Level Zero Fundamental Representations over Quantized Affine Algebras

Level Zero Fundamental Representations over Quantized Affine Algebras

and Demazure Modules

By

MasakiKashiwara∗

Abstract

LetW(k) be the finite-dimensional irreducible module over a quantized affine algebraUq(g) with the fundamental weightk as an extremal weight. We show that its crystalB(W(k)) is isomorphic to the Demazure crystalB⁻(−Λ0+k). This is derived from the following general result: for a dominant integral weightλand an integral weightµ, there exists a unique homomorphismU_q⁻(g)(uλ⊗uµ)→V(λ+µ) that sendsuλ⊗uµtouλ+µ. HereV(λ) is the extremal weight module withλas an extremal weight, anduλ∈V(λ) is the extremal weight vector of weightλ.

§1. Introduction

The finite-dimensional representations of quantized affine algebras U_q(g) are extensively studied in connection with exactly solvable models. It is ex- pected that there exists a “good” finite-dimensional U_q(g)-module W(mk) with a multiplem_k of a fundamental weight_k as an extremal weight. This module is good in the sense that it is irreducible and it has a crystal base and moreover a global basis.

In the untwisted case, its conjectural character formula is given by Kirillov–

Reshetikhin ([17], see also [16]), and its conjectural fusion construction is given by Kuniba–Nakanishi–Suzuki ([18]). It is proved by Nakajima ([22]) that the

Communicated by T. Kawai. Received January 27, 2004. Revised August 10, 2004.

1991 Mathematics Subject Classification. Primary 20G05; Secondary 17B37.

Key words: Crystal bases, extremal modules, fundamental representations, Demazure modules.

∗Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan This research is partially supported by Grant-in-Aid for Scientific Research (B1)13440006, Japan Society for the Promotion of Science.

fusion construction gives irreducible modules with the expected character in the simply laced case, and by Chari ([3]) in some cases.

It is also expected that any “good” finite-dimensional U_q(g)-module is a tensor product of modules of the above type.

It is also conjectured in [4, 5] that the U_q(g)-modules W(mk) has a perfect crystal of level if and only if m = c^∨_k (c^∨_k := max(1,2/(α_k, α_k))).

Moreover it is conjectured that the crystal baseB(W(c^∨_kk)) is isomorphic to the Demazure crystalB⁻(−Λ0+c^∨_kk) if we forget the 0-arrows. Here, for an integral weight λ,B^±(λ) denotes the crystal for the U_q^±(g)-module generated by the extremal vector with weightλ. They are proved in certain cases ([7, 8]).

More general relations of perfect crystals and Demazure crystals are discussed in [6].

In this paper we show thatB(W(_k)) is isomorphic to the Demazure crys- talB⁻(−Λ0+k), or equivalentlyB(W(−k)) is isomorphic to the Demazure crystalB⁺(Λ0−k) (Corollary 4.8).

The main ingredient is the following theorem, which the author started to study in order to answer a question raised by Miwa et al:

Theorem 3.3. Let U_q⁻(g)be a quantized affine algebra. Letλ∈P⁺ be a dominant integral weight andµ∈P an integral weight. Then there exists a unique homomorphism V(λ)⊗V(µ)⊃ U_q⁻(g)(uλ⊗uµ) −−−→V(λ+µ) that sendsuλ⊗uµ touλ+µ. Moreover this morphism is compatible with global bases.

Here V(λ) is the extremal weight module with λas an extremal weight, anduλ∈V(λ) is the extremal weight vector of weightλ.

§2. Review on Crystal Bases and Global Bases

In this section, we shall review briefly the quantized universal enveloping algebras and crystal bases. We refer the reader to [9, 10, 13, 14, 15, 19].

§2.1. Quantized universal enveloping algebras

We shall define the quantized universal enveloping algebraUq(g). Assume that we are given the following data.

P : a freeZ-module (called a weight lattice), I: an index set (for simple roots),

α_i∈P fori∈I (called a simple root),

hi∈P^∗:= Hom_Z(P,Z) (called a simple coroot), (·,·) :P×P →Q a bilinear symmetric form.

We shall denote by ·,· :P^∗×P →Zthe canonical pairing.

The data above are assumed to satisfy the following axioms.

(αi, αi)>0 for anyi∈I,

(αi, αj)0 for anyi, j∈Iwith i=j, h_i, λ=2(α_i, λ)

(αi, αi) for anyi∈Iandλ∈P. (2.1)

Let us take a positive integerdsuch that (αi, αi)/2∈Zd⁻¹for anyi∈I.

Now letqbe an indeterminate and set

K=Q(qs) whereqs=q^1/d. (2.2)

We define its subringsA0,A_∞ andAas follows.

A0={f /g;f, g∈Q[qs], g(0)= 0}, A_∞=

f /g;f, g∈Q[q_s⁻¹], g(0)= 0 , A=Q[qs, qs−1].

(2.3)

Definition 2.1. The quantized universal enveloping algebra U_q(g) is the algebra overKgenerated by the symbolsei, fi(i∈I) andq(h) (h∈d⁻¹P^∗) with the following defining relations.

(1) q(h1)q(h2) =q(h1+h2) forh1, h2∈d⁻¹P^∗, andq(h) = 1 forh= 0.

(2) q(h)eiq(h)⁻¹ = q^h,αiei and q(h)fiq(h)⁻¹ = q^−h,αifi for any i ∈I andh∈d⁻¹P^∗.

(3) [ei, fj] =δij

ti−t⁻_i¹

qi−q_i⁻¹ fori,j∈I. Hereqi=q^(αi,αi)/2andti=q(^(αi₂^,αi)hi).

(4) (Serre relation) For i=j, b

k=0

(−1)^ke^(k)_i eje^(b_i ^−k)= b k=0

(−1)^kf_i^(k)fjf_i^(b−k)= 0.

Hereb= 1− hi, αjand

e^(k)_i =e^k_i/[k]i!, f_i^(k)=f_i^k/[k]i!, [k]_i= (q^k_i −q_i^−k)/(q_i−q_i⁻¹), [k]_i! = [1]_i· · ·[k]_i.

Fori∈I, we denote byU_q⁻(g)_i the subalgebra of U_q⁻(g) generated bye_i, f_i andq(h) (h∈d⁻¹P^∗).

Let us denote byW the Weyl group, the subgroup ofGL(P) generated by the simple reflectionssi: si(λ) =λ− hi, λαi.

Let ∆⊂Q:=

iZαi be the set of roots. Let ∆^±:= ∆∩Q_± be the set of positive and negative roots, respectively. HereQ_±:=±

iZ0α_i. Let ∆^re be the set of real roots, and set ∆^re_±:= ∆_±∩∆^re.

§2.2. Braid group action on integrable modules

Theq-analogue of the action of the Weyl group is introduced in [19, 23].

We define aq-analog of the exponential function by exp_q(x) =

∞ n=0

q^n(n−1)/2xⁿ [n]! . (2.4)

This satisfies the following equations:

exp_q(x) exp_q(y) = exp_q(x+y) ifxy=q²yx, exp_q(x) exp_q−1(y) =_∞

n=0

1 [n]!

n−1

ν=0(q^νx+q^−νy) if [x, y] = 0, exp_q(x) exp_q−1(−x) = 1,

exp_q(x) =

1 + (1−q²)x

exp_q(q²x), exp_q(x) =_∞

n=0

1 +q²ⁿ(1−q²)x

for|q|<1, (2.5)

Fori∈I, we set S_i= exp_q−1

i

(q_i⁻¹e_it⁻_i ¹) exp_q−1 i

(−f_i) exp_q−1 i

(q_ie_it_i)q_i^hi(hi+1)/2

= exp_q−1

i (−q⁻_i ¹fiti) exp_q−1

i (ei) exp_q−1

i (−qifit⁻_i¹)q_i^hi(hi+1)/2. (2.6)

We regardSias an endomorphism of integrableUq(g)-modules, andq^h_i^i(hi+1)/2 acts on the weight space of weightλby the multiplication ofq_i^{hi,λ(hi,λ+1)/2}. On the (l+1)-dimensional irreducible representation ofU_q(g)_iwith a high- est weight vectoru^(l)₀ andu^(l)_k =f_i^(k)u^(l)₀ ,

Si(u^(l)_k ) = (−1)^l−kq_i^(l−k)(k+1)u^(l)_l−k, (2.7)

Hence,S_i sends the weight space of weightλto the weight space of weights_iλ.

By the above formula, we have

Siu^(l)_l =u^(l)₀ and Siu^(l)₀ = (−qi)^lu^(l)_l . (2.8)

Since{S_i} satisfies the braid relations, we can extend the actions ofS_ion integrable modules to the action of the braid group by

Sww=Sw◦Sw ifl(ww) =l(w) +l(w), S_si =S_i.

§2.3. Braid group action on U_q⁻(g) We define the ring automorphismTi ofUq(g) by

T_i(q) =q (2.9)

Ti(q(h)) =q(sih), (2.10)

Ti(ei) =−fiti, (2.11)

T_i(f_i) =−t⁻_i¹e_i, (2.12)

Ti(ej) =

−hi,αj k=0

(−1)^kq_i^−ke⁽_i^{−hi,αj−k)}eje^(k)_i , (2.13)

Ti(fj) =

−hi,αj k=0

(−1)^kq_i^kf_i^(k)fjf_i^{(−hi,αj−k)}fori=j.

(2.14)

Then it is well-defined, and it satisfies

Ti(P)u=SiP S_i⁻¹u (2.15)

for anyP ∈Uq(g) and any element uof an integrableUq(g)-module.

The operatorTi is invertible and its inverse is given as follows.

T_i⁻¹(q(h)) =q(s_ih), (2.16)

T_i⁻¹(ei) =−t⁻_i¹fi, (2.17)

T_i⁻¹(fi) =−eiti, (2.18)

T_i⁻¹(e_j) =

−h_i,α_j k=0

(−1)^kq⁻_i ^ke^(k)_i e_je⁽_i^{−hi,αj−k)}, (2.19)

T_i⁻¹(fj) =

−h_i,α_j k=0

(−1)^kq^k_if_i^{(−hi,αj−k)}fjf_i^(k). (2.20)

We can extend the actionTi to the action of the braid group by T_ww=T_w◦T_w ifl(ww) =l(w) +l(w),

Ts_i =Ti.

The following proposition is proved in [19].

Proposition 2.2. Forw∈W andi, j∈Isuch thatwα_i=α_j,we have Twei=T_w⁻₋₁¹ ei=ej and Twfi=T_w⁻₋₁¹ fi =fj.

§2.4. Crystals

We shall not review the notion of crystals, but refer the reader to [9, 10, 13, 15]. For a subsetJ of I, let us denote byU_q(g_J) the subalgebra of U_q⁻(g) generated bye_j,f_j(j∈J) andq(h) (h∈d⁻¹P^∗). We say that a crystalBover U_q⁻(g) is aregular crystal if, for anyJ⊂I of finite-dimensional type,Bis, as a crystal overUq(gJ), isomorphic to a crystal base associated with an integrable Uq(gJ)-module.

By [13], the Weyl group W acts on any regular crystal. This action S is given by

Ss_ib=

f˜_i^hi,wt(b)b ifhi,wt(b)0,

˜

e⁻_i ^hi,wt(b)b ifhi,wt(b)0.

Let us denote byU_q⁻(g) (resp. U_q⁺(g)) the subalgebra ofU_q⁻(g) generated by thefi’s (resp. by theei’s). ThenU_q⁻(g) has a crystal base denoted byB(∞) ([10]). A unique vector of B(∞) with weight 0 is denoted by u_∞. Similarly U_q⁺(g) has a crystal base denoted by B(−∞), and a unique vector ofB(−∞) with weight 0 is denoted byu_−∞.

Letψbe the ring automorphism ofU_q⁻(g) that sendsq_s,e_i,f_iandq(h) to qs,fi,eiandq(−h). It induces bijectionsU_q⁻(g)−→^∼ U_q⁺(g) andB(∞)−→^∼ B(−∞) by whichu_∞, ˜e_i, ˜f_i,ε_i, ϕ_i, wt correspond tou_−∞, ˜f_i, ˜e_i, ϕ_i,ε_i, −wt.

Let Uq(g) be the modified quantized universal enveloping algebra ⊕λ∈P

U_q⁻(g)aλ (see [13]). The elements aλ, the projectors to the weight λ-space, satisfyaλ·aµ=δλ,µaλ andaλP =P a_λ−wt(P)forP ∈U_q⁻(g).

ThenUq(g) has a crystal base (L(Uq(g)), B(Uq(g))). As a crystal,B(Uq(g)) is regular and isomorphic to

λ∈P

B(∞)⊗T_λ⊗B(−∞).

Here,T_λ is the crystal consisting of a single element t_λ withε_i(t_λ) =ϕ_i(t_λ) =

−∞and wt(t_λ) =λ.

Let∗ be the anti-involution ofU_q⁻(g) that sendsq(h) toq(−h), andq_s,e_i, fi to themselves. The involution ∗ofU_q⁻(g) induces an involution∗ onB(∞), B(−∞), B(Uq(g)). Then ˜e^∗_i = ∗ ◦˜ei ◦ ∗, etc. give another crystal structure

onB(∞),B(−∞),B(U_q(g)). We call it the star crystal structure. These two crystal structures onB(U_q(g)) are compatible, andB(U_q(g)) may be considered as a crystal overg⊕g, which corresponds to theU_q⁻(g)-bimodule structure on Uq(g). Hence, for example,S_w^∗, the Weyl group action onB(Uq(g)) with respect to the star crystal structure is a crystal automorphism ofB(Uq(g)) with respect to the original crystal structure. In particular, the two Weyl group actionsSw

andS_w^∗ commute with each other.

§2.5. Global bases

Recall thatA0⊂Kis the subring ofK consisting of rational functions in q_swithout pole atq_s= 0. Let−be the automorphism ofKsendingq_stoq_s⁻¹. Then A₀ coincides with the ringA_∞ of rational functions regular atq_s =∞. SetA:=Q[qs, qs−1]. Let V be a vector space over K,L0 anA-submodule of V,L_∞ anA_∞- submodule, andVAa A-submodule. SetE:=L0∩L_∞∩VA.

Definition 2.3 [10]. We say that (L0, L_∞, VA) is balanced if each of L0, L_∞ andVA generatesV as aK-vector space, and if one of the following equivalent conditions is satisfied.

(i) E→L0/qsL0 is an isomorphism.

(ii) E→L_∞/qs−1L_∞is an isomorphism.

(iii) (L0∩VA)⊕(qs−1L_∞∩VA)→VA is an isomorphism.

(iv) A₀⊗_QE →L₀, A_∞⊗_QE →L_∞, A⊗_QE →V_A and K⊗_QE →V are isomorphisms.

Let−be the ring automorphism ofU_q⁻(g) sendingq_s,q(h),e_i,f_i toq_s⁻¹, q(−h),ei,fi.

Let U_q⁻(g)A be the A-subalgebra of U_q⁻(g) generated by e⁽ⁿ⁾_i , f_i⁽ⁿ⁾ and q(h) (h∈d⁻¹P^∗).

LetM be aU_q⁻(g)-module. Let−be an involution ofM satisfying (au)⁻ =

¯

a¯ufor anya∈U_q⁻(g) andu∈M. We call in this paper such an involution abar involution. Let (L(M), B(M)) be a crystal base of an integrableU_q⁻(g)-module M.

LetMA be aU_q⁻(g)A-submodule ofM such that

(MA)⁻=MA, and (u−u)∈(qs−1)MA for everyu∈MA. (2.21)

Definition 2.4. A U_q⁻(g)-module M endowed with (L(M), B(M), M_A,−) as above is called with a global basis, if (L(M), L(M)⁻, MA) is balanced,

In such a case, letG: L(M)/qsL(M)−→^∼ E:=L(M)∩L(M)⁻∩MAbe the inverse of E−→^∼ L(M)/q_sL(M). Then {G(b);b ∈ B(M)} forms a basis of M. We call this basis a (lower) global basis. The global basis enjoys the following properties (see [10, 11]):

(i) G(b) =G(b) for anyb∈B(M).

(ii) For anyn∈Z0,{G(b);ε_i(b)n} is a basis of theA-submodule

mn

f_i^(m)MA.

(iii) for anyi∈Iandb∈B(M), we have

f_iG(b) = [1 +ε_i(b)]_iG( ˜f_ib) +

b

F_b,bⁱ G(b).

Here the sum ranges over b ∈ B(M) such that ε_i(b) > 1 +ε_i(b). The coefficientF_b,bⁱ belongs toq_sq¹_i^−εi(b)Q[q_s]. Similarly fore_iG(b).

LetM andN beU_q⁻(g)-modules with global bases. We say that aU_q⁻(g)- morphismf:M →N iscompatible with global bases if it satisfies the following conditions:

(i) If uis a global basis vector ofM, thenf(u) is a global basis vector of N or 0.

(ii) If a pair of global basis vectors uand v of M satisfies f(u) = f(v) = 0, thenu=v.

These conditions are equivalent to the following set of conditions:

(a) f commutes with the bar involutions.

(b) f sendsL(M) toL(N) andMA toNA.

(c) The induced morphismf:L(M)/qsL(M)→L(N)/qsL(N) sendsB(M) to B(N)∪ {0}.

(d) Ker(f) is generated by a part of the global basis ofM.

In such a case,f(M) has a global basis, and we have B(M)⊃B(f(M))⊂B(N).

If f is a monomorphism then B(M) B(f(M)) ⊂ B(N), and if f is an epimorphism thenB(M)⊃B(f(M))B(N).

§2.6. Extremal vectors

LetM be an integrableU_q⁻(g)-module. A non-zero vectoru∈M of weight λ∈P is calledextremal(see [13]), if we can find a subsetF of non-zero weight vectors inM containinguand satisfying the following properties:

(2.22)

ifv∈F andisatisfyhi,wt(v)0, theneiv= 0 andf_i^(hi,wt(v))v∈F, ifv∈F andisatisfyh_i,wt(v)0, thenf_iv= 0 and e⁽_i^−hi,wt(v))v∈F, The Weyl groupW acts on the set of extremal vectors by

ifhi,wt(u)0, thenS_s^norm_i u=f_i^(hi,wt(u))u, ifh_i,wt(u)0, thenS_s^norm

i u=e⁽_i^−hi,wt(u))u.

(2.23)

We have wt(S_w^normu) = wwt(u) for w ∈ W. Note that, by (2.7), S_w^normu is equal toSwuup to a non-zero constant multiple.

Similarly, for a vectorb of a regular crystalB with weightλ, we say that b is an extremal vector if it satisfies the following similar conditions:

ifw∈W andi∈I satisfyh_i, wλ0, then ˜e_iS_wb= 0, ifw∈W andi∈I satisfyh_i, wλ0 then ˜f_iS_wb= 0.

(2.24)

Forλ∈P, let us denote byV(λ) theU_q⁻(g)-module generated byu_λwith the defining relation thatuλ is an extremal vector of weightλ. This is in fact infinitely many linear relations on uλ.

For a dominant weight λ, V(λ) is an irreducible highest weight module with highest weightλ, andV(−λ) is an irreducible lowest weight module with lowest weight−λ.

We proved in [13]¹thatV(λ) has a global basis (L(λ), B(λ)). We denote by the same letteruλ the element ofB(λ) corresponding touλ∈V(λ). Moreover U_q⁻(g)aλ→V(λ) (aλ→uλ) is compatible with global bases. Hence the crystal

1In [13], it is denoted by V^max(λ), because I thought there would be a natural Uq⁻(g)- module whose crystal base is the connected component ofB(λ).

B(λ) is isomorphic to the subcrystal of B(∞)⊗t_λ ⊗B(−∞) consisting of vectorsbsuch thatb^∗is an extremal vector of weight−λ. By this embedding, uλ∈B(λ) corresponds tou_∞⊗tλ⊗u_−∞.

Note that

U_q⁺(g)u_λ=

b∈B(λ)∩

u_∞⊗t_λ⊗B(−∞)KG(b).

(2.25)

For anyw∈W,u_λ→S_w^norm₋₁ u_wλ gives an isomorphism ofU_q⁻(g)-modules:

V(λ)−→^∼ V(wλ).

This is compatible with global bases. Similarly, lettingS_w^∗ be the Weyl group action onB(Uq(g)) with respect to the star crystal structure and regardingB(λ) as a subcrystal ofB(Uq(g)),S_w^∗:B(Uq(g))−→^∼ B(Uq(g)) induces an isomorphism of crystals

S_w^∗:B(λ)−→^∼ B(wλ).

(2.26)

This coincides with the crystal isomorphism induced byV(λ)−→^∼ V(wλ). Note that we have

SwS_w^∗(u_∞⊗tλ⊗u_−∞) =u_∞⊗twλ⊗u_−∞.

§2.7. Global bases of tensor products

Let us recall the following results proved by Lusztig ([19]). Let Oint be the category of integrable U_q⁻(g)-modules which are a direct sum of V(λ)’s (λ ∈ P⁺). Similarly let Oint⁻ be the category of integrable U_q⁻(g)-modules which are a direct sum ofV(λ)’s (λ∈P⁻). LetM andN beU_q⁻(g)-modules.

Assume that M and N have bar involutions, and that either M ∈ Oint or N ∈Oint⁻. Then there exists a unique bar involution onM⊗N such that

(u⊗v)⁻ = ¯u⊗v¯ for everyu∈M andv∈N such that either

uis a highest weight vector orvis a lowest weight vector.

Assume further that M and N have a global basis. Then M ⊗N has a crystal base (L(M⊗N), B(M⊗N)) := (L(M)⊗A₀L(N), B(M)⊗B(N)), and anA-form (M⊗N)A=MA⊗ANA. ThenM⊗N has a global basis; namely (L(M ⊗N), L(M⊗N)⁻,(M ⊗N)_A) is balanced. In particular, V(λ)⊗V(µ) has a global basis either ifλis dominant or if−µis dominant.

Letλ∈P. Then for any pair of dominant integral weightsξand η such thatλ=ξ−η,U_q⁻(g)a_λ→V(ξ)⊗V(−η) (a_λ→u_ξ⊗u_−η) is compatible with global bases. Conversely the global basis of U_q⁻(g)aλ is characterized by the above property.

Lemma 2.5. Forλ∈P⁺ andµ∈P,

U_q⁻(g)a_λ+µ→V(λ)⊗U_q⁻(g)a_µ (a_λ+µ→u_λ⊗a_µ) (2.27)

is compatible with global bases.

Proof. For dominant integral weights ξ and η such that µ = ξ−η, we have a diagram of morphisms compatible with crystal basses except the dotted arrow:

Uq(g)aλ+µ ////

V(λ+ξ)⊗ _ V(−η)

V(λ)⊗U_q⁻(g)aµ ////V(λ)⊗V(ξ)⊗V(−η) Hence the dotted arrow is compatible with crystal bases.

This morphism (2.27) induces an embedding of crystals

B(U_q⁻(g)aλ+µ)→B(λ)⊗B(U_q⁻(g)aµ) forλ∈P⁺ andµ∈P.

There exists an embedding B(∞) → B(λ)⊗B(∞)⊗T₋λ, and the above morphism coincides with the composition

B(U_q⁻(g)aλ+µ)B(∞)⊗Tλ+µ⊗B(−∞)→B(λ)⊗B(∞)

⊗T_−λ⊗T_λ+µ⊗B(−∞)

B(λ)⊗B(∞)⊗Tµ⊗B(−∞)B(λ)⊗B(U_q⁻(g)aµ).

§2.8. Demazure modules

LetM be an integrable U_q⁻(g)-module with a global basis (L(M),B(M), MA,−). Let N be a U_q⁺(g)-submodule of M. We say that N is compatible with the global basis of M if there exists a subset B(N) of B(M) such that N =⊕b∈B(N)KG(b).

It is shown in [12] that

˜

e_iB(N) ⊂ B(N)∪ {0}, and U_q⁻(g)N = U_q⁻(g)N is also compatible with the global basis.

(2.28)

Namely there exists a subset B(U_q⁻(g)N) ofB(M) such that U_q⁻(g)N =

b∈B(U_q⁻(g)N)

KG(b).

Moreover we have B(U_q⁻(g)N) =

f˜i1· · ·f˜imb;m0, i1, . . . im∈I, b∈B(N) \ {0}.

Forλ∈P, theU_q^±(g)-submoduleU_q^±(g)uλofV(λ) is compatible with the global basis ofV(λ) (see (2.25)).

We set

B^±(λ) =B(U_q^±(g)u_λ).

RegardingB(λ) as a subset of B(U_q⁻(g)aλ) =B(∞)⊗tλ⊗B(−∞), we have B⁺(λ) =B(λ)∩

u_∞⊗tλ⊗B(−∞)

and B⁻(λ) =B(λ)∩

B(∞)⊗tλ⊗u_−∞

. The subsetB⁺(λ) satisfies the following properties.

Lemma 2.6.

(i) ˜eiB⁺(λ)⊂B⁺(λ)∪ {0}.

(ii) For any b∈B⁺(λ),if εi(b)>0,thenf˜ib∈B⁺(λ)∪ {0}. Or equivalently, for any i-stringS of B(λ), S∩B⁺(λ)is either S itself, the empty set or the set consisting of the highest weight vector of S. Here an i-string is a connected component with respect to the crystal structure overU_q⁻(g)i.

This is a consequence of the following lemma. Note that B(U_q⁺(g)aλ) = u_∞⊗Tλ⊗B(−∞).

Lemma 2.7.

(i) ˜eiB(U_q⁺(g)aλ)⊂B(U_q⁺(g)aλ)∪ {0}.

(ii) For any b∈ B(U_q⁺(g)aλ), if εi(b)>0, then f˜ib ∈B(U_q⁺(g)aλ)∪ {0}. Or equivalently, for any i-stringS of B(U_q⁻(g)aλ), S∩B(U_q⁺(g)aλ)is either S itself, the empty set or the set consisting of the highest weight vector of S.

Proof. The first property is evident. In order to prove (ii), write b = u_∞⊗t_λ⊗b with b ∈ B(−∞). Thenε_i(b) = max(0, ε_i(t_λ⊗b)), and hence 0 =ϕ_i(u_∞)< ε_i(t_λ⊗b). We have therefore ˜f_ib=u_∞⊗t_λ⊗f˜_ib.

Similar results hold forB⁻(λ) andB(U_q⁻(g)aλ).

Proposition 2.8. For β ∈ ∆^re₊ and λ ∈P, assume (β, λ) 0. Then we have

Ss_βuλ∈U_q⁻(g)uλ and Ss_βuλ∈B⁻(λ).

Proof. We shall argue by the induction of ht(β). Let us take i∈I such that hi, β > 0. If β = αi then the assertion is trivial. Otherwise we have γ:=si(β)∈∆^re₊. Since (γ, siλ) = (β, λ)0, the induction hypothesis implies that

SsγSiuλ∈U_q⁻(g)Siuλ. (2.29)

Ifhi, λ0, we have

U_q⁻(g)u_λ⊃U_q⁻(g)S_iu_λ⊃U_q⁻(g)S_sγS_iu_λ=S_iS_sβu_λ SinceU_q⁻(g)uλ is anUq(g)i-module, it contains Ss_βuλ.

Now assume thathi, λ<0. Thenhi, sβλ=hi, λ−β^∨, λhi, β<0.

By (2.29), we have

S_iS_sβS_i(u_∞⊗t_siλ⊗u_−∞) =S_sγ(u_∞⊗t_siλ⊗u_−∞)

∈B(∞)⊗ts_iλ⊗u_−∞. Applying ˜e^∗_i^max, we have (here ˜e^∗_i^maxb= ˜e^∗_i^ε∗i(b)b and ˜f_i^maxb= ˜f_i^ϕi(b)b)

SiSs_β(u_∞⊗tλ⊗u_−∞) = ˜e^∗_i^maxSiSs_βSi(u_∞⊗ts_iλ⊗u_−∞)

∈˜e^∗_i^max(B(∞)⊗t_siλ⊗u_−∞), and hence

Ss_β(u_∞⊗tλ⊗u_−∞) = ˜fimaxSiSs_β(u_∞⊗tλ⊗u_−∞)

∈f˜_i^maxe˜^∗_i^max

B(∞)⊗t_siλ⊗u_−∞

⊂f˜_i^max

n0

B(∞)⊗t_λ⊗e˜ⁿ_iu_−∞

⊂B(∞)⊗tλ⊗u_−∞. The last inclusion follows from

f˜_i^max(b1⊗tλ⊗b2) =b₁⊗tλ⊗f˜_i^maxb2 for some b₁∈B(∞).

§2.9. Affine case

Until now, we have assumed thatgis a symmetrizable Kac-Moody algebra.

From now on, we assume further thatU_q⁻(g) is a quantizedaffinealgebra.

2.9.1. Extended Weyl groups We take a weight latticeP of rank rk(g)+1 and an inner product onP as in [14]. We sett^∗=Q⊗P, which is canonically determined by the Dynkin diagram.

Let us defineδ∈

iZ0αi andc∈

iZ0hi by {λ∈

iZαi;hi, λ= 0 for everyi∈I}=Zδ, {h∈

iZhi;h, αi= 0 for everyi∈I}=Zc.

(2.30)

By the inner product of t^∗, we sometimes identify t^∗ and its dual. Note that the inner product ont^∗ is so normalized thatδandc correspond by this iden- tification.

Forα∈∆^re, we setc_α:= max(1,(α, α)/2)∈Z. Then we have (α+Zδ)∩∆ =α+cαZ.

Let us denote byP_cl the quotient spaceP/(P∩Qδ), and let us denote by cl :P →P_clthe canonical projection. Let us denote by P_cl^∗ the dual lattice of Pcl, i.e.P_cl^∗ = Ker(δ:P^∗→Z) = (

iQhi)∩P^∗.

Similarly toPcl, we definet^∗_cl:=t^∗/Qδ, and let cl :t^∗→t^∗_clbe the canonical projection. Definet^∗0:= Ker(c:t^∗ →Q), and t^∗_cl⁰= cl(t^∗0). The dimension of t^∗_cl⁰ is equal to rk(g)−1. The inner product of t^∗ induces a positive definite inner product ont^∗_cl⁰.

Let us denote by O(t^∗) the orthogonal group, and O(t^∗)_δ :=

{g∈O(t^∗) ;gδ=δ} the isotropy subgroup at δ. Then there is an exact se- quence

1−−−→t^∗_cl⁰−−→^t O(t^∗)δ cl₀

−−−→O(t^∗_cl⁰)−−−→1.

Heret:tcl0→O(t^∗)δ is given by

t(cl(ξ))(λ) =λ+ (λ, δ)ξ−(λ, ξ)δ−(ξ, ξ)

2 (λ, δ)δ forξ∈t^∗0 andλ∈t^∗. Let us set Wcl = cl0(W). Then Wcl is the Weyl group of the root system

∆cl:= cl(∆^re)⊂t^∗_cl⁰. We define the extended Weyl groupWby W:={w∈O(t^∗)δ;w∆ = ∆ and cl0(w)∈Wcl}.

Then we have a commutative diagram with the exact rows:

1 //Q //

_

W //

_

Wcl //1

1 //P_ //W //

_

Wcl //

_

1

1 //t^∗_cl^{0 t} //O(t^∗)_δ ^cl0 //O(t^∗_cl⁰) //1 HereP andQare given by

P =P_cl⁰∩P_cl^∨0 and Q=Qcl∩Q^∨_cl, where

P_cl⁰:=

λ∈t^∗_cl⁰;hi, λ ∈Zfor everyi∈I , P_cl^0∨:=

λ∈t^∗_cl⁰; (αi, λ)∈Zfor everyi∈I , Qcl:=

i∈I

Zcl(αi), Q^∨_cl:=

i∈I

Zcl(h_i).

The Weyl groupW is a normal subgroup ofW, andWis a semi-direct product of W and Aut₀(Dyn) :={ι;ιis a Dynkin diagram automorphism such that} cl₀(ι)∈W_cl.

P /Q−→^∼ W /W −→^∼ Aut0(Dyn).

Remark2.9.

(i) Ifgis untwisted, then (α, α)/21 for everyα∈∆^re and P =P_cl^0∨⊂P_cl⁰, Q=Q^∨_cl⊂Qcl.

(ii) Ifgis the dual of an untwisted affine algebra, then (α, α)/21 for every α∈∆^re and

P =P_cl⁰ ⊂P_cl^0∨, Q=Q_cl⊂Q^∨_cl. (iii) Ifg=A⁽²⁾_2n, then we have (α, α)/2 = 1/2, 1 or 2, and

P =Q=P_cl⁰ =P_cl^0∨=Qcl=Q^∨_cl=

α∈∆^re

Zcl(α) =

α∈∆^re,(α,α)/2=1

Zcl(α).