðmÞ ¼ ulr þ1 ðmÞ ¼ 1 and we have set mr ¼ lr1 olplr Yl appears one time in Et ðmÞ: In particular Pm0 ;m ðtÞ ¼ ta where aAZ is given by M ¼ ta m0 that is to say MM 1 ¼ t2a m01 m0 ¼ t2a m1 m: So we compute: ˜ 1 ˜ ˜ 1 ¼ t2R mm1 : & MM 1 ¼ t2R t4R A˜ 1 lR yAl1 mAlR yAl1 m Let us look at another example: g ¼ sl2 and m ¼ Y˜ 20 Y˜ 2 : We have Et ðmÞ ¼ Lt ðmÞ þ t2 Lt ðm0 Þ;

Pm0 ;m ðtÞ ¼ t2 ;

inv where m0 ¼ tY˜ 20 Y˜ 2 A˜ 1 1 ABm and

˜ 1 ˜ 1 ˜ ˜ Lt ðmÞ ¼ Ft ðY˜ 0 ÞFt ðY˜ 0 Y˜ 2 Þ ¼ Y˜ 0 ð1 þ tA˜ 1 1 ÞY0 Y2 ð1 þ tA3 ð1 þ tA1 ÞÞ; ˜ 1 Lt ðm0 Þ ¼ Ft ðm0 Þ ¼ tY˜ 20 Y˜ 2 A˜ 1 1 ð1 þ tA1 Þ:

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Let suppose that C ¼ B2 and m ¼ Y˜ 2;0 Y˜ 1;5 : The formulas for Et ðY˜ 2;0 Þ and Et ðY˜ 1;5 Þ are given in the appendix. We have Et ðmÞ ¼ Lt ðmÞ þ t1 Lt ðm0 Þ;

Pm0 ;m ðtÞ ¼ t1 ;

inv ˜ 1 where m0 ¼ tY˜ 2;0 Y˜ 1;5 A˜ 1 2;2 A1;4 ABm and

˜ 1 ˜ 1 ˜ 1 ˜ 1 Lt ðm0 Þ ¼ Ft ðm0 Þ ¼ tY˜ 2;0 Y˜ 1;5 A˜ 1 2;2 A1;4 ð1 þ tA1;2 ð1 þ tA2;4 ð1 þ tA1;6 ÞÞÞ:

7. Questions and conjectures 7.1. Positivity of coefficients Proposition 7.1. If g is of type An ðnX1Þ; the coefficients of wq;t ðYi;0 Þ are in N½t7 : Proof. We show that for all iAI the hypothesis of Proposition 5.15 for m ¼ Yi;0 are verified; in particular the property Q of Section 5.5 will P be verified. Let i be in I: For jAI; let us write EðYi;0 Þ ¼ mABj lj ðmÞEj ðmÞAKj where lj ðmÞAZ: Let D be the set D ¼ fmonomials of Ej ðmÞ=jAI; mABj ; lj ðmÞa0g: It suffices to prove that for jAI; mABj -D ) uj ðmÞp1 (because Proposition 5.15 implies that for all iAI; Ft ðY˜ i;0 Þ ¼ p1 ðEðYi;0 ÞÞÞ: As EðYi;0 Þ ¼ F ðYi;0 Þ; Yi;0 is the unique dominant Y-monomial in EðYi;0 Þ: So for a monomial mAD there is a finite sequence fm0 ¼ Yi;0 ; m1 ; y; mR ¼ mg such that for all 1prpR; there is r0 or and jAI such that mr0 ABj and for r0 or00 pr; mr00 is a 1 monomial of Ej ðmr0 Þ and mr00 m1 r00 1 AfAj;l =lAZg: Such a sequence is said to be adapted to m: Suppose there is jAI and mABj -D such that uj ðmÞX2: So there is m0 pm in D-Bj such that uj ðmÞ ¼ 2: So we can consider m0 AD such that there is j0 AI; m0 ABj0 ; uj0 ðmÞX2 and for all m0 om0 in D we have 8jAI; m0 ABj ) uj ðm0 Þp1: Let us write Y ðjÞ m0 ¼ Yj0 ;ql Yj0 ;qm m0 ; jaj0

Q u ðm Þ ðjÞ where for jaj0 ; m0 ¼ lAZ Yj;lj;l 0 : In a finite sequence adapted to m0 ; a term Yj0 ;ql or Yj0 ;qm must come from a Ej0 þ1 ðm1 Þ or a Ej0 1 ðm1 Þ: So for example we have m1 om0 Q ðjÞ in D of the form m1 ¼ Yj0 ;qm Yj0 þ1;ql1 jaj0 ;j0 þ1 m1 : In all cases we get a monomial m1 om0 in D of the form Y ðjÞ m1 : m1 ¼ Yj1 ;qm1 Yj1 þ1;ql1 jaj1 ;j1 þ1

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But the term Yj1 þ1;ql1 cannot come from a Ej1 ðm2 Þ because we would have uj1 ðm2 ÞX2: So we have m2 om1 in D of the form:

m2 ¼ Yj2 ;qm2 Yj2 þ2;ql2

Y

ðjÞ

m2 :

jaj2 ;j2 þ1;j2 þ2

This term must come from a Ej2 1 ; Ej2 þ3 : By induction, we get mN om0 in D of the form mN ¼ Y1;qmN Yn;qlN

Y

ðjÞ

mN ¼ Y1;qmN Yn;qlN :

ja1;y;n

It is a dominant monomial of DCDYi;0 which is not Yi;0 : It is impossible (proof of Lemma 5.12). & An analog result is also geometrically proved by Nakajima for the ADE-case in [12] (it is also algebraically for AD-cases proved in [13]). Those results and the explicit formulas in n ¼ 1; 2-cases (see the appendix) suggest: Conjecture 7.2. The coefficients of Ft ðY˜ i;0 Þ ¼ wq;t ðYi;0 Þ are in N½t7 : In particular for mAB; the coefficients of Et ðmÞ would be in N½t7 ; moreover, wq;t ðYi;0 Þ and wq ðYi;0 Þ would have the same monomials, the t-algorithm would stop and Imðwq;t ÞCYt : At the time he wrote this paper the author does not know a general proof of the conjecture. However a case by case investigation seems possible: the cases G2 ; B2 ; C2 are checked in the appendix and the cases F4 ; Bn ; Cn ðnp10Þ have been checked on a computer. So a combinatorial proof for series Bn ; Cn ðnX2Þ analog to the proof of Proposition 7.1 would complete the picture.

7.2. Decomposition in irreducible modules Proposition 6.10 suggests: Conjecture 7.3. For mAB we have pþ ðLt ðmÞÞ ¼ LðmÞ: In the ADE-case Conjecture 7.3 is proved by Nakajima with the help of geometry [12]. In particular this conjecture implies that the coefficients of pþ ðLt ðmÞÞ are nonnegative. It gives a way to compute explicitly the decomposition of a standard

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module in irreducible modules, because Conjecture 7.3 implies: EðmÞ ¼ LðmÞ þ

X

Pm0 ;m ð1ÞLðm0 Þ:

m0 om

In particular we would have Pm0 ;m ð1ÞX0: In Section 6.4.2 we have studied some examples: *

In Proposition 6.10 for g ¼ sl2 and mAB such that 8lAZ; ul ðmÞp1: we have pþ ðLt ðmÞÞ ¼ F ðmÞ ¼ LðmÞ and X

EðmÞ ¼

Lðm0 Þ:

m0 AB=m0 pm

*

For g ¼ sl2 and m ¼ Y˜ 20 Y˜ 2 : we have pþ ðLt ðmÞÞ ¼ F ðY0 ÞF ðY0 Y2 Þ ¼ LðmÞ and: EðY02 Y2 Þ ¼ LðY02 Y2 Þ þ LðY0 Þ:

*

Note that LðY02 Y2 Þ has two dominant monomials Y02 Y2 and Y0 because Y02 Y2 is irregular (Lemma 4.5). For C ¼ B2 and m ¼ Y˜ 2;0 Y˜ 1;5 : The pþ ðLt ðY˜ 2;0 Y˜ 1;5 ÞÞ has non-negative coefficients and the conjecture implies EðY2;0 Y1;5 Þ ¼ LðY2;0 Y1;5 Þ þ LðY1;1 Þ:

7.3. Further applications and generalizations We hope to address the following questions in the future: Our presentation of deformed screening operators as commutators leads to the definition of iterated deformed screening operators, for example in order 2: " S˜ j;i;t ðmÞ ¼

X

# ˜ Sj;l ; Si;t ðmÞ :

lAZ

Some generalizations of the approach used in this article would be possible: (a) the theory of q-characters at roots of unity [7] suggests a generalization to the case qN ¼ 1: (b) in this article, we decided to work with Yt which is a quotient of Yu : The same construction with Yu will give characters with an infinity of parameters of P deformation tr ¼ expð l40 h2l qlr cl Þ ðrAZÞ: (c) our construction is independent of representation theory and could be established for other generalized Cartan matrices (in particular for affine cases).

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Acknowledgments The author would like to thank M. Rosso for encouragements and precious comments on a previous version of this paper, I.B. Frenkel for having encouraged him in this direction, E. Frenkel for encouragements, useful discussions and references, E. Vasserot for very interesting explanations about [15], O. Schiffmann for valuable comments and his kind hospitality in Yale university, and T. Schedler for help on programming.

Appendix There are 5 types of semi-simple Lie algebra of rank 2: A1 A1 ; A2 ; C2 ; B2 ; G2 (see for example [9]). In each case we give the formula for Eð1Þ; Eð2ÞAK and we see that the hypothesis of Proposition 5.15 is verified. In particular, we have Et ðY˜ 1;0 Þ ¼ p1 ðEð1ÞÞ; Et ðY˜ 2;0 Þ ¼ p1 ðEð2ÞÞAKt : Following [5], we represent the Eð1Þ; Eð2ÞAK as a I Z-oriented colored tree. For wAK the tree Gw is defined as follows: the set of vertices is the set of Y-monomials of w: We draw an arrow of color ði; lÞ from m1 to m2 if m2 ¼ A1 i;l m1 and if in the P decomposition w ¼ mABi mm Li ðmÞ there is MABi such that mM a0 and m1 ; m2 appear in Li ðMÞ:   2 0 and r1 ¼ r2 ¼ 1 (note that in this case the computaA1 A1 -case: C ¼ 0 2 tions keep unchanged for all r1 ; r2 Þ:

 A2 -case: C ¼

2 1

 1 : It is symmetric, r1 ¼ r2 ¼ 1: 2

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 C2 ; B2 -case (dual cases): we use C ¼ B2 ¼



2 2

 2 1 G2 -case: C ¼ and r1 ¼ 1; r2 ¼ 3: 3 2 First fundamental representation:

1 2

 and r1 ¼ 1; r2 ¼ 2:

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Second fundamental representation:

References [1] N. Bourbaki, Groupes et alge`bres de Lie, Hermann, Paris, 1968 (Chapitres IV–VI). [2] V. Chari, A. Pressley, Quantum affine algebras and their representations, in: Representations of Groups, Banff, AB, 1994, CMS Conference Proceeding, Vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 59–78. [3] V. Chari, A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994. [4] E. Frenkel, N. Reshetikhin, Deformations of W -algebras associated to simple Lie algebras, Comm. Math. Phys. 197 (1) (1998) 1–32. [5] E. Frenkel, N. Reshetikhin, The q-characters of representations of quantum affine algebras and deformations of W -algebras, in: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Contemporary Mathematics, vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp. 163–205.

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[6] E. Frenkel, E. Mukhin, Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras, Comm. Math. Phy. 216 (1) (2001) 23–57. [7] E. Frenkel, E. Mukhin, The q-characters at roots of unity, Adv. Math. 171 (1) (2002) 139–167. [8] D. Hernandez, t-analogues des ope´rateurs d’e´crantage associe´s aux q-caracte`res, Internat. Math. Res. Not. 2003 (8) (2003) 451–475. [9] V. Kac, Infinite Dimensional Lie Algebras, 3rd Edition, Cambridge University Press, Cambridge, 1990. [10] D. Kazhdan, G. Lusztig, Representations of coxeter groups and Hecke algebras, Invent. Math. 53 (1979) 165–184. [11] H. Nakajima, T-analogue of the q-characters of finite dimensional representations of quantum affine algebras, in: Physics and Combinatorics, 2000 (Nagoya), World Scientific, River Edge, NJ, 2001, pp. 196–219. [12] H. Nakajima, Quiver varieties and t-analogs of q-characters of quantum affine algebras, Preprint arXiv:math.QA/0105173. [13] H. Nakajima, t-analogs of q-characters of quantum affine algebras of type An ; Dn ; in: Combinatorial and Geometric Representation Theory, Contemporary Mathematics, Vol. 325, Amer. Math. Soc., Providence, RI, 2003, pp. 141–160. [14] M. Rosso, Repre´sentations des groupes quantiques, Se´minaire Bourbaki exp. No. 744, Aste´risque 201–203, 443-83, SMF, 1992. [15] M. Varagnolo, E. Vasserot, Perverse Sheaves and Quantum Grothendieck Rings, Preprint arXiv:math.QA/0103182.