Generalized Weyl modules and Demazure submodules of level-zero extremal weight modules

0.25 MB·

by  Fumihiko Nomoto

GENERALIZED WEYL MODULES AND DEMAZURE SUBMODULES OF
LEVEL-ZERO EXTREMAL WEIGHT MODULES.
FUMIHIKONOMOTO
7
1
0
2

n Abstract. WestudyarelationshipbetweenthegradedcharactersofgeneralizedWeylmod-
a ules W , w ∈W, over the positive part of the affine Lie algebra and those of specific quo-
wλ
J tients V−(λ)/X−(λ), w ∈ W, of the Demazure submodules V−(λ) of the extremal weight
w w w
9 modules V(λ) over the quantum affine algebra, where W is the finite Weyl group and λ
2 is a dominant weight. More precisely, we prove that a specific quotient of the Demazure

submodule is a quantumanalog of a generalized Weylmodule.
]
A
Mathematics Subject Classification2010: Primary05E05;Secondary33D52,17B37,20G42.
Q
. 1. Introduction
h
t Let g be a finite-dimensional simple Lie algebra over C, and g the untwisted affine Lie
a aff
m algebra associated to g. In [OS], Orr-Shimozono gave a formula for the specialization at t = 0
[ of nonsymmetric Macdonald polynomials in terms of quantum alcove paths; they also gave

a similar formula for the specialization at t = ∞. Recently, for an arbitrary element w of
1
the finite Weyl group W, Feigin-Makedonskyi [FM] introduced a generalized Weyl module,
v
7 denoted by W , over the positive part n of the affine Lie algebra g , and proved that its
wλ aff aff
7 graded character equals the graded character Ct(w◦λ) of the set QB(w;t(w λ)) of quantum
3 w ◦
8 alcove paths, where λ is a dominant weight for g; here, w◦ is the longest element of W, and
0 t(w λ) is an element of the extended affine Weyl group, which gives the translation by w λ.
◦ ◦
.
1 Soon afterward, in [NNS], we proved that the graded character of the specific quotient
0 module V−(λ)/X−(λ) of V−(λ) is identical to the graded character gch QLS(λ) of the set
w w w w
7
QLS(λ) of quantum Lakshmibai-Seshadri (QLS) paths of shape λ; here, V(λ) is the extremal
1
: weight module of extremal weight λ over the quantum affine algebra Uv(gaff), and Vw−(λ) is
v
the Demazure submodule of V(λ) ([NNS]). In particular, in the case w = w , this graded
i ◦
X character turned out to be the specialization E (q,∞) of the nonsymmetric Macdonald
w◦λ
r polynomials E (q,t) at t = ∞.
a w◦λ
The purpose of this paper is to reveal the relationship between the graded character of the
generalized Weyl modules W over n and that of the quotient module V−(λ)/X−(λ) of
wλ aff w w
V−(λ) for w ∈ W. More precisely, we prove the following.
w
Theorem 1 (=Theorem 5.1.3). Let λ be a dominant weight, and w ∈ W. Then, there holds
the following equality:
w (gchW )= gch(V− (λ)/X− (λ)).
◦ wλ w◦ww◦ w◦ww◦
Here, ·i denotes the substitution q 7→ q−1, and the Weyl group action on Q(q)[P] is defined
by: w·eµ := ewµ, w ∈ W, µ ∈ P, where P is the weight lattice for g.
For the case w = e, Lenart-Naito-Sagaki-Schilling-Shimozono [LNSSS2] constructed a bi-
jection Ξ : QB(e;t(w λ)) → QLS(λ) in order to prove that the graded character of QLS(λ) is
◦
identical to the specialization E (q,0) at t = 0. In this paper, we generalize their construc-
w◦λ
tion to an arbitrary w ∈ W, and prove Theorem 1 by means of the constructed bijection.
1
2 F.NOMOTO
This paper is organized as follows. In Section 2, we recall some basic facts about the
quantum Bruhat graph. In Section 3, we review the definitions of quantum alcove paths
and their graded characters. In Section 4, we recall the definition of QLS paths, and then
define some variants of their graded characters. In Section 5, we give a bijection between the
set QB(w;t(w λ)) of quantum alcove paths and the set QLS(λ) of QLS paths that preserves
◦
weights and degrees. Using this bijection, we prove that w (Ct(w◦λ)) = gch QLS(λ), and
◦ w w◦ww◦
hence Theorem 1.
2. (Parabolic) Quantum Bruhat Graph
Let g be a finite-dimensional simple Lie algebra over C, I the vertex set for the Dynkin
diagram of g, and {α } (resp., {α∨} ) the set of simple roots (resp., coroots) of g. Then
i i∈I i i∈I
h = Cα∨ is a Cartan subalgebra of g, with h∗ = Cα the dual space of h, and
i∈I i i∈I i
h∗ = Rα its real form; the canonical pairing between h and h∗ is denoted by h·,·i :
hR∗ ×Lh →i∈IC. Liet Q = Zα ⊂ h∗ denote the rootLlattice, Q∨ = Zα∨ ⊂ h the
corootLlattice, and P = i∈IZ̟i ⊂ h∗R the weight lattice of g, where thei̟∈I, i ∈i I, arRe the
Pi∈I i R P i
fundamental weights for g, i.e., h̟ ,α∨i = δ for i,j ∈ I; we set P+ := Z ̟ , and call
P i j ij i∈I ≥0 i
an elements λ of P+ a dominant (integral) weight. Let us denote by ∆ the set of roots, and
P
by ∆+ (resp., ∆−) the set of positive (resp., negative) roots. Also, let W := hs | i ∈ Ii be the
i
Weyl group of g, where s , i ∈I, are the simple reflections acting on h∗ and on h as follows:
i
s ν = ν −hν,α∨iα , ν ∈h∗,
i i i
s h= h−hα ,hiα∨, h∈ h;
i i i
we denote the identity element and the longest element of W by e and w , respectively. If
◦
α ∈ ∆ is written as α = wα for w ∈ W and i ∈ I, we define α∨ to be wα∨; we often
i i
identify sα with sα∨. For u∈ W, the length of u is denoted by ℓ(u), which coincides with the
cardinality of the set ∆+∩u−1∆−.
Definition 2.1 ([BFP, Definition 6.1]). The quantum Bruhat graph, denoted by QBG(W),
is the directed graph with vertex set W whose directed edges are labeled by positive roots as
β
follows. For u,v ∈ W, and β ∈ ∆+, an arrow u −→ v is a directed edge of QBG(W) if the
following hold:
(1) v = us , and
β
(2) either (2a): ℓ(v) = ℓ(u)+1 or (2b): ℓ(v) = ℓ(u)−2hρ,β∨i+1,
where ρ := 1 α. An edge satisfying (2a) (resp., (2b)) is called a Bruhat (resp., quan-
2 α∈∆+
tum) edge.
P
Remark 2.2. The quantum Bruhat graph defined above is a “right-handed” version, while the
one defined in [BFP] is a “left-handed” version. Note that the results of [BFP] used in this
paper (such as Proposition 2.4) are unaffected by this difference (cf. [Po]).
β
For an edge u −→ v of QBG(W), we set
β
0 if u −→ v is a Bruhat edge,
wt(u→ v):=
β
( β∨ if u −→ v is a quantum edge.
Also, for u,v ∈ W, we take a shortest directed path u = x −γ→1 x −γ→2 ··· −γ→r x = v in
0 1 r
QBG(W), and set
wt(u ⇒ v) := wt(x → x )+···+wt(x → x )∈ Q∨;
0 1 r−1 r
GENERALIZED WEYL MODULES AND LEVEL-ZERO DEMAZURE MODULES 3
we know from [Po, Lemma 1(2),(3)] that this definition does not depend on the choice of
a shortest directed path from u to v in QBG(W). For a dominant weight λ ∈ P+, we set
wt (u ⇒ v) := hλ, wt(u ⇒ v)i, and call it the λ-weight of a directed path from u to v in
λ
QBG(W).
β −w◦β
Lemma 2.3. If x −→ y is a Bruhat (resp., quantum) edge of QBG(W), then yw −−−→ xw
◦ ◦
is also a Bruhat (resp., quantum) edge of QBG(W).
Proof. This follows easily from equalities ℓ(y)− ℓ(x) = ℓ(xw )−ℓ(yw ) and hρ,−w β∨i =
◦ ◦ ◦
hρ,β∨i. (cid:3)
Let w ∈ W. We take (and fix) reduced expressions w = s ···s and w w−1 = s ···s ;
i1 ip ◦ i−q i0
note that w = s ···s s ···s is also a reduced expression for the longest element w .
◦ i−q i0 i1 ip ◦
Now, we set
(2.1) β := s ···s α , −q ≤ k ≤ p;
k ip ik+1 ik
we have {β ,...,β ,...,β } = ∆+. Then we define a total order ≺ on ∆+ by:
−q 0 p
(2.2) β ≺ β ≺ ··· ≺ β ;
−q −q+1 p
note that this total order is a reflection order; see Remark 5.2.1.
Proposition 2.4 ([BFP, Theorem 6.4]). Let u,v ∈ W.
(1) There exists a unique directed path from u to v in QBG(W) for which the edge labels
are strictly increasing (resp., strictly decreasing) in the total order ≺ above.
(2) The unique label-increasing (resp., label-decreasing) path
u= u −γ→1 u −γ→2 ··· −γ→r u = v
0 1 r
from u to v in QBG(W) is a shortest directed path from u to v. Moreover, it is
lexicographically minimal (resp., lexicographically maximal) among allshortest directed
paths from u to v; that is, for an arbitrary shortest directed path
γ′ γ′ γ′
u= u′ −→1 u′ −→2 ··· −→r u′ = v
0 1 r
from u to v in QBG(W), there exists some 1 ≤ j ≤ r such that γ ≺ γ′ (resp.,
j j
γ ≻ γ′), and γ = γ′ for 1 ≤ k ≤ j −1.
j j k k
For a subset S ⊂ I, we set W := hs | i ∈ Si; notice that S may be an empty set
S i
∅. We denote the longest element of W by w (S). Also, we set ∆ := Q ∩ ∆, where
S ◦ S S
Q := Zα , and ∆+ := ∆ ∩∆+, ∆− := ∆ ∩∆−. For w ∈W, we denote the minimal-
S i∈S i S S S S
length coset representative for the coset wW in W/W by ⌊w⌋, and for a subset X ⊂ W,
S S
we setP⌊X⌋ := {⌊w⌋ | w ∈ X} ⊂ WS, where WS := ⌊W⌋ is the set of minimal-length coset
representatives for the cosets in W/W
S
Definition 2.5 ([LNSSS1, Section 4.3]). The parabolic quantum Bruhat graph, denoted by
QBG(WS), is the directed graph with vertex set WS whose directed edges are labeled by
positive roots in ∆+\∆+ as follows. For u,v ∈ WS, and β ∈ ∆+\∆+, an arrow u −→β v is a
S S
directed edge of QBG(WS) if the following hold:
(1) v = ⌊us ⌋ and
β
(2) either (2a): ℓ(v) = ℓ(u)+1 or (2b): ℓ(v) = ℓ(u)−2hρ−ρ ,β∨i+1,
S
where ρ = 1 α. An edge satisfying (2a) (resp., (2b)) is called a Bruhat (resp.,
S 2 α∈∆+
S
quantum) edge.
P
4 F.NOMOTO
For an edge u −→β v in QBG(WS), we set
β
0 if u−→ v is a Bruhat edge,
wtS(u → v) :=
β
( β∨ if u−→ v is a quantum edge.
Also, for u,v ∈ WS, we take a shortest directed path p : u = x −γ→1 x −γ→2 ··· −γ→r x = v in
0 1 r
QBG(WS) (such a directed path always exists by [LNSSS1, Lemma 6.12]), and set
wtS(p) := wtS(x → x )+···+wtS(x → x )∈ Q∨;
0 1 r−1 r
we know from [LNSSS1, Proposition 8.1] that if q is another shortest directed path from u to
v in QBG(WS), then wtS(p)−wtS(q) ∈Q∨ := Z α∨.
S i∈S ≥0 i
Now, for a dominant weight λ ∈ P+, we set
P
S = S := {i ∈I | hλ,α∨i =0}.
λ i
By the remark justabove, for u,v ∈ WS, thevalue hλ, wtS(p)i does not dependon thechoice
of a shortest directed path p from u to v in QBG(WS); this value is called the λ-weight of a
directed path from u to v in QBG(WS). Moreover, we know from [LNSSS2, Lemma 7.2] that
the value hλ, wtS(p)i is equal to the value wt (x ⇒ y) = hλ, wt(x ⇒ y)i for all x ∈ uW
λ S
and y ∈ vW . In view of this fact, for u,v ∈ WS, we also write wt (u ⇒ v) for the value
S λ
hλ, wtS(p)i by abuse of notation. Using this notation, we have
(2.3) wt (x ⇒ y) = wt (⌊x⌋ ⇒ ⌊y⌋)
λ λ
for all x,y ∈ W.
Definition 2.6 ([LNSSS2, Section 3.2]). Let λ ∈P+ beadominant weight and σ ∈ Q∩[0,1],
and set S = S . We denote by QBG (W) (resp., QBG (WS) ) the subgraph of QBG(W)
λ σλ σλ
(resp., QBG(WS))withthesamevertex setbuthavingonlytheedges: u −→β v withσhλ,β∨i ∈
Z.
β
Lemma 2.7 ([LNSSS2, Lemma 6.1]). Let σ ∈ Q∩[0,1]; notice that σ may be 1. If u −→ v is a
directed edge of QBG (W), then there exists a directed path from ⌊u⌋ to ⌊v⌋ in QBG (WS).
σλ σλ
Also, for u,v ∈ W, let ℓ(u ⇒ v) denote the length of a shortest directed path in QBG(W)
from u to v. For w ∈ W, following [BFP], we define the w-tilted Bruhat order ≤ on W as
w
follows: for u,v ∈ W,
def
u≤ v ⇔ ℓ(v ⇒ w) = ℓ(v ⇒ u)+ℓ(u ⇒ w);
w
the w-tilted Bruhat order on W is a partial order with the unique minimal element w.
Lemma 2.8 ([LNSSS1, Theorem 7.1], [LNSSS2, Lemma 6.5]). Let u,v ∈ WS, and w ∈ W .
S
(1) There exists a unique minimal element in the coset vW in the uw-tilted Bruhat order
S
≤ . We denote it by min(vW ,≤ ).
uw S uw
(2) There exists a unique directed path from some x ∈ vW to uw in QBG(W) whose edge
S
labels are increasing in the total order ≺ on ∆+ defined in (2.2), and lie in ∆+\∆+.
S
This path begins with min(vW ,≤ ).
S uw
(3) Let σ ∈ Q∩[0,1], and λ ∈ P+ a dominant weight. If there exists a directed path from
v to u in QBG (WS), then the directed path in (2) is one in QBG (W).
σλ σλ
GENERALIZED WEYL MODULES AND LEVEL-ZERO DEMAZURE MODULES 5
3. Quantum Alcove paths and their graded characters
Inthis section, following [OS, Sections 4 and5] and [FM, Section 1], we recall the definition
and some of the properties of the graded characters of quantum alcove paths.
Let g denote the finite-dimensional simple Lie algebra whose root datum is dual to that of
g; the set of simple roots is {α∨} ⊂ h, and the set of simple coroots is {α } ⊂ h∗; We
i i∈I i i∈I
denoteethe set of roots of g by ∆ = {α∨ | α ∈ ∆}, and the set of positive (resp., negative)
roots by ∆+ (resp., ∆−).
We consider the untwisteed affienization of the root datum of g. Let us denote by ∆ the
aff
set of all reeal roots, aend by ∆+ (resp., ∆− ) the set of all positive (resp., negative) real roots.
aff aff
Then we have ∆ = {α∨ + aδ | α ∈ ∆,a ∈ Z}, with δ the (eprimitive) null root. eWe set
aff
α∨ := δ −ϕ∨, where ϕ ∈ ∆e denotes thee highest short root, and set I := I ⊔{0}. Then,
0 aff
{α∨} is theeset of simple roeots. Also, for β ∈ h⊕Cδ,ewe define deg(β) ∈ C and β ∈ h by:
i i∈Iaff
e
(3.1) β = β+deg(β)δ.e
WedenotetheWeylgroupofgbyW;weidentifyWeandW throughtheidentificationofthe
simple reflections of the same index for each i ∈ I. For ν ∈ h∗, let t(ν) denote the translation
in h∗: t(ν)γ = γ + ν for γ ∈ eh∗. Tfhe correspondfing affine Weyl group and the extended
affine Weyl group are defined by W := t(Q)⋊W and W := t(P)⋊W, respectively. Also,
aff ext
we define s : h∗ → h∗ by ν 7→ ν −(hν,ϕ∨i−1)ϕ. Then, W = hs | i ∈ I i; note that
0 aff i aff
s = t(ϕ)s . The extended affinefWeyl group W actsfon h⊕Cδ as linear transformations,
0 ϕ ext
and on h∗ as affine transformations: for v ∈ W, t(ν)∈ t(P),f
f e
vt(ν)(β +rδ) = vβ +(r−hν,βi)δ, β ∈ h,r ∈ C,
vt(ν)γ = vν +vγ, γ ∈h∗.
e e
An element u∈ W can be written as
ext
(3.2) u= t(wt(u))dir(u),
f
with wt(u) ∈ P and dir(u) ∈ W, according to the decomposition W = t(P) ⋊ W. For
ext
w ∈ W , we denote the length of w by ℓ(w), which equals # ∆+ ∩w−1∆− . Also, we set
ext aff aff
f
Ω := {w ∈W | ℓ(w) = 0}. (cid:16) (cid:17)
ext
f e e
Let µ ∈ P, and denote by m ∈ W the shortest element in the coset t(µ)W. We take a
µ ext
reduced expfression m = us ···s ∈ W = Ω⋉W , where u ∈ Ω and ℓ ,...,ℓ ∈ I .
µ ℓ1 ℓL ext aff 1 L aff
f
Remark 3.1 ([M, Section 2.4]). For a dominant weight λ ∈ P+, we have
f f
(3.3) m = t(w λ)
w◦λ ◦
and
(3.4) dir(t(w λ)) = e.
◦
Let w ∈W. For each J = {j < j < j < ··· < j }⊂ {1,...,L}, we define an alcove path
1 2 3 r
p = wm = z ,z ,...,z ;β˜ ,...,β˜ as follows: we set β˜ := s ···s α∨ ∈ ∆+ for
J µ 0 1 r j1 jr k ℓL ℓk+1 ℓk aff
(cid:16) (cid:17)
e
6 F.NOMOTO
1 ≤ k ≤ L, and set
z = wm ,
0 µ
z = wm s ,
1 µ β˜j1
z = wm s s ,
2 µ β˜j1 β˜j2
.
.
.
z = wm s ···s .
r µ β˜j1 β˜jr
Also, following [OS, Section 3.3], we set B(w;m ) := {p | J ⊂ {1,...,L}} and end(p ) :=
µ J J
z ∈ W . Then we define QB(w;m ) to be the following subset of B(w;m ):
r ext µ µ
∨
f − β˜ji+1
p ∈ B(w;m ) dir(z )−−−−−−−→ dir(z ) is a directed edge of QBG(W), 0 ≤ i≤ r−1 .
 J µ (cid:12) i (cid:16) (cid:17) i+1 
(cid:12)
 
(cid:12)
(cid:12)
Remark 3.2 ([M, ((cid:12)(cid:12)2.4.7)]). If j ∈{1,...,L}, then − β˜j ∨ ∈ ∆+. 
(cid:16) (cid:17)
For p ∈ QB(w;m ), we define qwt(p ) as follows. Let J− ⊂ J denote the set of those
J µ J
∨
− β˜ji
indices j ∈J for which dir(z ) −−−−−→ dir(z ) is a quantum edge. Then we set
i i−1 (cid:16) (cid:17) i
qwt(p ) := β˜ .
J j
j∈J−
X
Now, following [FM, Definition 1.9], let Cmµ denote the graded character
w
(3.5) qdeg(qwt(pJ))ewt(end(pJ)).
pJ∈QXB(w;mµ)
of QB(w;m ).
µ
Remark 3.3 ([OS]; see also [FM]). For µ ∈ P, we denote by E (q,t) the nonsymmetric
µ
Macdonald polynomial, and by E (q,0) (resp., E (q,∞)) the specialization lim E (q,t)
µ µ t→0 µ
(resp., lim E (q,t)) at t = 0 (resp., t = ∞).
t→∞ µ
(1) For the special case w = e, it holds that
E (q,0) = Cmµ.
µ e
(2) For the special case w = w , it holds that
◦
E (q−1,∞) = w Cmµ;
µ ◦ w◦
namely,
E (q−1,∞) = qdeg(qwt(pJ))ew◦wt(end(pJ)).
µ
pJ∈QXB(w◦;mµ)
Let g denote the affine Lie algebra associated to g, and let g = n ⊕h ⊕n− be its
aff aff aff aff aff
triangular decomposition.
Remark 3.4. We should warn the reader that the root datum of the affine Lie algebra g is
aff
not necessarily dual to that of the untwisted affine Lie algebra associated to g with the set
∆ of real roots, though the root datum of g is dual to that of g. In particular, for the index
aff
0 ∈ Iaff, the simple coroot α∨0 = c−θ∨, with θ ∈ ∆+ the highest root of g and cethe canonical
e e
GENERALIZED WEYL MODULES AND LEVEL-ZERO DEMAZURE MODULES 7
central element of g , does not agree with the simple root δ −ϕ∨ at the beginning of this
aff
section, which is denoted by α∨ there.
0
e
Definition 3.5 ([FM, Definition 2.1]). Let λ be a dominant weight, and w ∈ W. Then the
generalized Weyl module W is the cyclic n -module with a generator v and following
ww◦λ aff
relations:
h⊗tkv = 0 for all h∈ h,k > 0,
(f ⊗t)v = 0 for α ∈ w∆−∩∆−,
α
(e ⊗1)v = 0 for α ∈ w∆−∩∆+,
α
(f ⊗t)−hw◦λ,α∨iv = 0 for α ∈ ∆+ such that wα ∈ ∆−,
wα
(e ⊗1)−hw◦λ,α∨iv = 0 for α ∈ ∆+ such that wα ∈ ∆+,
wα
where e , f , α ∈ ∆+, denote the Chevalley generators.
α −α
We can see the generalized Weyl module W as n ⊕h-module by: hv = hw λ, hiv,
ww◦λ aff ◦
h ∈ h; hence the module W is a h-weighted module. Also, the module W has degree
ww◦λ ww◦λ
grading defined by two conditions: deg(v) = 0, and the operator of the form x ⊗tk ∈ n
aff
increases the degree by k. Then the graded character of the generalized Weyl module W
ww◦λ
is defined by:
gchW := dimW [γ,k]qkeγ;
ww◦λ ww◦λ
here, W [γ,k] is the subspace of WX whose element has degree k and h-weight γ.
ww◦λ ww◦λ
Feigin-Makedonskyi proved that the graded character gchW of W is identical to the
ww◦λ ww◦λ
graded character Ct(w◦λ)(= Cmw◦λ) ([FM, Theorem 2.21]).
w w
4. Quantum Lakshmibai-Seshadri paths and some variants of their graded
characters
4.1. Quantum Lakshmibai-Seshadri paths.
Definition 4.1.1 ([LNSSS2, Definition 3.1]). Let λ ∈ P+ be a dominant weight, and set
S := S = {i ∈ I | hλ,α∨i = 0}. A pair η = (w ,w ,...,w ;σ ,σ ,...,σ ) of a sequence
λ i 1 2 s 0 1 s
w ,...,w ofelementsinWS suchthatw 6= w for1 ≤ k ≤ s−1andanincreasingsequence
1 s k k+1
0 = σ < ··· < σ = 1 of rational numbers, is called a quantum Lakshmibai-Seshadri (QLS)
0 s
path of shape λ if
(C) for every 1 ≤ i ≤ s−1, there exists a directed path from w to w in QBG (WS).
i+1 i σiλ
Let QLS(λ) denote the set of all QLS paths of shape λ.
Remark 4.1.2. As in [LNSSS3, Definition 3.2.2 and Theorem 4.1.1], condition (C) can be
replaced by:
(C’) for every 1 ≤ i ≤ s−1, there exists a shortest directed path in QBG(WS) from w
i+1
to w that is also a directed path in QBG (WS).
i σiλ
The set QLS(λ) provides a realization of the crystal basis of a particular quantum Weyl
module Wv(λ) over Uv′(gaff), where Uv′(gaff) denotes the quantum affine algebra without the
degreeoperator. (see[LNSSS3,Theorem4.1.1], [NS1,Theorem3.2], [N,Remark2.15]). More-
over, QLS(λ) ∼= i∈IQLS(̟i)⊗mi as Uv′(gaff)-crystal, where λ = i∈Imi̟i; in particular,
#QLS(λ) = (#QLS(λ))mi.
i∈IN P
Q
8 F.NOMOTO
4.2. Some variants of graded characters. Letλ ∈ P+ beadominantweight, andw ∈ W.
For η = (w ,...,w ;σ ,...,σ ) ∈ QLS(λ), we set
1 s 0 s
s
wt(η) := (σ −σ )w λ ∈ P,
i i−1 i
i=1
X
s−1
Deg∗(η) := (1−σ )wt (w ⇒ w ),
i λ i+1 i
i=1
X
s−1
Deg (η) := σ wt (w ⇒ w ),
∗ i λ i+1 i
i=1
X
s−1
Degw(η) := Deg∗(η)+wt (w ⇒ w ) = (1−σ )wt (w ⇒ w ),
λ 1 0 i λ i+1 i
i=0
X
s
Deg (η) := Deg (η)+wt (w ⇒ w ) = σ wt (w ⇒ w ),
w ∗ λ s+1 s i λ i+1 i
i=1
X
where we set w := w and w := w. Note that by Remark 4.1.2, we have σ wt (w ⇒
0 s+1 i λ i+1
w )∈ Z for 1 ≤ i≤ s−1. Hence it follows that Deg∗(η),Deg (η),Degw(η),Deg (η) ∈ Z ;
i ≥0 ∗ w ≥0
notice that σ =0, and σ = 1.
0 s
Also, we set
gchwQLS(λ) := q−Degw(η)ewt(η),
η∈QLS(λ)
X
gch QLS(λ) := q−Degw(η)ewt(η).
w
η∈QLS(λ)
X
Let λ = m ̟ ∈ P+ be a dominant weight, and V(λ) the extremal weight module
i∈I i i
of extremal weight λ over the quantum affine algebra Uv(gaff), and by Vw−(λ) the Demazure
P
submodule of V(λ); see [NS2].
In [NNS], we proved that gchV−(λ) of the Demazure submoduleV−(λ) of V(λ) is identical
w w
to
mi −1
(1−q−r) gch QLS(λ).
w
!
i∈I r=1
YY
Moreover,thegradedcharacterofthespecific(finite-dimensional)quotientmoduleV−(λ)/X−(λ)
w w
of Vw−(λ) over the positive part Uv+(gaff) of Uv(gaff) is identical to gchwQLS(λ); see [NNS,
(4.26) and (5.7)] for the definitions of the Demazure submodule V−(λ) and the specific quo-
w
tient V−(λ)/X−(λ).
w w
Remark 4.2.1. Let λ be a dominant weight λ ∈ P+. We know the following:
(1) ifw = e,thengch (QLS(λ)) =E (q−1,0)([LNSSS2,Lemma7.7andTheorem7.9]);
e w◦λ
(2) if w = w , then gch (QLS(λ)) = E (q,∞) ([NNS, Theorem 3.2.7]),
◦ w◦ w◦λ
where E (q−1,0) and E (q,∞) are specializations of the nonsymmetric Macdonald poly-
w◦λ w◦λ
nomial E (q,t) at t = 0 and t = ∞, respectively.
w◦λ
4.3. Lusztig involution. Let λ ∈ P+ be a dominant weight, and set S := S = {i ∈
λ
I | hλ,α∨i = 0}. In this subsection, we state the relation between the graded characters
i
gch QLS(λ) and gchwQLS(λ) for w ∈ W.
w
GENERALIZED WEYL MODULES AND LEVEL-ZERO DEMAZURE MODULES 9
For η = (w ,...,w ;σ ,...,σ ) ∈ QLS(λ), we define T(η) to be (⌊w w ⌋,...,⌊w w ⌋;1−
1 s 0 s ◦ s ◦ 1
σ ,...,1−σ ). The following follows from [LNSSS2, Section 4.5].
s 0
Lemma 4.3.1.
(1) Let w ,w ∈W. Then, wt (w w ⇒ w w )= wt (w ⇒ w ).
1 2 λ ◦ 1 ◦ 2 λ 2 1
(2) Let η = (w ,...,w ;σ ,...,σ )∈ QLS(λ). Then we have
1 s 0 s
(a) T(η) ∈ QLS(λ);
(b) wt(T(η)) = w wt(η);
◦
(c) Deg (T(η)) = Degw◦w(η).
w
Proof. Parts (1), (2a), and (2b) are proved in [LNSSS2, Section 4.5]. Let us prove part (2c).
It follows from [LNSSS2, Corollary 4.8] that
Deg (T(η)) = Deg∗(η).
∗
(2.3)
Also, by part (1), we have wt (w ⇒ ⌊w w ⌋) = wt (w ⇒ w w ) = wt (w ⇒ w w). From
λ ◦ s λ ◦ s λ s ◦
these, we see that
Deg (T(η)) = Deg (T(η))+wt (w ⇒ w w )
w ∗ λ ◦ s
= Deg∗(η)+wt (w ⇒ w w) = Degw◦w(η),
λ s ◦
as desired. This proves the lemma. (cid:3)
The operator T above is an involution on QLS(λ), called the Lusztig involution. By using
Lemma 4.3.1, we deduce that
(4.1) gch QLS(λ) = w (gchw◦wQLS(λ)) = q−Degw◦w(η)ew◦wt(η)
w ◦
η∈QLS(λ)
X
for w ∈ W.
5. Relationship between the two graded characters
5.1. Relationship between the graded characters gchW and gch(V−(λ)/X−(λ)).
wλ w w
We define an involution ·i on Q(q) by q = q−1. and set f = f eµ for f = f eµ
µ∈P µ µ∈P µ
with f ∈ Q(q).
µ
P P
Theorem 5.1.1. Let λ ∈ P+ be a dominant weight, and w ∈ W. Then we have
Ct(w◦λ) = gchww◦QLS(λ).
w
In particular, #QB(w,t(w λ)) = #QLS(λ) (see Proposition 5.3.4).
◦
We will give a proof of Theorem 5.1.1 in Section 5.3. By combining Theorem 5.1.1 with
(4.1), we obtain the following theorem.
Theorem 5.1.2. Let λ ∈ P+ be a dominant weight, and w ∈ W. Then we have
w Ct(w◦λ) = gch QLS(λ).
◦ w w◦ww◦
(cid:18) (cid:19)
Because Ct(w◦λ) equals the graded character gchW of the generalized Weyl module W ,
w wλ wλ
and gch QLS(λ) equals the graded character gch(V− (λ)/X− (λ)) of the quotient
w◦ww◦ w◦ww◦ w◦ww◦
module V− (λ)/X− (λ) of the Demazure module V− (λ), we obtain the following.
w◦ww◦ w◦ww◦ w◦ww◦
Theorem 5.1.3. Let λ ∈ P+ be a dominant weight, and w ∈ W. Then we have
(5.1) w gchW = gch(V− (λ)/X− (λ)).
◦ wλ w◦ww◦ w◦ww◦
(cid:0) (cid:1)
10 F.NOMOTO
5.2. Reduced expressions for t(w λ) and a total order on ∆+ ∩ t(w λ)−1∆− . Let
◦ aff ◦ aff
λ ∈ P+ be a dominant weight, and set λ := w λ, S := S = {i ∈ I | hλ,α∨i = 0}.
− ◦ λ i
For µ ∈ Wλ, we denote by v(µ) ∈ WS the minimal-length coset reperesentative for tehe coset
{w ∈ W |wλ = µ}inW/W . Sincew isthelongestelementofW,wehavew = v(λ )w (S)
S ◦ ◦ − ◦
and ℓ(w )= ℓ(v(λ ))+ℓ(w (S)).
◦ − ◦
In this subsection, we recall a particular reduced expression for m (= t(λ ) by (3.3))
λ− −
with respect to a fixed total order on ∆+ ∩t(w λ)−1∆− , and review some of its properties
aff ◦ aff
from [NNS].
We fix reduced expressions e e
(5.2) v(λ )= s ···s ,
− i1 iM
(5.3) w (S)= s ···s
◦ iM+1 iN
for v(λ ) and w (S), respectively. Then
− ◦
(5.4) w = s ···s
◦ i1 iN
is a reduced expression for w . We set β := s ···s α , 1 ≤ j ≤ N. Then we have
◦ j iN ij+1 ij
∆+\∆+ = {β ,...,β } and ∆+ = {β ,...,β }. We fix a total order on ∆+ such that
S 1 M S M+1 N
(5.5) β ≻ β ≻ ··· ≻ β ≻ β ≻ ··· ≻ β .
1 2 M M+1 N
∈∆+\∆+ ∈∆+
S S
Remark 5.2.1. The total|order ≺{zabove is}a r|eflection{zorder o}n ∆+ in the sense that if
α,β,γ ∈ ∆+ with γ∨ = α∨+β∨, then α≺ γ ≺ β or β ≺ γ ≺ α.
Now, we define an injective map
Φ : ∆+ ∩t(λ )−1∆− → Q × ∆+\∆+ ,
aff − aff ≥0 S
hλ−,β(cid:0)i−deg(β(cid:1)) ∨
eβ = β +deg(eβ)δ 7→ ,w◦β .
hλ ,βi
(cid:18) − (cid:19)
Here we note that hλ ,βi > 0, hλ ,βi−edeg(β) ≥ 0, and w β∨ ∈ ∆+ \∆+ since hλ ,βi =
− − ◦ S −
hλ,w βi> 0; recall from [M, (2.4.7) (i)] that
◦
(5.6) ∆+ ∩t(λ )−1∆− = {α+aδ | α ∈ ∆−,a ∈ Z, and 0 < a ≤ hλ ,α∨i}.
aff − aff −
Let us consider the lexicographic order < on Q ×(∆+ \∆+) induced by the usual total
e e e ≥0 S
order on Q and the reverse order of the restriction to ∆+\∆+ of the total order ≺ on ∆+
≥0 S
above; that is, for (a,α),(b,β) ∈Q ×(∆+\∆+),
≥0 S
(a,α) < (b,β) if and only if a < b, or a = b and α ≻ β.
Then we denote by ≺′ the total order on ∆+ ∩ t(λ )−1∆− induced by the lexicographic
aff − aff
order on Q ×(∆+\∆+) through the injective map Φ.
≥0 S
The proof of the following proposition is tehe same as thaet of [NNS, Proposition 3.1.8].
Proposition 5.2.2. With the notation and setting above, let us write ∆+ ∩ t(λ )−1∆−
aff − aff
as {γ ≺′ ··· ≺′ γ }. Then, there exists a unique reduced expression t(λ ) = us ···s
1 L − ℓ1 ℓL
for t(λ ), with u ∈ Ω and {ℓ ,...,ℓ } ⊂ I , such that β˜ = γ for 1e≤ j ≤ L, wheere
− 1 L aff j j
β˜ = s ···s α∨, 1 ≤ j ≤ L.
j ℓL ℓj+1 ℓj
In the following, we use the reduced expression t(λ ) = us ···s for t(λ ) give by this
− ℓ1 ℓL −
proposition. The proof of the following lemma is the same as that of [NNS, Lemma 3.1.10].