On Level Zero Representations of Quantized Affine Algebras, [pdf]

55 Pages·2014·0.49 MB·English

ON LEVEL ZERO REPRESENTATIONS OF
QUANTIZED AFFINE ALGEBRAS
MASAKI KASHIWARA
Abstract. We study the properties of level zero modules over
quantizedaffinealgebras. Theproofoftheconjectureonthecyclic-
ity of tensor products by Akasaka and the present author is given.
Several properties of modules generated by extremal vectors are
proved. The weights of a module generated by an extremal vector
are contained in the convex hull of the Weyl group orbit of the
extremal weight. The universal extremal weight module with level
zero fundamental weight as an extremal weight is irreducible, and
isomorphic to the affinization of an irreducible finite-dimensional
module.
Contents
1. Introduction 2
2. Review on crystal bases 4
2.1. Quantized universal enveloping algebras 4
2.2. Crystals 5
2.3. Schubert decomposition of crystal bases 6
2.4. Global bases 7
3. Extremal weight modules 8
3.1. Extremal vectors 8
3.2. Dominant weights 9
4. Affine quantum algebras 10
4.1. Affine root systems 11
4.2. Affinization 16
4.3. Simple crystals 17
5. Affine extremal weight modules 17
5.1. Extremal vectors—affine case 17
5.2. Fundamental representations 19
1991 Mathematics Subject Classification. 17B37.
Key words and phrases. crystal base, quantized affine algebra.
This work benefits from a “Chaire Internationale de Recherche Blaise Pascal de
l’Etat et de la R´egion d’Ile-de-France, g´er´ee par la Fondation de l’Ecole Normale
Sup´erieure”.
1
2 M. KASHIWARA
6. Existence of Global bases 25
6.1. Regularized modified operators 25
6.2. Existence theorem 26
7. Universal R-matrix 29
8. Good modules 31
9. Main theorem 36
10. Combinatorial R-matrices 38
11. Energy function 40
12. Fock space 42
12.1. Some properties of good modules 42
12.2. Wedge spaces 45
12.3. Global basis of the Fock space 47
13. Conjectural structure of V(λ) 51
Appendix A. 52
Appendix B. Formulas for the crystal B(U˜ (g))) 53
q
References 54
1. Introduction
In this paper, we study the level zero representations of quantum
affine algebras. This paper is divided into three parts, on extremal
weight modules, on the conjecture in [1] on the cyclicity of the tensor
products of fundamental representations, and on the global basis of the
Fock space.
In [12], as a generalization of highest weight vectors, the notion of
extremal weight vectors is introduced, and it is shown that the uni-
versal module generated by an extremal weight vector has favorable
properties: this has a crystal base, a global basis, etc. The main pur-
pose of the first part (§ 2—§ 5) is to study such modules in the affine
case and to prove the following two properties.
(a) If a module is generated by an extremal vector with weight λ,
then all the weights of this module are contained in the convex
hull of the Weyl group orbit of λ.
(b) Any module generated by an extremal vector with a level zero
fundamental weight ̟ is irreducible, and isomorphic to the
i
affinization of an irreducible finite-dimensional module W(̟ )
i
(seeTheorem5.17andProposition5.16foranexactstatement).
QUANTIZED AFFINE ALGEBRAS 3
In the second part, we shall prove the following theorem 1, which is
(1) (1)
conjectured in [1] and proved in the case of A and C .
n n
Theorem. If a /a has no pole at q = 0 (ν = 1,...,m−1), then
ν ν+1
W(̟ ) ⊗···⊗W(̟ ) is generated by the tensor product of the
i1 a1 im am
extremal vectors.
In the course of the proof, one uses the global basis on the tensor
products of the affinizations of W(̟ ), especially the fact that the
iν
transformation matrix between the global basis of the tensor products
and the tensor products of global bases is triangular.
Among the consequences of this theorem (see § 9), we mention here
the following one. Under the conditions of the theorem above, there is
a unique homomorphism up to a constant multiple
W(̟ ) ⊗···⊗W(̟ ) −→ W(̟ ) ⊗···⊗W(̟ ) ,
i1 a1 im am im am i1 a1
and its image is an irreducible U′(g)-module. This phenomenon is
q
analogous to the morphism from the Verma module to the dual Verma
module. Conversely, combining with a result of Drinfeld ([4]), any
irreducibleintegrableU′(g)-moduleisisomorphictotheimageforsome
q
{(i ,a ),...,(i ,a )}. Moreover, {(i ,a ),...,(i ,a )} is unique up
1 1 m m 1 1 m m
to a permutation.
In the third part (§ 12), we prove the existence of the global basis
on the Fock space.
The plan of the paper is as follows. In § 2 –§ 4, we review some of
the known results of crystal bases. Then, in § 5, we give a proof of (a)
and (b).
In § 6, we prove a sufficient condition for a module to admit a global
basis: very roughly speaking, it is enough to have a global basis in
the extremal weight spaces. In § 7, we review the universal R-matrix
and the universal conjugation operator. After introducing the notion
of good modules (rudely speaking, a module with a global basis), we
shall prove in § 9 the above theorem in the framework of good modules
After preparations in § 10–§ 11 on the combinatorial R-matrix and
the energy function, we shall prove in § 12 the properties of good mod-
ules which are postulated for the existence of the wedge products and
the Fock space in [13]. Finally, we shall show that the Fock space ad-
mits a global basis. In the case of the vector representation of g = A(1),
n
the global basis of the corresponding Fock space is already constructed
by B. Leclerc and J.-Y. Thibon [14] (see also [15, 21]).
1M. Varagnolo–E. Vasserot (Standard modules of quantum affine algebras,
math.QA/0006084) prove the same conjecture in the simply-laced case by a dif-
ferent method.
4 M. KASHIWARA
In the last section, we present conjectures on the structure of V(λ).
Acknowledgements. The author would like to thank Anne Schilling
who kindly provided a proof of the formula (6.2).
2. Review on crystal bases
In this section, we shall review very briefly the quantized universal
enveloping algebra and crystal bases. We refer the reader to [8, 9, 12].
2.1. Quantized universal enveloping algebras. We shall define
the quantized universal enveloping algebra U (g). Assume that we are
q
given the following data.
P : a free Z-module (called a weight lattice)
I : an index set (for simple roots)
α ∈ P for i ∈ I (called a simple root)
i
h ∈ P∗ = Hom (P,Z) (called a simple coroot)
i Z
(·, ·): P ×P → Q a bilinear symmetric form.
We shall denote by h·, ·i: P∗ ×P → Z the canonical pairing.
The data above are assumed to satisfy the following axioms.
(2.1) (α ,α ) > 0 for any i ∈ I,
i i
2(α ,λ)
i
(2.2) hh ,λi = for any i ∈ I and λ ∈ P,
i
(α ,α )
i i
(2.3) (α ,α ) ≤ 0 for any i, j ∈ I with i 6= j.
i j
Let us choose a positive integer d such that (α ,α )/2 ∈ Zd−1 for
i i
any i ∈ I. Now let q be an indeterminate and set
(2.4) K = Q(q ) where q = q1/d.
s s
Definition 2.1. The quantized universal enveloping algebra U (g) is the
q
algebra over K generated by the symbols e ,f (i ∈ I) and q(h) (h ∈
i i
d−1P∗) with the following defining relations.
(1) q(h) = 1 for h = 0.
(2) q(h )q(h ) = q(h +h ) for h ,h ∈ d−1P∗.
1 2 1 2 1 2
(3) q(h)e q(h)−1 = qhh,αiie and q(h)f q(h)−1 = q−hh,αiif for any
i i i i
i ∈ I and h ∈ d−1P∗.
t −t−1
(4) [e ,f ] = δ i i for i, j ∈ I. Here q = q(αi,αi)/2 and t =
i j ijq −q−1 i i
i i
q((αi,αi)h ).
2 i
QUANTIZED AFFINE ALGEBRAS 5
(5) (Serre relation) For i 6= j,
b b
(−1)ke(k)e e(b−k) = (−1)kf(k)f f(b−k) = 0.
i j i i j i
k=0 k=0
X X
Here b = 1−hh ,α i and
i j
e(k) = ek/[k] ! , f(k) = fk/[k] ! ,
i i i i i i
[k] = (qk −q−k)/(q −q−1) , [k] ! = [1] ···[k] .
i i i i i i i i
Sometimes we need an algebraically closed field containing K, for
example
(2.5) K = C((q1/n)),
n
[
b
and to consider U (g) as an algebra over K.
q
We denote by U (g) the subalgebra of U (g) over Q[q ±1] generated
q Q q s
by the e(n)’s, the f(n)’s (i ∈ I) and qh (h ∈bd−1P∗).
i i
Let us denote by W the Weyl group, the subgroup of GL(P) gener-
ated by the simple reflections s : s (λ) = λ−hh ,λiα .
i i i i
Let∆ ⊂ Q = Zα bethesetofroots. Let∆± = ∆∩Q betheset
i i ±
of positive and negative roots, respectively. Here Q = ± Z α .
± i ≥0 i
P
Let ∆re be the set of real roots. ∆re = ∆ ∩∆re.
± ±
P
2.2. Crystals. We shall not review the notion of crystals, but refer
the reader to [8, 9, 12]. We say that a crystal B over U (g) is a regular
q
crystal if, for any J⊂I such that {α ;i ∈ J} is of finite-dimensional
i
type, B is, as a crystal over U (g ), isomorphic to the crystal bases
q J
associated with an integrable U (g )-module. Here U (g ) is the sub-
q J q J
algebra of U (g) generated by e , f (j ∈ J) and qh (h ∈ d−1P∗). By
q j j
[12], the Weyl group W acts on any regular crystal. This action S is
given by
f˜hhi,wt(b)ib if hh ,wt(b)i ≥ 0,
S b = i i
si (e˜i−hhi,wt(b)ib if hhi,wt(b)i ≤ 0.
Let us denote by U−(g) (resp. U+(g)) the subalgebra of U (g) gen-
q q q
erated by the f ’s (resp. by the e ’s). Then U−(g) has a crystal base
i i q
denoted by B(∞) ([9]). The unique weight vector of B(∞) with weight
0 is denoted by u . Similarly U+(g) has a crystal base denoted by
∞ q
B(−∞), and the unique weight vector of B(−∞) with weight 0 is
denoted by u .
−∞
6 M. KASHIWARA
Let ψ be the ring automorphism of U (g) that sends q , e , f and
q s i i
q(h) to q , f , e and q(−h). It gives a bijection B(∞) ≃ B(−∞) by
s i i
˜ ˜
which u , e˜, f , ε , ϕ , wt corresponds to u , f , e˜, ϕ , ε , −wt.
∞ i i i i −∞ i i i i
Let us denote by U˜ (g) the modified quantized universal envelop-
q
ing algebra ⊕ U (g)a (see [12]). Then U˜ (g) has a crystal base
λ∈P q λ q
B(U˜ (g)). As a crystal, B(U˜ (g)) is regular and isomorphic to
q q
B(∞)⊗T ⊗B(−∞).
λ
λ∈P
G
Here, T is the crystal consisting of a single element t with ε (t ) =
λ λ i λ
ϕ (t ) = −∞ and wt(t ) = λ.
i λ λ
Let ∗ be the anti-involution of U (g) that sends q(h) to q(−h), and
q
q , e , f to themselves. The involution ∗ of U (g) induces an involu-
s i i q
tion ∗ on B(∞), B(−∞), B(U˜ (g)). Then e˜∗ = ∗ ◦ e˜ ◦ ∗, etc. give
q i i
another crystal structure on B(∞), B(−∞), B(U˜ (g)). We call it the
q
star crystal structure. In the case of B(U˜ (g)), these two crystal struc-
q
tures are compatible, and B(U˜ (g)) may be considered as a crystal
q
over g⊕g. Hence, for example, S∗, the Weyl group action on B(U˜ (g))
w q
with respect to the star crystal structure is a crystal automorphism of
B(U˜ (g)) with respect to the original crystal structure. In particular,
q
the two Weyl group actions S and S∗ commute with each other.
w w′
The formulas concerning with B(U˜ (g)) are given in Appendix B.
q
Note that we have always
(2.6) ε (b)+ϕ∗(b) = ε∗(b)+ϕ (b) ≥ 0 for any b ∈ B(∞).
i i i i
2.3. Schubert decomposition of crystal bases. For w ∈ W with a
reduced expression s ···s , we define the subset B (∞) of B(∞) by
i1 iℓ w
(2.7) B (∞) = {f˜a1···f˜aℓu ;a ,...,a ∈ Z }.
w i1 iℓ ∞ 1 ℓ ≥0
Then B (∞) does not depend on the choice of a reduced expression.
w
We refer the reader to [11] on the details of B (∞) and its relationship
w
with the Demazure module.
We have ([11])
(i) B (∞)∗ = B (∞).
w w−1
(ii) If w′ ≤ w, then B ∞) ⊂ B (∞).
w′( w
˜
(iii) If s w < w, then f B (∞) ⊂ B (∞).
i i w w
(iv) e˜B (∞) ⊂ B (∞)⊔{0}.
i w w
(v) If both b and f˜b belong to B (∞), then all f˜kb (k ≥ 0) belong
i w i
to B (∞).
w
QUANTIZED AFFINE ALGEBRAS 7
Here ≤ is the Bruhat order. Set
B (∞) = B (∞)\ B (∞) .
w w w′
w′<w
(cid:0) [ (cid:1)
P. Littelmann ([16]) showed
B(∞) = B (∞).
w
w∈W
G
We have
(2.8) B (∞)∗ = B (∞).
w w−1
(2.9) If s w < w, then e˜maxB (∞) ⊂ B (∞),
i i w siw
˜
f B (∞) ⊂ B (∞).
i w w
In particular, ε (b) > 0 for any b ∈ B (∞).
i w
Here, we use the notation e˜maxb = e˜εi(b)b.
i i
2.4. Global bases. LetA ⊂ K bethesubringofK consistingofratio-
nal functions in q without pole at q = 0. Let − be the automorphism
s s
of K sending q to q −1. Set K := Q[q ,q −1]. Let V be a vector
s s Q s s
space over K, L an A-submodule of V, L an A- submodule, and V
0 ∞ Q
a K -submodule. Set E := L ∩L ∩V .
Q 0 ∞ Q
Definition 2.2 ([9]). We say that (L ,L ,V ) is balanced if each of
0 ∞ Q
L , L and V generates V as a K vector space, and if the following
0 ∞ Q
equivalent conditions are satisfied.
(i) E → L /q L is an isomorphism.
0 s 0
(ii) E → L /q −1L is an isomorphism.
∞ s ∞
(iii) (L ∩V )⊕(q −1L ∩V ) → V is an isomorphism.
0 Q s ∞ Q Q
(iv) A⊗ E → L , A⊗ E → L , K ⊗ E → V and K⊗ E → V
Q 0 Q ∞ Q Q Q Q
are isomorphisms.
Let − be the ring automorphism of U (g) sending q , qh, e , f to
q s i i
q −1, q−h, e , f .
s i i
Let U (g) be the K -subalgebra of U (g) generated by e(n), f(n) and
q Q Q q i i
qh (h ∈ P∗).
n
Let M be a U (g)-module. Let − be an involution of M satisfying
q
(cid:8) (cid:9)
(au)− = a¯u¯ for any a ∈ U (g) and u ∈ M. We call in this paper
q
such an involution a bar involution. Let (L,B) be a crystal base of an
integrable U (g)-module M.
q
Let M be a U (g) -submodule of M such that
Q q Q
(2.10) (M )− = M , and (u−u) ∈ (q −1)M for every u ∈ M .
Q Q s Q Q
8 M. KASHIWARA
Definition 2.3. If (L,L,M ) is balanced, we say that M has a global
Q
basis.
∼
In such a case, let G: L/q L −→ E := L ∩ L ∩ M be the inverse
s Q
∼
of E −→ L/q L. Then {G(b);b ∈ B} forms a basis of M. We call
s
this basis a (lower) global basis. The global basis enjoys the following
properties ([9, 10]):
(i) G(b) = G(b) for any b ∈ B.
(ii) For any n ∈ Z , {G(b);ε (b) ≥ n} is a basis of the K -
≥0 i Q
(m)
submodule f M .
m≥n i Q
(iii) for any i ∈ I and b ∈ B, we have
P
f G(b) = [1+ε (b)] G(f˜b)+ Fi G(b′).
i i i i b,b′
b′
X
Here the sum ranges over b′ ∈ B such that ε (b′) > 1 + ε (b).
i i
The coefficient Fi belongs to q q1−εi(b′)Q[q ].
b,b′ s i s
Similarly for e G(b).
i
3. Extremal weight modules
3.1. Extremal vectors. Let M be an integrable U (g)-module. A
q
vector u ∈ M of weight λ ∈ P is called extremal (see [1, 12]), if we can
find vectors {u } satisfying the following properties:
w w∈W
(3.1) u = u for w = e,
w
(3.2) if hh ,wλi ≥ 0, then e u = 0 and f(hhi,wλi)u = u ,
i i w i w siw
(3.3) if hh ,wλi ≤ 0, then f u = 0 and e(−hhi,wλi)u = u .
i i w i w siw
Hence if such {u } exists, then it is unique and u has weight wλ. We
w w
denote u by S u.
w w
Similarly, for a vector b of a regular crystal B with weight λ, we
say that b is an extremal vector if it satisfies the following similar
conditions: we can find vectors {b } such that
w w∈W
(3.4) b = b for w = e,
w
(3.5) if hh ,wλi ≥ 0 then e˜b = 0 and f˜hhi,wλib = b ,
i i w i w siw
(3.6) if hh ,wλi ≤ 0 then f˜v = 0 and e˜−hhi,wλib = b .
i i w i w siw
Then b must be S b.
w w
For λ ∈ P, let us denote by V(λ) the U (g)-module generated by
q
u with the defining relation that u is an extremal vector of weight
λ λ
λ. This is in fact infinitely many linear relations on u . We proved in
λ
QUANTIZED AFFINE ALGEBRAS 9
[12] 2 that V(λ) has a global crystal base (L(λ),B(λ)). Moreover the
crystal B(λ) is isomorphic to the subcrystal of B(∞) ⊗ t ⊗ B(−∞)
λ
consisting of vectors b such that b∗ is an extremal vector of weight −λ.
We denote by the same letter u the element of B(λ) corresponding to
λ
u ∈ V(λ). Then u ∈ B(λ) corresponds to u ⊗t ⊗u .
λ λ ∞ λ −∞
Note that, for b ⊗t ⊗b ∈ B(∞)⊗t ⊗B(−∞) belonging to B(λ),
1 λ 2 λ
one has
ε∗(b ) ≤ max(hh ,λi,0) and ϕ∗(b ) ≤ max(−hh ,λi,0)
(3.7) i 1 i i 2 i
for any i ∈ I.
For any w ∈ W, u 7→ S u gives an isomorphism of U (g)-
λ w−1 wλ q
modules:
∼
V(λ) −→V(wλ).
Similarly, letting S∗ be the Weyl group action on B(U˜ (g)) with re-
w q
spect to the star crystal structure and regarding B(λ) as a subcrystal
of B(U˜ (g)), S∗ : B(U˜ (g)) −∼→ B(U˜ (g)) induces an isomorphism of
q w q q
crystals
S∗ : B(λ) −∼→B(wλ).
w
For a dominant weight λ, V(λ) is an irreducible highest weight mod-
ule of highest weight λ, and V(−λ) is an irreducible lowest weight
module of lowest weight −λ.
3.2. Dominant weights.
Definition 3.1. For a weight λ ∈ P and w ∈ W, we say that λ is w-
dominant (resp. w-regular) if hβ,λi ≥ 0 (resp. hβ,λi 6= 0) for any
β ∈ ∆re ∩w−1∆re. If λ is w-dominant and w-regular, we say that λ is
+ −
regularly w-dominant.
If w = s ···s is a reduced expression, then we have
iℓ i1
∆re ∩w−1∆re = {s ···s α ;1 ≤ k ≤ ℓ}.
+ − i1 ik−1 ik
Hence λ is w-dominant (resp. w-regular) if and only if
hh ,s ···s λi ≥ 0
(3.8) ik ik−1 i1
(resp. hh ,s ···s λi 6= 0).
k ik−1 i1
Conversely one has the following lemma.
Lemma 3.2. For i ,...,i ∈ I, and a weight λ, assume that
1 l
hh ,s s ···s λi > 0 for k = 1,...,l.
ik ik−1 ik−1 i1
Then w = s ···s is a reduced expression.
il i1
2In [12], it is denoted by Vmax(λ), because I thought there would be a natural
U (g)-module whose crystal base is the connected component of B(λ).
q
10 M. KASHIWARA
Proof. By the induction on l, we may assume that s ···s is a
il−1 i1
reduced expression. If l(w) < l, then there exists k with 1 ≤ k ≤ l−1
such that s ···s (h ) = −h . Hence
il−1 ik+1 ik il
hh ,s ···s λi = −hh ,s s ···s λi < 0,
il il−1 i1 ik ik−1 ik−1 i1
which is a contradiction. Q.E.D.
This lemma implies the following lemma.
Lemma 3.3. Let w , w ∈ W and let λ be an integral weight. If
1 2
λ is regularly w -dominant and w λ is regularly w -dominant, then
2 2 1
ℓ(w w ) = ℓ(w ) + ℓ(w ) and λ is regularly w w -dominant. Here
1 2 1 2 1 2
ℓ: W → Z is the length function.
Proposition 3.4. Let λ ∈ P and b ∈ B (∞), b ∈ B (−∞). If
1 w1 2 w2
b := b ⊗t ⊗b belongs to B(λ), then one has:
1 λ 2
(i) λ is regularly w -dominant and −λ is regularly w -dominant,
1 2
(ii) ℓ(w w−1) = ℓ(w )+ℓ(w ),
1 2 1 2
(iii) One has
S∗ (b ⊗t ⊗b ) ∈ B (∞)⊗t ⊗u ,
w2 1 λ 2 w1w2−1 w2λ −∞
S∗ (b ⊗t ⊗b ) ∈ u ⊗t ⊗B (−∞).
w1 1 λ 2 ∞ w1λ w2w1−1
More generally if w = w′w′′ with ℓ(w ) = ℓ(w′)+ℓ(w′′), then
1 1
S∗ (b ⊗t ⊗b ) ∈ B (∞)⊗t ⊗B (−∞).
w′′ 1 λ 2 w′ w′′λ w2w′′−1
Proof. Assume w s < w . Then c := ε∗(b ) > 0 by (2.9). Hence
1 i 1 i 1
hh ,λi ≥ c > 0 by (3.7). We have e˜∗maxb ∈ B¯ (∞).
i i 1 w1si
(3.9) b′ = S∗(b ⊗t ⊗b ) = (e˜∗maxb )⊗t ⊗(e˜∗hhi,λi−cb ).
i 1 λ 2 i 1 siλ i 2
Hence, λ is regularly w -dominant by the induction on the length of
1
w . The other statement in (i) is similarly proved.
1
(ii) follows from (i) and the preceding lemma.
In (3.9), e˜∗hhi,λi−cb belongs to B (−∞), since (ii) implies w s >
i 2 w2si 2 i
w . Repeating this, we obtain (iii). Q.E.D.
2
4. Affine quantum algebras
In the sequel we assume that g is affine.