Super duality and Crystal bases for quantum orthosymplectic superalgebras

0.49 MB·English

by  Jae-Hoon Kwon

SUPER DUALITY AND CRYSTAL BASES FOR QUANTUM
ORTHOSYMPLECTIC SUPERALGEBRAS
3
JAE-HOONKWON
1
0
2
Abstract. We introduce a semisimple tensor category Oint(m|n) of modules
q
n
over an quantum orthosymplectic superalgebra. It is a natural counterpart of
a
J thecategory of finitelydominated integrable modules overthequantumclassical
9 (super) algebra of typeBm+n, Cm+n, Dm+n or B(0,m+n) from a viewpoint of
super duality. Weshow that a highest weight module in Oint(m|n) has a unique
q
]
A crystalbasewhenitcorrespondstoahighestweightmoduleoftypeBm+n,Cm+n
or B(0,m+n) under super duality. An explicit description of the crystal graph
Q
is given in terms of a new combinatorial object called orthosymplectic tableaux.
.
h
t
a
m
[ Contents

1
1. Introduction 1
v
6 2. Orthosymplectic Lie superalgebra g 5
m|n
5
3. Super duality and a semisimple tensor category of g -modules 10
7 m|n
1 4. Category Oint(m|n) over the quantum superalgebra U (g ) 19
q q m|n
.
1 5. Crystal base of a q-deformed Fock space 22
0
3 6. Orthosymplectic tableaux of type B and C 29
1 7. Character of a highest weight module in Oint(m|n) 37
: q
v 8. Crystal base of a highest weight module in Oint(m|n) 44
i q
X References 52
r
a
1. Introduction
1.1. TheKashiwara’scrystalbase[23]isacertainnicebasisatq = 0ofamoduleM
over the quantized enveloping algebra associated to a symmetrizable Kac-Moody al-
gebra, which still contains rich combinatorial information on M, andit has beenone
of the most important and successful tools in representation theory of the quantum
groups.
This work was supported by Basic Science Research Program through the National Research
Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No.
2012-0002607).
1
2 JAE-HOONKWON
For a contragredient Lie superalgebra g [18], a theoretical background of crystal
bases for the quantum superalgebra U (g) is given by Benkart, Kang and Kashiwara
q
in [2]. The existence of a crystal base is shown when g is a general linear Lie super-
algebra gl [2] and a queer Lie superalgebra q [14, 15] for a special class of finite
m|n n
dimensional modules appearing in a tensor power of the natural representation V of
U (g), often called polynomial representations. We should remark that the crystal
q
base theory for these two Lie superalgebras is not parallel to that of a symmetriz-
able Kac-Moody algebra due to the same substantial difficulties encountered when
we consider the representations of classical Lie superalgebras compared to those of
Lie algebras. For example, a finite dimensional representation of g is not semisimple
in general. Indeed, the above results for g = gl and q are based on the semisim-
m|n n
plicity of V⊗r, and closely related with the Schur-Weyl-Sergeev dualities [31]. Also,
there is a work on crystal bases of a family of infinite dimensional representations
of D(2|1,α) [34].
There is another important class in classical Lie superalgebras called orthosym-
plectic Lie superalgebras. However, there is little known about the existence of its
crystal bases except for osp , which is a Kac-Moody superalgebra of type B(0,r),
1|2r
and where we can apply the crystal base theory developed in [17].
In this paper, we construct for the first time crystal bases of a large family of
semisimple modules over a quantum orthosymplectic Lie superalgebra. We also
prove the uniqueness of these crystal bases and give a combinatorial model for the
associated crystals.
1.2. Let us explain our results in more details. Our first step is to find a nice
semisimplecategoryofmodulesoveraquantumorthosymplecticsuperalgebra. Since
a tensor power of the natural representation of an orthosymplectic Lie superalgebra
is not semisimple in general, we take a completely different approach inspired by a
recent work of Cheng, Lam and Wang on super duality [7].
Super duality is an equivalence between a parabolic BGG category O(m + ∞)
of modules over the classical Lie algebras g and a category O(m|∞) of mod-
m+∞
ules over the basic classical Lie superalgebras g of infinite rank, where g =
m|∞
gl,b,b•,c,d. (From now on, we use g as a symbol representing the type of a Lie su-
peralgebra.) It was originally introduced in [8, 10] as a conjecture in case of general
linear Lie superalgebras and later proved by Cheng and Lam [6]. Then the duality
for orthosymplectic Lie superalgebras was established by Cheng, Lam and Wang
[7]. One of its most remarkable and powerful features is that super duality reveals
a natural connection with the Kazhdan-Lusztig theory of Lie algebras.
3
We consider the semisimple subcategory Oint(m|∞) of O(m|∞) equivalent to the
subcategory Oint(m+∞) of integrable modules in O(m+∞) under super duality.
It is known that Oint(m|∞) is the category of polynomial modules when g is
m|∞
a general linear Lie superalgebra, that is, g = gl, [9]. Motivated by this fact, we
prove that when g is orthosymplectic, that is, g = b,b•,c,d, Oint(m|∞) is a full
m|∞
subcategory of O(m|∞) such that the weights of each object are polynomial with
respect to a suitably chosen dual basis of the Cartan subalgebra (Theorem 3.7). We
have a similar result for a category Oint(m|n) of modules over g of finite rank,
m|n
where Oint(m|n) is obtained from Oint(m|∞) by applying the truncation functor.
We should note that Oint(m|∞) is not characterized only by locally nilpotent ac-
tions of positive simple root vectors since the odd isotropic root vectors are always
nilpotent on g -modules.
m|n
Unlike the case of g = gl, the irreducible modules in Oint(m|n) are infinite di-
mensional when g = b,b•,c,d and n > 0, which were called oscillator modules and
studied via Howe duality in [5]. But one may still regard Oint(m|n) as a natural
counterpartofthecategoryOint(m+n)offinitedimensionalmodulesoverg offi-
m+n
niterank,sincebothofthemareobtainedfromtwoequivalentcategories Oint(m|∞)
and Oint(m+∞) by truncation, respectively (see the diagram below, where F is
the super duality functor and tr is truncation functor).
n
O(m+∞) ∼ // O(m|∞)
F
Oint(mS+∞) ∼ // Oint(Sm|∞)
F
trn trn
(cid:15)(cid:15) (cid:15)(cid:15)
Oint(m+n) Oint(m|n)
Then we consider q-deformations of g -modules in Oint(m|n) for g = b,b•,c,d.
m|n
More precisely, based on our characterization of Oint(m|n), we consider a category
Oint(m|n) of U (g )-modules with the same conditions on weights. By using the
q q m|n
method of classical limit and the semisimplicity of Oint(m|n), we show the following
(Theorem 4.3).
Theorem. Oint(m|n) is a semisimple tensor category equivalent to Oint(m|n).
q
An irreducible highest weight module in Oint(m|n) is parametrized by P(g) ,
q m|n
which is a set of pairs (λ,ℓ) of a partition and a positive integer, and its highest
weight is denoted by Λ (λ,ℓ). Let L (g ,Λ (λ,ℓ)) be an irreducible highest
m|n q m|n m|n
weight U (g )-module with highest weight Λ (λ,ℓ).
q m|n m|n
4 JAE-HOONKWON
The notion of a crystal base of a module in Oint(m|n) can be defined following [2]
q
for g = b,b•,c,d. We first consider a U (g )-module V with a crystal base, which
q m|n q
is a Fock space over a q-deformed Clifford-Weyl algebra (cf.[11]). An irreducible
summand of V is a highest weight module, which corresponds to a fundamental
q
weight U (g )-module under super duality, and each highest weight module in
q m+n
Oint(m|n) can be embedded into V⊗M for some M ≥ 1.
q q
Now, we consider the cases when g = b,b•,c. To describe the connected compo-
nent in the crystal of V⊗M includingahighest weight vector with weight Λ (λ,ℓ),
q m|n
weintroduceanewcombinatorialobjectcalledorthosymplectictableaux(depending
on g), which is partly motivated by [32]. Let T (λ,ℓ) denote the set of orthosym-
m|n
plectic tableaux of shape(λ,ℓ). We show that the character of T (λ,ℓ) is equal to
m|n
that of L (g ,Λ (λ,ℓ)), and T (λ,ℓ) is a connected crystal, where the crystal
q m|n m|n m|n
structure on T (λ,ℓ) is naturally induced from that on V⊗M. Using these facts,
m|n q
we prove the following, which is the main result in this paper (Theorem 8.8).
Theorem. Each highest weight module in Oint(m|n) has a unique crystal base for
q
g =b,b•,c. Moreover, the crystal of L (g ,Λ (λ,ℓ)) is isomorphic to T (λ,ℓ)
q m|n m|n m|n
for (λ,ℓ) ∈ P(g) .
m|n
We remark that an orthosymplectic tableau is defined over an arbitrary linearly
ordered Z -graded set A and its main advantage is compatibility with super du-
2
ality functor F and truncation functor tr . More precisely, the set of orthosym-
n
plectic tableaux of shape (λ,ℓ) gives both irreducible characters in Oint(m + n)
q
and Oint(m|n) with suitable choices of A. The Schur positivity of the character of
q
orthosymplectic tableaux of shape (λ,ℓ) plays a crucial role in proving this compat-
ibility. Also, when A is a finite set with even elements, we have an explicit bijection
between orthosymplectic tableaux and Kashiwara-Nakashima tableaux [26] of type
B and C.
1.3. The paper is organized as follows. In Section 2, we recall the definition of
orthosymplectic Lie superalgebras g based on [9]. In Section 3, we briefly review
m|n
super duality and present a simple characterization of Oint(m|n). In Section 4, we
define Oint(m|n) and show that it is a semisimple tensor category. In Section 5, we
q
review the notion of a crystal base for a quantum superalgebra [2] and prove the
existence of a crystal base of a q-deformed Fock space V . Then we introduce our
q
main combinatorial object T (λ,ℓ) in Section 6, and show that its character gives
m|n
an irreduciblecharacter in Oint(m|n) for g = b,b•,c in Section 7. Finally, in Section
q
8, we show that L (g ,Λ (λ,ℓ)) has a unique crystal base for g = b,b•,c and
q m|n m|n
(λ,ℓ) ∈ P(g) , whose crystal is isomorphic to T (λ,ℓ).
m|n m|n
5
Acknowledgement The author would like to thank S.-J. Cheng and W. Wang
for valuable discussion on super duality and allowing him the draft of their book
[9]. This work was first announced at the 2nd Mini Symposium on Representation
Theory in Jeju, Korea, December 2012. He thanks S.-J. Kang for his kind invitation
and interest in this work.
2. Orthosymplectic Lie superalgebra g
m|n
In this section, let us briefly recall some necessary background on Lie superal-
gebras (see [9, 18] for more details). Our exposition is based on [9] with a little
modification. We assume that the base field is C.
2.1. General linear Lie superalgebras. Throughoutthis paper, wefix apositive
integer m and let
I = {k,−k|1 ≤ k ≤ m}∪ 1Z,
m 2
I = {k,−k|1 ≤ k ≤ m}∪Z,
m
e
I = {k,−k|1 ≤ k ≤ m}∪ 1 +Z ∪{0},
m 2
(cid:0) (cid:1)
where I is a linearly ordered Z -graded set with
m 2
··· < −3 < −1< −1 < −1 <··· < −m <0 < m < ··· < 1 < 1 < 1 < 3 < ...,
e 2 2 2 2
I ⊃ {k,−k|1 ≤ k ≤ m}∪Z×, I = 1 +Z,
m m 2
0 1
(cid:16) (cid:17) (cid:16) (cid:17)
(the degreee of 0 will be specified later) and the leinear orderings and Z2-gradings on
the other sets are induced from those on I . For a ∈ I , |a| denotes the degree of
m m
a. We put I+ = {a ∈ I+ |a > 0} and I× = {a ∈ I|a 6= 0} for I ⊂ I .
m m m
e e
For I⊂ I , we denote by V the superspace with basis {v |a ∈ I}, where the Z -
m I a 2
e e e
gradingisinducedfromI . Letgl(V )bethegenerallinearLiesuperalgebraoflinear
m I
e
endomorphisms on V vanishing on v ’s except for finitely many a’s. We identify
I a
e
gl(V ) with the space of matrices (a ) spanned by the elementary matrices E .
I ij i,j∈I i,j
WeassumethatVI isasubspaceofVeIm andgl(VI) ⊂ gl(VeIm). Letgl(VI)bethecentral
extension of gl(V ) by a one-dimensional center CK with respect to the 2-cocycle
I
α(A,B) = Str([J,A]B), where Str is the supertrace with Str(ab) = (−1)|i|a
ij i∈I ii
and J = E .
i≤0 i,i P
For n ∈ Z ∪{∞}, we put
P >0
J = {a ∈ I |m ≤ a ≤ n},
m+n m
J = {a ∈ I |m ≤ a ≤ n− 1}.
m|n m 2
We assume that J = J = {m,...,1}.
m+0 m|0
6 JAE-HOONKWON
2.2. Orthosymplectic Lie superalgebras. Suppose that the degree of 0 ∈ I is
m
1. Define a skew-supersymmetric bilinear form (·|·) on Ve by
Im e
(v |v ) =0, (v |v )= −(−1)|a||b|(v |v )= δ ,
±a ±b a −b −b a ab
(2.1)
(v |v ) = 1, (v |v )= 0,
0 0 0 ±a
for a,b ∈ I+. For I ⊂ I , letspo(V )bethesubalgebraof gl(V ) preservingtheskew-
m m I I
supersymmetricbilinearformonV inducedfrom(2.1). Thenwedefineb• ,b• ,
I m+∞ m|∞
e e
c and c to be the central extensions of spo(V ) induced from gl(V ) when I
m+∞ m|∞ I I
is I , I , I× and I×, respectively.
m m m m
b
Next, suppose that the degree of 0 ∈ I is 0. Define a supersymmetric bilinear
m
form (·|·) on Ve by
Im e
(v |v ) = 0, (v |v ) =(−1)|a||b|(v |v ) = δ ,
±a ±b a −b −b a ab
(2.2)
(v |v ) = 1, (v |v ) = 0,
0 0 0 ±a
for a,b ∈ I+. For I ⊂ I , let osp(V ) be the subalgebra of gl(V ) preserving the
m m I I
supersymmetric bilinear form on V induced from (2.2). Then we define b ,
I m+∞
e e
b , d and d to be the central extensions of osp(V ) induced from gl(V )
m|∞ m+∞ m|∞ I I
when I is I , I , I× and I×, respectively.
m m m m
b
From now on, we assume that g is a symbol, which denotes one of b, b•, c and
d. Let U(g ) and U(g ) be the enveloping superalgebras associated to g
m+∞ m|∞ m+∞
and g , respectively.
m|∞
Let h (resp. h ) be the Cartan subalgebra of g (resp. g ) spanned
m+∞ m|∞ m+∞ m|∞
by K and E := E −E for a ∈ J (resp. J ), and let h∗ (resp.
a a,a −a,−a m+∞ m|∞ m+∞
h∗ ) be the restricted dual of h (resp. h ) spanned by Λ and δ for
m|∞ m+∞ m|∞ m a
a ∈ J (resp. J ), where hE ,δ i = δ , hK,δ i = 0, hE ,Λ i = 0 for a,b and
m+∞ m|∞ b a ab a a m
hK,Λ i = r with r = 1 for g = c and r = 1 otherwise. Here h·, ·i denotes the
m 2
natural pairing on h ×h∗ or h ×h∗ .
m+∞ m+∞ m|∞ m|∞
Let I = {m,...,1,0}∪Z . Then the set of simple roots Π = {α |i ∈
m+∞ >0 m+∞ i
I },thesetofsimplecorootsΠ∨ = {α∨|i ∈ I }andtheDynkindiagram
m+∞ m+∞ i m+∞
associated to the Cartan matrix (hα∨,α i) of g are listed below (the
i j i,j∈Im+∞ m+∞
simple roots are with respect to a Borel subalgebra spanned by the uppertriangular
matrices):
• b•
m+∞
7
−δ , if i= m, −2E +2K, if i= m,
m m
δ −δ , if i= k (6= m), E −E , if i= k (=6 m),
αi =  k+1 k α∨i =  k+1 k
δ −δ , if i= 0, E −E , if i= 0,
 1 1  1 1
δ −δ , if i∈ Z , E −E , if i∈ Z .
i i+1 >0 i i+1 >0
 
 
 
 
 
 
y⇐=(cid:13) (cid:13) ··· (cid:13) (cid:13) (cid:13) (cid:13) ···
αm αm−1 αm−2 α1 α0 α1 α2
( ydenotes a non-isotropic odd simple root.)
• c ,
m+∞
−2δ , if i = m, −E +K, if i= m,
m m
δ −δ , if i = k (6= m), E −E , if i= k (=6 m),
αi =  k+1 k α∨i =  k+1 k
δ −δ , if i = 0, E −E , if i= 0,
 1 1  1 1
δ −δ , if i ∈Z , E −E , if i∈ Z .
i i+1 >0 i i+1 >0
 
 
 
 
 
 
(cid:13)=⇒(cid:13) (cid:13) ··· (cid:13) (cid:13) (cid:13) (cid:13) ···
αm αm−1 αm−2 α1 α0 α1 α2
• b
m+∞
−δ , if i= m, −2E +2K, if i= m,
m m
δ −δ , if i= k (6= m), E −E , if i= k (=6 m),
αi =  k+1 k α∨i =  k+1 k
δ −δ , if i= 0, E −E , if i= 0,
 1 1  1 1
δ −δ , if i∈ Z , E −E , if i∈ Z .
i i+1 >0 i i+1 >0
 
 
 
 
 
 
(cid:13)⇐=(cid:13) (cid:13) ··· (cid:13) (cid:13) (cid:13) (cid:13) ···
αm αm−1 αm−2 α1 α0 α1 α2
• d
m+∞
8 JAE-HOONKWON
−δ −δ , if i = m, −E −E +2K, if i = m,
m m−1 m m−1
δ −δ , if i = k (6= m), E −E , if i = k (=6 m),
αi =  k+1 k α∨i =  k+1 k
δ −δ , if i = 0, E −E , if i = 0,
 1 1  1 1
δ −δ , if i ∈ Z , E −E , if i ∈ Z .
i i+1 >0 i i+1 >0
 
 
 
 
 
 αm 
(cid:13)
@
@
(cid:13) (cid:13) ··· (cid:13) (cid:13) (cid:13) (cid:13) ···
(cid:0)(cid:0)αm−2 αm−3 α1 α0 α1 α2
(cid:13)
αm−1
Let I = {m,...,1,0} ∪ 1 +Z . Then the set of simple roots Π =
m|∞ 2 ≥0 m|∞
{β |i ∈ I }, the set of simple coroots Π∨ = {β∨|i ∈ I } and the Dynkin
i m|∞ (cid:0) (cid:1) m|∞ i m|∞
diagram associated to the Cartan matrix (hβ∨,β i) of g are listed below
i j i,j∈Im|∞ m|∞
(the simple roots are with respect to a Borel subalgebra spanned by the upper
triangular matrices):
• b•
m|∞
−δ , if i= m, −2E +2K, if i = m,
m m
δ −δ , if i= k (6= m), E −E , if i = k (=6 m),
βi = δk+−1δ ,k if i= 0, βi∨ = Ek++1E ,k if i = 0,
 1 1  1 1
2 2
δ −δ , if i∈ 1 +Z , E −E , if i ∈ 1 +Z .
 i i+1 2 ≥0  i i+1 2 ≥0
 
 
 
 
 
y⇐=(cid:13) (cid:13) ··· (cid:13) (cid:13) (cid:13) ···
βm βm−1 βm−2 β1 Nβ0 β12 β32
( denotes an isotropic odd simple root.)
• c
mN|∞
−2δ , if i= m, −E +K, if i = m,
m m
δ −δ , if i= k (6= m), E −E , if i = k (=6 m),
βi = δk+−1δ ,k if i= 0, βi∨ = Ek++1E ,k if i = 0,
 1 1  1 1
2 2
δ −δ , if i∈ 1 +Z , E −E , if i ∈ 1 +Z .
 i i+1 2 ≥0  i i+1 2 ≥0
 
 
 
 
 
9
(cid:13)=⇒(cid:13) (cid:13) ··· (cid:13) (cid:13) (cid:13) ···
βm βm−1 βm−2 β1 Nβ0 β12 β32
• b
m|∞
−δ , if i= m, −2E +2K, if i = m,
m m
δ −δ , if i= k (6= m), E −E , if i = k (=6 m),
βi = δk+−1δ ,k if i= 0, βi∨ = Ek++1E ,k if i = 0,
 1 1  1 1
2 2
δ −δ , if i∈ 1 +Z , E −E , if i ∈ 1 +Z .
 i i+1 2 ≥0  i i+1 2 ≥0
 
 
 
 
 
(cid:13)⇐=(cid:13) (cid:13) ··· (cid:13) (cid:13) (cid:13) ···
βm βm−1 βm−2 β1 Nβ0 β12 β32
• d
m|∞
−δ −δ , if i= m, −E −E +2K, if i = m,
m m−1 m m−1
δ −δ , if i= k (6= m), E −E , if i = k (=6 m),
βi = δk+−1δ ,k if i= 0, βi∨ = Ek++1E ,k if i = 0,
 1 1  1 1
2 2
δ −δ , if i∈ 1 +Z , E −E , if i ∈ 1 +Z .
 i i+1 2 ≥0  i i+1 2 ≥0
 
 
 
 
 
βm
(cid:13)
@
@
(cid:13) (cid:13) ··· (cid:13) (cid:13) (cid:13) ···
(cid:0)(cid:0)βm−2 βm−3 β1 β0 β1 β2
(cid:13) N
βm−1
Note that α = β for i= m,...,1.
i i
We assume that h∗ and h∗ have symmetric bilinear forms (·|·) given by
m+∞ m|∞
(λ|δ ) = s (−1)|a|E −K,λ , (Λ |Λ ) = 0,
a a m m
(cid:10) (cid:11)
for a,b ∈ J or J and λ ∈ h∗ or h∗ . Here we assume s = 2 for
m+∞ m|∞ m+∞ m|∞
g = b,b•, and s = 1 otherwise. We have (δ |δ ) = s(−1)|a|δ for a,b, and hence
a b ab
10 JAE-HOONKWON
(α |α ), (β |β ) ∈ 2Z for i ∈ I and j ∈ I . Let
i i j j m+∞ m|∞
1 if i = m and g = b•,b,d,
 2 if i = m and g = c,


(2.3) si =  2 if i ∈ {m−1,...,1,0}∪Z>0 and g = b•,b,
 1 if i ∈ {m−1,...,1,0}∪Z and g = c,d,
 >0
−2 if i ∈ 1 +Z and g = b•,b,
 2 ≥0

−1 if i ∈ 12 +Z≥0 and g = c,d.



Then sihα∨i ,λi =(αi|λ) for i ∈ Im+∞, λ ∈ h∗m+∞, and sjhβj∨,µi = (βj|µ) for
j ∈ I , µ ∈h∗ .
m|∞ m|∞
For n ≥ 0, we put I = {i ∈ I |(α |δ ) 6= 0 for some a ∈J } and
m+n m+∞ i a m+n
I = {i ∈ I |(β |δ ) 6= 0 for some a ∈J }. Let g (resp. g ) be
m|n m|∞ i a m|n m+n m|n
the subalgebra of g (resp. g ) generated by the root vectors E for γ ∈
m+∞ m|∞ ±γ
Π := {α |i ∈ I } (resp. Π := {β |i ∈ I }) and K. The Cartan
m+n i m+n m|n i m|n
subalgebra h (resp. h ) of g (resp. g ) is spanned by K and E for
m+n m|n m+n m|n a
a ∈ J (resp. J ).
m+n m|n
3. Super duality and a semisimple tensor category of g -modules
m|n
Throughout the paper, a module M over a superalgebra U is understood to be
a supermodule, that is, M = M ⊕ M with U M ⊂ M for i,j ∈ Z . If U
0 1 i j i+j 2
has a comultiplication ∆, then we have a U-module structure on M ⊗ N via ∆
for U-modules M and N, where we have a superalgebra structure on U ⊗U with
multiplication (u ⊗u )(v ⊗v ) = (−1)|u2||v1|(u v )⊗(u v ) (|u| denotes the degree
1 2 1 2 1 1 2 2
of a homogeneous element u∈ U).
3.1. Super duality. Let us briefly recall the super duality for orthosymplectic Lie
superalgebras [7]. Let P denote the set of partitions. For λ = (λ ) ∈ P, let
i i≥1
λ′ = (λ′) be the conjugate of λ.
i i≥1
Let l be the standard Levi subalgebra of g corresponding to {α | ∈
m+∞ m+∞ i
J } for some J with Z ⊂ J ⊂ I \{0}. Let
m+∞ m+∞ >0 m+∞ m+∞
(1) c ∈ C and λ+ := (λ ,λ ,...) ∈ P,
P+ = Λ= cΛ + λ δ 1 2
m+∞  m a∈XJm+∞ a a(cid:12)(cid:12) (2) hα∨i ,Λi ∈ Z≥0 for i∈ Jm+∞ 
(cid:12)
(cid:12)
be the setof l -dominant integr(cid:12)al weights in h∗ . For Λ ∈ P+ ,let
m+∞ m+∞ m+∞
L(l ,Λ) betheirreduciblel -modulewithhighest weight Λ, and L(g ,Λ)
m+∞ m+∞ m+∞