**Symmetric Crystals for** gl

_{∞}*Dedicated to Professor Heisuke Hironaka on the occasion of*
*his seventy-seventh birthday*

By

NaoyaEnomoto* ^{∗}*and MasakiKashiwara

^{∗∗}**Abstract**

In the preceding paper, we formulated a conjecture on the relations between
certain classes of irreducible representations of aﬃne Hecke algebras of type B and
symmetric crystals for gl* _{∞}*. In the present paper, we prove the existence of the
symmetric crystal and the global basis forgl

*.*

_{∞}**§****1.** **Introduction**

Lascoux-Leclerc-Thibon ([LLT]) conjectured the relations between the rep-
resentations of Hecke algebras of*type A*and the crystal bases of the aﬃne Lie
algebras of type A. Then, S. Ariki ([A]) observed that it should be understood
in the setting of aﬃne Hecke algebras and proved the LLT conjecture in a more
general framework. Recently, we presented the notion of symmetric crystals
and conjectured that certain classes of irreducible representations of the aﬃne
Hecke algebras of*type B*are described by symmetric crystals forgl* _{∞}*([EK]).

The purpose of the present paper is to prove the existence of symmetric
crystals in the case ofgl* _{∞}*.

Let us recall the Lascoux-Leclerc-Thibon-Ariki theory. Let H^{A}* _{n}* be the
aﬃne Hecke algebra of type A of degree

*n*. Let K

^{A}

*be the Grothendieck group*

_{n}Communicated by T. Kawai. Received May 14, 2007. Revised November 18, 2007.

2000 Mathematics Subject Classiﬁcation(s): Primary 17B37; Secondary 20C08.

Key words: crystal bases, aﬃne Hecke algebras, LLT conjecture.

The second author is partially supported by Grant-in-Aid for Scientiﬁc Research (B) 18340007, Japan Society for the Promotion of Science.

*∗*Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606–8502, Japan.

e-mail: henon@kurims.kyoto-u.ac.jp

*∗∗*e-mail: masaki@kurims.kyoto-u.ac.jp

c 2008 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

Naoya Enomoto and Masaki Kashiwara

of the abelian category of ﬁnite-dimensional H^{A}* _{n}*-modules, and K

^{A}=

*⊕*

*n0*K

^{A}

*. Then it has a structure of Hopf algebra by the restriction and the induction.*

_{n}The set *I* = C* ^{∗}* may be regarded as a Dynkin diagram with

*I*as the set of vertices and with edges between

*a∈I*and

*ap*

^{2}

_{1}. Here

*p*1is the parameter of the aﬃne Hecke algebra usually denoted by

*q*. Letg

*be the associated Lie algebra, andg*

_{I}

^{−}*the unipotent Lie subalgebra. Let*

_{I}*U*

*I*be the group associated to g

^{−}*. Hence g*

_{I}*is isomorphic to a direct sum of copies of*

_{I}*A*

^{(1)}

*if*

_{−1}*p*

^{2}

_{1}is a primitive -th root of unity and to a direct sum of copies of gl

*if*

_{∞}*p*1 has an inﬁnite order. ThenC

*⊗*K

^{A}is isomorphic to the algebra

*O*(

*U*

*I*) of regular functions on

*U*

*I*. Let

*U*

*q*(g

*) be the associated quantized enveloping algebra. Then*

_{I}*U*

_{q}*(g*

^{−}*) has an upper global basis*

_{I}*{*G

^{up}(

*b*)

*}*

*. By specializing*

_{b∈B(∞)}C[*q, q** ^{−1}*]G

^{up}(

*b*) at

*q*= 1, we obtain

*O*(

*U*

*I*). Then the LLTA-theory says that the elements associated to irreducible H

^{A}-modules corresponds to the image of the upper global basis.

In [EK], we gave analogous conjectures for aﬃne Hecke algebras of type
B. In the type B case, we have to replace*U*_{q}* ^{−}*(g

*) and its upper global basis with symmetric crystals (see*

_{I}*§*2.3). It is roughly stated as follows. Let H

^{B}

*be the aﬃne Hecke algebra of type B of degree*

_{n}*n*. Let K

^{B}

*be the Grothendieck group of the abelian category of ﬁnite-dimensional modules over H*

_{n}^{B}

*, and K*

_{n}^{B}=

*⊕**n0*K^{B}* _{n}*. Then K

^{B}has a structure of a Hopf bimodule over K

^{A}. The group

*U*

*I*has the anti-involution

*θ*induced by the involution

*a→a*

*of*

^{−1}*I*=C

*. Let*

^{∗}*U*

_{I}*be the*

^{θ}*θ*-ﬁxed point set of

*U*

*I*. Then

*O*(

*U*

_{I}*) is a quotient ring of*

^{θ}*O*(

*U*

*I*).

The action of*O*(*U**I*)C*⊗*K^{A} onC*⊗*K^{B}, in fact, descends to the action of
*O*(*U*_{I}* ^{θ}*).

We introduce *V**θ*(*λ*) (see*§*2.3), a kind of the *q*-analogue of *O*(*U*_{I}* ^{θ}*). The
conjecture in [EK] is then:

(i) *V**θ*(*λ*) has a crystal basis and a global basis.

(ii) K^{B}is isomorphic to a specialization of*V**θ*(*λ*) at*q*= 1 as an*O*(*U**I*)-module,
and the irreducible representations correspond to the upper global basis
of*V**θ*(*λ*) at*q*= 1.

*Remark.* In [KM], Miemietz and the second author gave an analogous
conjecture for the aﬃne Hecke algebras of type D.

In the present paper, we prove that*V**θ*(*λ*) has a crystal basis and a global
basis forg=gl* _{∞}* and

*λ*= 0.

More precisely, let*I*=Zodd be the set of odd integers. Let*α**i* (*i∈I*) be

the simple roots with

(*α**i**, α**j*) =

2 if*i*=*j*,

*−*1 if*i*=*j±*2,
0 otherwise.

Let*θ* be the involution of*I* given by*θ*(*i*) =*−i*. Let *B**θ*(gl* _{∞}*) be the algebra
over

**K**:=Q(

*q*) generated by

*E*

*i*,

*F*

*i*, and invertible elements

*T*

*i*(

*i∈I*) satisfying the following deﬁning relations:

(i) the*T**i*’s commute with each other,
(ii) *T**θ(i)*=*T**i* for any*i*,

(iii) *T**i**E**j**T*_{i}* ^{−1}*=

*q*

^{(α}

^{i}^{+α}

^{θ(i)}

^{,α}

^{j}^{)}

*E*

*j*and

*T*

*i*

*F*

*j*

*T*

_{i}*=*

^{−1}*q*

^{(α}

^{i}^{+α}

^{θ(i)}

^{,−α}

^{j}^{)}

*F*

*j*for

*i, j∈I*, (iv)

*E*

*i*

*F*

*j*=

*q*

^{−(α}

^{i}

^{,α}

^{j}^{)}

*F*

*j*

*E*

*i*+ (

*δ*

*i,j*+

*δ*

*θ(i),j*

*T*

*i*) for

*i, j∈I*,

(v) the*E**i*’s and the*F**i*’s satisfy the Serre relations (see Deﬁnition 2.1 (4)).

Then there exists a unique irreducible*B**θ*(gl* _{∞}*)-module

*V*

*θ*(0) with a generator

*φ*satisfying

*E*

*i*

*φ*= 0 and

*T*

*i*

*φ*=

*φ*(Proposition 2.11). We deﬁne the endomor- phisms

*E*

*i*and

*F*

*i*of

*V*

*θ*(0) by

*E**i**a*=

*n1*

*F*_{i}^{(n−1)}*a**n**,* *F**i**a*=

*n0*

*f*_{i}^{(n+1)}*a**n**,*
when writing

*a*=

*n0*

*F*_{i}^{(n)}*a**n* with*E**i**a**n*= 0*.*

Here *F*_{i}^{(n)} = *F*_{i}^{n}*/*[*n*]! is the divided power. Let **A**_{0} be the ring of functions
*a∈***K** which do not have a pole at *q*= 0. Let *L**θ*(0) be the**A**_{0}-submodule of
*V**θ*(0) generated by the elements*F**i*1*· · ·F**i**φ*(0,*i*1*, . . . , i**∈I*). Let B* _{θ}*(0)
be the subset of

*L*

*θ*(0)

*/qL*

*θ*(0) consisting of the

*F*

*i*1

*· · ·F*

*i*

*φ*’s. In this paper, we prove the following theorem.

**Theorem** (Theorem 4.15).

(i) *F**i**L**θ*(0)*⊂L**θ*(0) *andE**i**L**θ*(0)*⊂L**θ*(0),
(ii) B* _{θ}*(0)

*is a basis ofL*

*θ*(0)

*/qL*

*θ*(0),

(iii) *F**i*B* _{θ}*(0)

*⊂*B

*(0), and*

_{θ}*E*

*i*B

*(0)*

_{θ}*⊂*B

*(0)*

_{θ}*{*0

*},*

Naoya Enomoto and Masaki Kashiwara

(iv) *F**i**E**i*(*b*) =*bfor anyb∈*B* _{θ}*(0)

*such thatE*

*i*

*b*= 0, and

*E*

*i*

*F*

*i*(

*b*) =

*bfor any*

*b∈*B

*(0).*

_{θ}By this theorem, B* _{θ}*(0) has a similar structure to the crystal structure.

Namely, we have operators *F**i*: B* _{θ}*(0)

*→*B

*(0) and*

_{θ}*E*

*i*: B

*(0)*

_{θ}*→*B

*(0)*

_{θ}*{*0

*}*, which satisfy (iv). Moreover

*ε*

*i*(

*b*) := max

*n∈*Z0*|E*_{i}^{n}*b∈*B* _{θ}*(0) is ﬁnite.

We call it the*symmetric crystal*associated with (*I, θ*). Contrary to the usual
crystal case,*E**θ(i)**b* may coincide with*E**i**b*in the symmetric crystal case.

Let*−*be the bar operator of*V**θ*(0). Namely,*−*is a unique endomorphism
of *V**θ*(0) such that *φ*= *φ*, *av* = ¯*av*¯ and *F**i**v* =*F**i**v*¯ for*a* *∈* **K** and *v* *∈* *V**θ*(0).

Here ¯*a*(*q*) = *a*(*q** ^{−1}*). Let

*V*

*θ*(0)

**be the smallest submodule of**

_{A}*V*

*θ*(0) over

**A**:=Q[

*q, q*

*] such that it contains*

^{−1}*φ*and is stable by the

*F*

_{i}^{(n)}’s.

Then we prove the existence of global basis:

**Theorem** (Theorem 5.5).

(i) *For any* *b∈* B* _{θ}*(0), there exists a unique G

^{low}

*(*

_{θ}*b*)

*∈V*

*θ*(0)

_{A}*∩L*

*θ*(0)

*such*

*that*G

^{low}

*(*

_{θ}*b*) = G

^{low}

*(*

_{θ}*b*)

*andb*= G

^{low}

*(*

_{θ}*b*) mod

*qL*

*θ*(0),

(ii) *{*G^{low}* _{θ}* (

*b*)

*}*

_{b∈B}*(0)*

_{θ}*is a basis of the*

**A**

_{0}

*-moduleL*

*θ*(0), the

**A-module**

*V*

*θ*(0)

_{A}*and the*

**K-vector space**

*V*

*θ*(0).

We call G^{low}* _{θ}* (

*b*) the

*lower global basis. The*

*B*

*(gl*

_{θ}*)-module*

_{∞}*V*

*θ*(0) has a unique symmetric bilinear form (

*•*

*,*

*) such that (*

^{•}*φ, φ*) = 1 and

*E*

*i*and

*F*

*i*are transpose to each other. The dual basis to

*{*G

^{low}

*(*

_{θ}*b*)

*}*

_{b∈B}*(0) with respect to (*

_{θ}

^{•}*,*

*) is called an*

^{•}*upper global basis.*

Let us explain the strategy of our proof of these theorems. We ﬁrst con-
struct a PBW type basis*{P**θ*(m)*φ}*m of*V**θ*(0) parametrized by the*θ*-restricted
multisegments m. Then, we explicitly calculate the actions of *E**i* and *F**i* in
terms of the PBW basis*{P**θ*(m)*φ}*m. Then, we prove that the PBW basis gives
a crystal basis by the estimation of the coeﬃcients of these actions. For this
we use a criterion for crystal bases (Theorem 4.1).

**§****2.** **General Deﬁnitions and Conjectures**

**§****2.1.** **Quantized universal enveloping algebras and**
**its reduced** *q***-analogues**

We shall recall the quantized universal enveloping algebra*U**q*(g). Let*I*be
an index set (for simple roots), and*Q*the freeZ-module with a basis*{α**i**}**i∈I*.

Let (^{•}*,** ^{•}*) :

*Q×Q→*Zbe a symmetric bilinear form such that (

*α*

*i*

*, α*

*i*)

*/*2

*∈*Z

*>0*

for any*i* and (*α*^{∨}_{i}*, α**j*)*∈*Z0 for *i*=*j* where *α*^{∨}* _{i}* := 2

*α*

*i*

*/*(

*α*

*i*

*, α*

*i*). Let

*q*be an indeterminate and set

**K**:=Q(

*q*). We deﬁne its subrings

**A**

_{0},

**A**

*and*

_{∞}**A**as follows.

**A**_{0}=*{f* *∈***K***|f* is regular at*q*= 0*},*
**A*** _{∞}*=

*{f*

*∈*

**K**

*|f*is regular at

*q*=

*∞},*

**A**=Q[*q, q** ^{−1}*]

*.*

**Deﬁnition 2.1.** The quantized universal enveloping algebra *U**q*(g) is
the**K-algebra generated by elements** *e**i**, f**i* and invertible elements *t**i* (*i* *∈* *I*)
with the following deﬁning relations.

(1) The*t**i*’s commute with each other.

(2) *t**j**e**i**t*^{−1}* _{j}* =

*q*

^{(α}

^{j}

^{,α}

^{i}^{)}

*e*

*i*and

*t*

*j*

*f*

*i*

*t*

^{−1}*=*

_{j}*q*

^{−(α}

^{j}

^{,α}

^{i}^{)}

*f*

*i*for any

*i, j∈I*. (3) [

*e*

*i*

*, f*

*j*] =

*δ*

*ij*

*t*

*i*

*−t*

^{−1}

_{i}*q**i**−q*_{i}* ^{−1}* for

*i*,

*j∈I*. Here

*q*

*i*:=

*q*

^{(α}

^{i}

^{,α}

^{i}^{)/2}. (4) (Serre relation) For

*i*=

*j*,

*b*
*k=0*

(*−*1)^{k}*e*^{(k)}_{i}*e**j**e*^{(b−k)}* _{i}* = 0

*,*

*b*

*k=0*

(*−*1)^{k}*f*_{i}^{(k)}*f**j**f*_{i}^{(b−k)}= 0*.*
Here *b*= 1*−*(*α*^{∨}_{i}*, α**j*) and

*e*^{(k)}* _{i}* =

*e*

^{k}

_{i}*/*[

*k*]

*!*

_{i}*, f*

_{i}^{(k)}=

*f*

_{i}

^{k}*/*[

*k*]

*!*

_{i}*,*

[*k*]* _{i}* = (

*q*

^{k}

_{i}*−q*

^{−k}*)*

_{i}*/*(

*q*

*i*

*−q*

^{−1}*)*

_{i}*,*[

*k*]

*! = [1]*

_{i}

_{i}*· · ·*[

*k*]

_{i}*.*

Let us denote by*U*_{q}* ^{−}*(g) (resp.

*U*

_{q}^{+}(g)) the

**K-subalgebra of**

*U*

*q*(g) generated by the

*f*

*i*’s (resp. the

*e*

*i*’s).

Let*e*^{}* _{i}* and

*e*

^{∗}*be the operators on*

_{i}*U*

_{q}*(g) (see [K1, 3.4]) deﬁned by [*

^{−}*e*

*i*

*, a*] = (

*e*

^{∗}

_{i}*a*)

*t*

*i*

*−t*

^{−1}

_{i}*e*

^{}

_{i}*a*

*q**i**−q*^{−1}* _{i}* (

*a∈U*

*q*

*(g))*

^{−}*.*

These operators satisfy the following formulas similar to derivations:

*e*^{}* _{i}*(

*ab*) =

*e*

^{}*(*

_{i}*a*)

*b*+ (Ad(

*t*

*i*)

*a*)

*e*

^{}

_{i}*b,*

*e*

^{∗}*(*

_{i}*ab*) =

*ae*

^{∗}

_{i}*b*+ (

*e*

^{∗}

_{i}*a*)(Ad(

*t*

*i*)

*b*)

*.*(2.1)

Naoya Enomoto and Masaki Kashiwara

Note that in [K1], the operator*e*^{}*i* was deﬁned. It satisﬁes*e*^{}*i* =*−◦e*^{}*i**◦−*, while
*e*^{∗}* _{i}* satisﬁes

*e*

^{∗}*=*

_{i}*∗ ◦e*

^{}

_{i}*◦ ∗*. They are related by

*e*

^{∗}*= Ad(*

_{i}*t*

*i*)

*◦e*

^{}*.*

_{i}The algebra*U*_{q}* ^{−}*(g) has a unique symmetric bilinear form (

*•*

*,*

*) such that (1*

^{•}*,*1) = 1 and

(*e*^{}_{i}*a, b*) = (*a, f**i**b*) for any*a, b∈U*_{q}* ^{−}*(g).

It is non-degenerate and satisﬁes (*e*^{∗}_{i}*a, b*) = (*a, bf**i*). The left multiplication of
*f**j**, e*^{}* _{i}* and

*e*

^{∗}*have the commutation relations*

_{i}*e*^{}_{i}*f**j* =*q*^{−(α}^{i}^{,α}^{j}^{)}*f**j**e*^{}* _{i}*+

*δ*

*ij*

*, e*

^{∗}

_{i}*f*

*j*=

*f*

*j*

*e*

^{∗}*+*

_{i}*δ*

*ij*Ad(

*t*

*i*)

*,*and both the

*e*

^{}*’s and the*

_{i}*e*

^{∗}*’s satisfy the Serre relations.*

_{i}**Deﬁnition 2.2.** The reduced*q*-analogue*B*(g) ofgis the**K-algebra gen-**
erated by*e*^{}* _{i}* and

*f*

*i*.

**§****2.2.** **Review on crystal bases and global bases**

Since*e*^{}* _{i}*and

*f*

*i*satisfy the

*q*-boson relation, any element

*a∈U*

_{q}*(g) can be uniquely written as*

^{−}*a*=

*n0*

*f*_{i}^{(n)}*a**n* with*e*^{}_{i}*a**n*= 0*.*

Here*f*_{i}^{(n)}= *f*_{i}* ^{n}*
[

*n*]

*!.*

_{i}**Deﬁnition 2.3.** We deﬁne the modiﬁed root operators *e**i* and *f**i* on
*U*_{q}* ^{−}*(g) by

*e**i**a*=

*n1*

*f*_{i}^{(n−1)}*a**n**,* *f**i**a*=

*n0*

*f*_{i}^{(n+1)}*a**n**.*
**Theorem 2.4**([K1]). *We deﬁne*

*L*(*∞*) =

*0, i*1*,...,i**∈I*

**A**_{0}*f*˜*i*1*· · ·f*˜*i* *·*1*⊂U**q** ^{−}*(g)

*,*B(

*∞*) =

*f*˜*i*1*· · ·f*˜*i**·*1 mod*qL*(*∞*)*|*0*, i*1*,· · ·, i**∈I* *⊂L*(*∞*)*/qL*(*∞*)*.*

*Then we have*

(i) *e**i**L*(*∞*)*⊂L*(*∞*) *andf**i**L*(*∞*)*⊂L*(*∞*),
(ii) B(*∞*)*is a basis of* *L*(*∞*)*/qL*(*∞*),

(iii) *f**i*B(*∞*)*⊂*B(*∞*)*ande**i*B(*∞*)*⊂*B(*∞*)*∪ {*0*}.*
*We call*(*L*(*∞*)*,*B(*∞*))*the* crystal basis*of* *U*_{q}* ^{−}*(g).

Let *−* be the automorphism of **K** sending *q* to *q** ^{−1}*. Then

**A**

_{0}coincides with

**A**

*.*

_{∞}Let *V* be a vector space over**K,***L*0 an**A**_{0}-submodule of*V*, *L**∞* an**A*** _{∞}*-
submodule, and

*V*

**A**an

**A-submodule. Set**

*E*:=

*L*0

*∩L*

*∞*

*∩V*

**A**.

**Deﬁnition 2.5** ([K1], [K2, 2.1]). We say that (*L*0*, L**∞**, V***A**) is*balanced*
if each of*L*0,*L**∞* and *V***A** generates*V* as a **K-vector space, and if one of the**
following equivalent conditions is satisﬁed.

(i) *E→L*0*/qL*0is an isomorphism,
(ii) *E→L**∞**/q*^{−1}*L**∞*is an isomorphism,

(iii) (*L*0*∩V***A**)*⊕*(*q*^{−1}*L**∞**∩V***A**)*→V***A** is an isomorphism,

(iv) **A**_{0}*⊗*_{Q}*E* *→L*0,**A**_{∞}*⊗*_{Q}*E→L**∞*, **A***⊗*_{Q}*E* *→V***A** and **K***⊗*_{Q}*E* *→V* are
isomorphisms.

Let*−*be the ring automorphism of*U**q*(g) sending*q*,*t**i*,*e**i*,*f**i*to *q** ^{−1}*,

*t*

^{−1}*,*

_{i}*e*

*i*,

*f*

*i*.

Let *U**q*(g)** _{A}** be the

**A-subalgebra of**

*U*

*q*(g) generated by

*e*

^{(n)}

*,*

_{i}*f*

_{i}^{(n)}and

*t*

*i*. Similarly we deﬁne

*U*

_{q}*(g)*

^{−}**.**

_{A}**Theorem 2.6.** (*L*(*∞*)*, L*(*∞*)^{−}*, U*_{q}* ^{−}*(g)

**)**

_{A}*is balanced.*

Let

G^{low}:*L*(*∞*)*/qL*(*∞*)*−→E** ^{∼}* :=

*L*(

*∞*)

*∩L*(

*∞*)

^{−}*∩U*

_{q}*(g)*

^{−}**be the inverse of**

_{A}*E−→L*

*(*

^{∼}*∞*)

*/qL*(

*∞*). Then

G^{low}(*b*)*|b∈*B(*∞*)

forms a basis
of*U**q** ^{−}*(g). We call it a (lower)

*global basis. It is ﬁrst introduced by G. Lusztig*([L]) under the name of “canonical basis” for the A, D, E cases.

**Deﬁnition 2.7.** Let

*{*G^{up}(*b*)*|b∈*B(*∞*)*}*
be the dual basis of

G^{low}(*b*)*|b∈*B(*∞*)

with respect to the inner product
(^{•}*,** ^{•}*). We call it the upper global basis of

*U*

*q*

*(g).*

^{−}Naoya Enomoto and Masaki Kashiwara

**§****2.3.** **Symmetric crystals**

Let *θ* be an automorphism of *I* such that *θ*^{2} = id and (*α**θ(i)**, α**θ(j)*) =
(*α**i**, α**j*). Hence it extends to an automorphism of the root lattice*Q*by*θ*(*α**i*) =
*α**θ(i)*, and induces an automorphism of*U**q*(g).

**Deﬁnition 2.8.** Let*B**θ*(g) be the **K-algebra generated by** *E**i*, *F**i*, and
invertible elements*T**i*(*i∈I*) satisfying the following deﬁning relations:

(i) the*T**i*’s commute with each other,
(ii) *T**θ(i)*=*T**i* for any*i*,

(iii) *T**i**E**j**T*_{i}* ^{−1}*=

*q*

^{(α}

^{i}^{+α}

^{θ(i)}

^{,α}

^{j}^{)}

*E*

*j*and

*T*

*i*

*F*

*j*

*T*

_{i}*=*

^{−1}*q*

^{(α}

^{i}^{+α}

^{θ(i)}

^{,−α}

^{j}^{)}

*F*

*j*for

*i, j∈I*, (iv)

*E*

*i*

*F*

*j*=

*q*

^{−(α}

^{i}

^{,α}

^{j}^{)}

*F*

*j*

*E*

*i*+ (

*δ*

*i,j*+

*δ*

*θ(i),j*

*T*

*i*) for

*i, j∈I*,

(v) the*E**i*’s and the*F**i*’s satisfy the Serre relations (Deﬁnition 2.1 (4)).

We set*E*_{i}^{(n)}=*E*_{i}^{n}*/*[*n*]* _{i}*! and

*F*

_{i}^{(n)}=

*F*

_{i}

^{n}*/*[

*n*]

*!.*

_{i}**Lemma 2.9.** *IdentifyingU*_{q}* ^{−}*(g)

*with the subalgebra ofB*

*(g)*

_{θ}*by the mor-*

*phismf*

*i*

*→F*

*i*

*, we have*

*T**i**a*=

Ad(*t**i**t**θ(i)*)*a*
*T**i**,*
(2.2)

*E**i**a*=

Ad(*t**i*)*a*

*E**i*+*e*^{}*i**a*+

Ad(*t**i*)(*e*^{∗}_{θ(i)}*a*)
*T**i*

(2.3)

*fora∈U*_{q}* ^{−}*(g).

*Proof.* The ﬁrst relation is obvious. In order to prove the second, it is
enough to show that if*a*satisﬁes (2.3), then*f**j**a*satisﬁes (2.3). We have

*E**i*(*f**j**a*) = (*q*^{−(α}^{i}^{,α}^{j}^{)}*f**j**E**i*+*δ**i,j*+*δ**θ(i),j**T**i*)*a*

=*q*^{−(α}^{i}^{,α}^{j}^{)}*f**j*(

Ad(*t**i*)*a*

*E**i*+*e*^{}_{i}*a*+

Ad(*t**i*)(*e*^{∗}_{θ(i)}*a*)
*T**i*)
+*δ**i,j**a*+*δ**θ(i),j*

Ad(*t**i**t**θ(i)*)*a*
*T**i*

= (

Ad(*t**i*)(*f**j**a*)

*E**i*+*e*^{}* _{i}*(

*f*

*j*

*a*) +

Ad(*t**i*)(*e*^{∗}* _{θ(i)}*(

*f*

*j*

*a*)

*T*

*i*

*.*

The following lemma can be proved in a standard manner and we omit the proof.

**Lemma 2.10.** *Let* **K[***T*_{i}* ^{±}*;

*i*

*∈*

*I*]

*be the commutative*

**K-algebra gener-**

*ated by invertible elements*

*T*

*i*(

*i*

*∈*

*I*)

*with the deﬁning relations*

*T*

*θ(i)*=

*T*

*i*

*.*

*Then the mapU*

_{q}*(g)*

^{−}*⊗*

**K[**

*T*

_{i}*;*

^{±}*i∈I*]

*⊗U*

_{q}^{+}(g)

*→ B*

*θ*(g)

*induced by the multipli-*

*cation is bijective.*

Let *λ∈P*+:=*{λ∈*Hom(*Q,*Q)*| α*^{∨}_{i}*, λ ∈*Z0 for any*i∈I}* be a domi-
nant integral weight such that*θ*(*λ*) =*λ*.

**Proposition 2.11.**

(i) *There exists aB**θ*(g)-module*V**θ*(*λ*)*generated by a non-zero vectorφ**λ**such*
*that*

(a) *E**i**φ**λ*= 0 *for any* *i∈I,*
(b) *T**i**φ**λ*=*q*^{(α}^{i}^{,λ)}*φ**λ* *for any* *i∈I,*

(c) *{u∈V**θ*(*λ*)*|E**i**u*= 0*for any* *i∈I}*=**K***φ**λ**.*

*Moreover such a* *V**θ*(*λ*)*is irreducible and unique up to an isomorphism.*

(ii) *there exists a unique symmetric bilinear form* (*•**,** ^{•}*)

*on*

*V*

*θ*(

*λ*)

*such that*(

*φ*

*λ*

*, φ*

*λ*) = 1

*and*(

*E*

*i*

*u, v*) = (

*u, F*

*i*

*v*)

*for any*

*i∈I*

*andu, v*

*∈V*

*θ*(

*λ*), and

*it is non-degenerate.*

*Remark* 2.12. Set *P**θ* = *{µ∈P|θ*(*µ*) =*µ}*. Then *V**θ*(*λ*) has a weight
decomposition

*V**θ*(*λ*) =

*µ∈P**θ*

*V**θ*(*λ*)_{µ}*,*
where *V**θ*(*λ*)* _{µ}* =

*u∈V**θ*(*λ*)*|T**i**u*=*q*^{(α}^{i}^{,µ)}*u*

. We say that an element *u* of
*V**θ*(*λ*) has a*θ*-weight*µ*and write wt* _{θ}*(

*u*) =

*µ*if

*u∈V*

*θ*(

*λ*)

*. We have wt*

_{µ}*(*

_{θ}*E*

*i*

*u*) = wt

*(*

_{θ}*u*) + (

*α*

*i*+

*α*

*θ(i)*) and wt

*(*

_{θ}*F*

*i*

*u*) = wt

*(*

_{θ}*u*)

*−*(

*α*

*i*+

*α*

*θ(i)*).

In order to prove Proposition 2.11, we shall construct two*B**θ*(g)-modules,
analogous to Verma modules and dual Verma modules.

**Lemma 2.13.** *LetU*_{q}* ^{−}*(g)

*φ*

^{}

_{λ}*be a freeU*

_{q}*(g)-module with a generator*

^{−}*φ*

^{}

_{λ}*.*

*Then the following action gives a structure of aB*

*(g)-module on*

_{θ}*U*

_{q}*(g)*

^{−}*φ*

^{}*:*

_{λ}

*T**i*(*aφ*^{}* _{λ}*) =

*q*

^{(α}

^{i}*(Ad(*

^{,λ)}*t*

*i*

*t*

*θ(i)*)

*a*)

*φ*

^{}

_{λ}*,*

*E*

*i*(

*aφ*

^{}*) =*

_{λ}*e*^{}_{i}*a*+*q*^{(α}^{i}* ^{,λ)}*Ad(

*t*

*i*)(

*e*

^{∗}

_{θ(i)}*a*)

*φ*

^{}

_{λ}*,*

*F*

*i*(

*aφ*

^{}*) = (*

_{λ}*f*

*i*

*a*)

*φ*

^{}

_{λ}(2.4)

*for anyi∈I* *anda∈U*_{q}* ^{−}*(g).

*Moreover* *B**θ*(g)*/*

*i∈I*

(*B**θ*(g)*E**i*+*B**θ*(g)(*T**i**−q*^{(α}^{i}* ^{,λ)}*))

*→U*

_{q}*(g)*

^{−}*φ*

^{}

_{λ}*is an iso-*

*morphism.*

Naoya Enomoto and Masaki Kashiwara

*Proof.* We can easily check the deﬁning relations in Deﬁnition 2.8 except
the Serre relations for the*E**i*’s.

For*i*=*j∈I*, set*S* =_{b}

*n=0*(*−*1)^{n}*E*_{i}^{(n)}*E**j**E*_{i}^{(b−n)}where*b*= 1*− h**i**, α**j*. It
is enough to show that the action of*S*on*U*_{q}* ^{−}*(g)

*φ*

^{}*is equal to 0. We can easily check that*

_{λ}*SF*

*k*=

*q*

^{−(bα}

^{i}^{+α}

^{j}

^{,α}

^{k}^{)}

*F*

*k*

*S*. Since

*Sφ*

^{}*= 0, we have*

_{λ}*SU*

_{q}*(g)*

^{−}*φ*

^{}*= 0.*

_{λ}Hence*U*_{q}* ^{−}*(g)

*φ*

^{}*has a*

_{λ}*B*

*(g)-module structure.*

_{θ}The last statement is obvious.

**Lemma 2.14.** *Let* *U*_{q}* ^{−}*(g)

*φ*

^{}

_{λ}*be a free*

*U*

_{q}*(g)-module with a generator*

^{−}*φ*

^{}

_{λ}*. Then the following action gives a structure of aB*

*θ*(g)-module on

*U*

*q*

*(g)*

^{−}*φ*

^{}*:*

_{λ}

*T**i*(*aφ*^{}* _{λ}*) =

*q*

^{(α}

^{i}*(Ad(*

^{,λ)}*t*

*i*

*t*

*θ(i)*)

*a*)

*φ*

^{}

_{λ}*,*

*E*

*i*(

*aφ*

^{}*) = (*

_{λ}*e*

^{}

_{i}*a*)

*φ*

^{}

_{λ}*,*

*F**i*(*aφ*^{}* _{λ}*) =

*f**i**a*+*q*^{(α}^{i}* ^{,λ)}*(Ad(

*t*

*i*)

*a*)

*f*

*θ(i)*

*φ*^{}* _{λ}*
(2.5)

*for anyi∈I* *anda∈U*_{q}* ^{−}*(g). Moreover, there exists a non-degenerate bilinear

*form*

^{•}*,*

*:*

^{•}*U*

_{q}*(g)*

^{−}*φ*

^{}

_{λ}*×U*

_{q}*(g)*

^{−}*φ*

^{}

_{λ}*→*

**K**

*such thatF*

_{i}*u, v*=

*u, E*

_{i}*v,E*

_{i}*u, v*=

*u, F*

*i*

*v,T*

*i*

*u, v*=

*u, T*

*i*

*vforu∈U*

_{q}*(g)*

^{−}*φ*

^{}

_{λ}*andv∈U*

_{q}*(g)*

^{−}*φ*

^{}

_{λ}*, andφ*

^{}

_{λ}*, φ*

^{}*= 1.*

_{λ}*Proof.* There exists a unique symmetric bilinear form (^{•}*,** ^{•}*) on

*U*

*q*

*(g) such that (1*

^{−}*,*1) = 1 and

*f*

*i*and

*e*

^{}*are transpose to each other. Let us deﬁne*

_{i}

^{•}*,*

*:*

^{•}*U*

_{q}*(g)*

^{−}*φ*

^{}

_{λ}*×U*

_{q}*(g)*

^{−}*φ*

^{}

_{λ}*→*

**K**by

*aφ*

^{}

_{λ}*, bφ*

^{}*= (*

_{λ}*a, b*) for

*a∈U*

_{q}*(g) and*

^{−}*b∈*

*U*

_{q}*(g). Then we can easily check*

^{−}*F*

_{i}*u, v*=

*u, E*

_{i}*v*,

*T*

_{i}*u, v*=

*u, T*

_{i}*v*. Since

*e*

^{∗}*is transpose to the right multiplication of*

_{i}*f*

*i*, we have

*E*

*i*

*u, v*=

*u, F*

*i*

*v*. Hence the action of

*E*

*i*,

*F*

*i*,

*T*

*i*on

*U*

*q*

*(g)*

^{−}*φ*

^{}*satisfy the deﬁning relations in Deﬁnition 2.8.*

_{λ}*Proof of Proposition*2.11. Since*E**i**φ*^{}* _{λ}*= 0 and

*φ*

^{}*has a*

_{λ}*θ*-weight

*λ*, there exists a unique

*B*

*θ*(g)-linear morphism

*ψ*:

*U*

_{q}*(g)*

^{−}*φ*

^{}

_{λ}*→U*

_{q}*(g)*

^{−}*φ*

^{}*sending*

_{λ}*φ*

^{}*to*

_{λ}*φ*

^{}*. Let*

_{λ}*V*

*θ*(

*λ*) be its image

*ψ*(

*U*

_{q}*(g)*

^{−}*φ*

^{}*).*

_{λ}(i) (c) follows from

*u∈U*_{q}* ^{−}*(g)

*|e*

^{}

_{i}*u*= 0 for any

*i*

=**K** and*U*_{q}* ^{−}*(g)

*φ*

^{}

_{λ}*⊃*

*V*

*θ*(

*λ*). The other properties (a), (b) are obvious. Let us show that

*V*

*θ*(

*λ*) is irreducible. Let

*S*be a non-zero

*B*

*θ*(g)-submodule. Then

*S*contains a non-zero vector

*v*such that

*E*

*i*

*v*= 0 for any

*i*. Then (c) implies that

*v*is a constant multiple of

*φ*

*λ*. Hence

*S*=

*V*

*θ*(

*λ*).

Let us prove (ii). For*u, u*^{}*∈U*_{q}* ^{−}*(g)

*φ*

^{}*, set ((*

_{λ}*u, u*

*)) =*

^{}*u, ψ*(

*u*

*). Then it is a bilinear form on*

^{}*U*

_{q}*(g)*

^{−}*φ*

^{}*which satisﬁes*

_{λ}((*φ*^{}_{λ}*, φ*^{}* _{λ}*)) = 1

*,*((

*F*

*i*

*u, u*

*)) = ((*

^{}*u, E*

*i*

*u*

*))*

^{}*,*((

*E*

*i*

*u, u*

*)) = ((*

^{}*u, F*

*i*

*u*

*))*

^{}*,*and ((

*T*

*i*

*u, u*

*)) = ((*

^{}*u, T*

*i*

*u*

*))*

^{}*.*

(2.6)

It is easy to see that a bilinear form which satisﬁes (2.6) is unique. Since
((*u*^{}*, u*)) also satisﬁes (2.6), ((*u, u** ^{}*)) is a symmetric bilinear form on

*U*

*q*

*(g)*

^{−}*φ*

^{}*. Since*

_{λ}*ψ*(

*u*

*) = 0 implies ((*

^{}*u, u*

*)) = 0, ((*

^{}*u, u*

*)) induces a symmetric bilinear form on*

^{}*V*

*θ*(

*λ*). Since (

*•*

*,*

*) is non-degenerate on*

^{•}*U*

_{q}*(g), ((*

^{−}*•*

*,*

*)) is a non-degenerate symmetric bilinear form on*

^{•}*V*

*θ*(

*λ*).

**Lemma 2.15.** *There exists a unique endomorphism* *−* *of* *V**θ*(*λ*) *such*
*thatφ**λ*=*φ**λ* *andav*= ¯*av*¯*,F**i**v*=*F**i**v*¯*for any* *a∈***K** *andv∈V**θ*(*λ*).

*Proof.* The uniqueness is obvious.

Let *ξ* be an anti-involution of *U*_{q}* ^{−}*(g) such that

*ξ*(

*q*) =

*q*

*and*

^{−1}*ξ*(

*f*

*i*) =

*f*

*θ(i)*. Let ˜

*ρ*be an element of Q

*⊗P*such that ( ˜

*ρ, α*

*i*) = (

*α*

*i*

*, α*

*θ(i)*)

*/*2. Deﬁne

*c*(

*µ*) =

(*µ*+ ˜*ρ, θ*(*µ*+ ˜*ρ*))*−*( ˜*ρ, θ*( ˜*ρ*))

*/*2 + (*λ, µ*) for*µ∈P*. Then it satisﬁes
*c*(*µ*)*−c*(*µ−α**i*) = (*λ*+*µ, α**θ(i)*)*.*

Hence*c* takes integral values on*Q*:=

*i*Zα*i*.

We deﬁne the endomorphism Φ of *U*_{q}* ^{−}*(g)

*φ*

^{}*by Φ(*

_{λ}*aφ*

^{}*) =*

_{λ}*q*

^{−c(µ)}*ξ*(

*a*)

*φ*

^{}*for*

_{λ}*a∈U*

_{q}*(g)*

^{−}*. Let us show that*

_{µ}Φ(*F**i*(*aφ*^{}* _{λ}*)) =

*F*

*i*Φ(

*aφ*

^{}*) for any*

_{λ}*a∈U*

_{q}*(g).*

^{−}(2.7)

For*a∈U*_{q}* ^{−}*(g)

*, we have Φ(*

_{µ}*F*

*i*(

*aφ*

^{}*)) = Φ*

_{λ}*f**i**a*+*q*^{(α}^{i}^{,λ+µ)}*af**θ(i)*

*φ*^{}_{λ}

=

*q*^{−c(µ−α}^{i}^{)}*ξ*(*a*)*f**θ(i)*+*q*^{−(α}^{i}*,λ+µ)−c(µ−α**θ(i)*)*f**i**ξ*(*a*)
*φ*^{}_{λ}*.*
On the other hand, we have

*F**i*Φ(*aφ*^{}* _{λ}*) =

*F*

*i*

*q*^{−c(µ)}*ξ*(*a*)*φ*^{}_{λ}

=*q*^{−c(µ)}

*f**i**ξ*(*a*) +*q*^{(α}^{i}^{,λ+θ(µ))}*ξ*(*a*)*f**θ(i)*

*φ*^{}_{λ}*.*

Therefore we obtain (2.7).

Hence Φ induces the desired endomorphism of*V**θ*(*λ*)*⊂U*_{q}* ^{−}*(g)

*φ*

^{}*. Hereafter we assume further that*

_{λ}there is no*i∈I* such that*θ*(*i*) =*i*.

We conjecture that*V**θ*(*λ*) has a crystal basis under this assumption. This means
the following. Since*E**i*and*F**i*satisfy the*q*-boson relation, any*u∈V**θ*(*λ*) can be

Naoya Enomoto and Masaki Kashiwara

uniquely written as*u*=

*n0**F*_{i}^{(n)}*u**n* with*E**i**u**n* = 0. We deﬁne the modiﬁed
root operators*E**i* and*F**i*by:

*E**i*(*u*) =

*n1*

*F*_{i}^{(n−1)}*u**n* and*F**i*(*u*) =

*n0*

*F*_{i}^{(n+1)}*u**n**.*

Let*L**θ*(*λ*) be the**A**_{0}-submodule of*V**θ*(*λ*) generated by*F**i*1*· · ·F**i**φ**λ* (0 and
*i*1*, . . . , i**∈I*), and let B* _{θ}*(

*λ*) be the subset

*F**i*1*· · ·F**i**φ**λ*mod*qL**θ*(*λ*)*|*0,*i*1*, . . . , i**∈I*

of*L**θ*(*λ*)*/qL**θ*(*λ*).

**Conjecture 2.16.** For a dominant integral weight*λ*such that*θ*(*λ*) =*λ*,
we have

(1) *F**i**L**θ*(*λ*)*⊂L**θ*(*λ*) and*E**i**L**θ*(*λ*)*⊂L**θ*(*λ*),
(2) B* _{θ}*(

*λ*) is a basis of

*L*

*θ*(

*λ*)

*/qL*

*θ*(

*λ*),

(3) *F**i*B* _{θ}*(

*λ*)

*⊂*B

*(*

_{θ}*λ*), and

*E*

*i*B

*(*

_{θ}*λ*)

*⊂*B

*(*

_{θ}*λ*)

*{*0

*}*,

(4) *F**i**E**i*(*b*) =*b* for any*b∈*B* _{θ}*(

*λ*) such that

*E*

*i*

*b*= 0, and

*E*

*i*

*F*

*i*(

*b*) =

*b*for any

*b∈*B

*(*

_{θ}*λ*).

As in [K1], we have

**Lemma 2.17.** *Assume Conjecture*2.16. Then we have
(i) *L**θ*(*λ*) =*{v∈V**θ*(*λ*)*|*(*L**θ*(*λ*)*, v*)*⊂***A**_{0}*},*

(ii) *Let* (*•**,** ^{•}*)

_{0}

*be the*Q

*-valued symmetric bilinear form on*

*L*

*θ*(

*λ*)

*/qL*

*θ*(

*λ*)

*induced by*(

*•*

*,*

*). Then B*

^{•}*(*

_{θ}*λ*)

*is an orthonormal basis with respect to*(

^{•}*,*

*)*

^{•}_{0}

*.*

Moreover we conjecture that*V**θ*(*λ*) has a global crystal basis. Namely we
have

**Conjecture 2.18.** The triplet (*L**θ*(*λ*)*, L**θ*(*λ*)^{−}*, V**θ*(*λ*)^{low}** _{A}** ) is balanced.

Here*V**θ*(*λ*)^{low}** _{A}** :=

*U*

_{q}*(g)*

^{−}

_{A}*φ*

*λ*. Its dual version is as follows.

Let us denote by*V**θ*(*λ*)^{up}** _{A}** the dual space

*v∈V**θ*(*λ*)*|*(*V**θ*(*λ*)^{low}_{A}*, v*)*⊂***A**
.
Then Conjecture 2.18 is equivalent to the following conjecture.

**Conjecture 2.19.** (*L**θ*(*λ*)*, c*(*L**θ*(*λ*))*, V**θ*(*λ*)^{up}** _{A}**) is balanced.

Here *c* is a unique endomorphism of *V**θ*(*λ*) such that *c*(*φ**λ*) = *φ**λ* and
*c*(*av*) = ¯*ac*(*v*), *c*(*E**i**v*) = *E**i**c*(*v*) for any *a* *∈* **K** and *v* *∈* *V**θ*(*λ*). We have
(*c*(*v** ^{}*)

*, v*) = (

*v*

^{}*,v*¯) for any

*v, v*

^{}*∈V*

*θ*(

*λ*).

Note that*V**θ*(*λ*)^{up}** _{A}** is the largest

**A-submodule**

*M*of

*V*

*θ*(

*λ*) such that

*M*is invariant by the

*E*

_{i}^{(n)}’s and

*M*

*∩*

**K**

*φ*

*λ*=

**A**

*φ*

*λ*.

By Conjecture 2.19, *L**θ*(*λ*)*∩c*(*L**θ*(*λ*))*∩V**θ*(*λ*)^{up}_{A}*→* *L**θ*(*λ*)*/qL**θ*(*λ*) is an
isomorphism. Let G^{up}* _{θ}* be its inverse. Then

*{*G

^{up}

*(*

_{θ}*b*)

*}*

_{b∈B}

_{θ}_{(λ)}is a basis of

*V*

*θ*(

*λ*), which we call the

*upper global basis*of

*V*

*θ*(

*λ*). Note that

*{*G

^{up}

*(*

_{θ}*b*)

*}*

_{b∈B}

_{θ}_{(λ)}is the dual basis to

*{*G

^{low}

*(*

_{θ}*b*)

*}*

_{b∈B}*(λ)with respect to the inner product of*

_{θ}*V*

*θ*(

*λ*).

We shall prove these conjectures in the caseg=gl* _{∞}*and

*λ*= 0.

**§****3.** **PBW Basis of** *V**θ*(0) **for** g=gl_{∞}

**§****3.1.** **Review on the PBW basis**
In the sequel, we set *I*=Zodd and

(*α**i**, α**j*) =

2 for*i*=*j*,

*−*1 for *j*=*i±*2,
0 otherwise,

and we consider the corresponding quantum group *U**q*(gl* _{∞}*). In this case, we
have

*q*

*i*=

*q*. We write [

*n*] and [

*n*]! for [

*n*]

*and [*

_{i}*n*]

*! for short.*

_{i}We can parametrize the crystal basis B(*∞*) by the multisegments. We
shall recall this parametrization and the PBW basis.

**Deﬁnition 3.1.** For*i, j∈I* such that *ij*, we deﬁne a segment*i, j*
as the interval [*i, j*] *⊂* *I*:=Zodd. A multisegment is a formal ﬁnite sum of
segments:

m=

*ij*

*m**ij**i, j*

with*m**i,j* *∈*Z_{0}. We call*m**ij* the multiplicity of a segment*i, j*. If*m**i,j* *>*0,
we sometimes say that*i, j*appears inm. We sometimes write*m**i,j*(m) for*m**i,j*.
We sometimes write*i*for *i, i*. We denote by *M*the set of multisegments.

We denote by*∅* the zero element (or the empty multisegment) of*M*.

**Deﬁnition 3.2.** For two segments *i*1*, j*1 and *i*2*, j*2, we deﬁne the
orderingPBW by the following:

*i*1*, j*1PBW*i*2*, j*2* ⇐⇒*

*j*1*> j*2

or

*j*1=*j*2 and*i*1*i*2*.*

Naoya Enomoto and Masaki Kashiwara

We call this ordering the*PBW-ordering.*

**Deﬁnition 3.3.** For a multisegment m, we deﬁne the element *P*(m)*∈*
*U*_{q}* ^{−}*(gl

*) as follows.*

_{∞}(1) For a segment*i, j*, we deﬁne the element*i, j ∈U*_{q}* ^{−}*(gl

*) inductively by*

_{∞}*i, i*=

*f*

*i*

*,*

*i, j*=*i, j−*2*j, j −qj, ji, j−*2 for*i < j*.
(2) For a multisegmentm=

*ij*

*m**ij**i, j*, we deﬁne

*P*(m) =*−→*

*i, j*^{(m}^{ij}^{)}*.*
Here the product *−→*

is taken over segments appearing inm from large to
small with respect to the PBW-ordering. The element *i, j*^{(m}^{ij}^{)} is the
divided power of *i, j*i.e.

*i, j*^{(n)}=

1

[*n*]!*i, j** ^{n}* for

*n >*0,

1 for*n*= 0,

0 for*n <*0.

Hence the weight of*P*(m) is equal to wt(m):=*−*

*ikj**m**i,j**α**k*: *t**i**P*(m)*t*^{−1}* _{i}* =

*q*

^{(α}

^{i}

^{,wt(m))}*P*(m).

**Theorem 3.4**([L]). *The set of elements{P*(m)*|*m*∈ M}is a***K-basis**
*ofU**q** ^{−}*(gl

*). Moreover this is an*

_{∞}**A-basis of**

*U*

*q*

*(gl*

^{−}*)*

_{∞}

_{A}*. We call this basis the*PBW basis

*ofU*

_{q}*(gl*

^{−}*).*

_{∞}**Deﬁnition 3.5.** For two segments *i*1*, j*1 and *i*2*, j*2, we deﬁne the
ordering_{cry} by the following:

*i*_{1}*, j*1_{cry}*i*_{2}*, j*2* ⇐⇒*

*j*1*> j*2

or

*j*1=*j*2 and*i*1*i*2*.*
We call this ordering the*crystal ordering.*

**Example 3.6.** The crystal ordering is diﬀerent from the PBW-ordering.

For example, we have*−*1*,*1*>*cry1*,*1*>*cry*−*1, while we have1*,*1*>*PBW

*−*1*,*1*>*PBW *−*1.