Representations of quantized function algebras and the transition matrices from Canonical bases to PBW bases

0.48 MB·English

by  Hironori Oya

REPRESENTATIONS OF QUANTIZED FUNCTION ALGEBRAS
AND THE TRANSITION MATRICES FROM CANONICAL
BASES TO PBW BASES
HIRONORIOYA
Abstract. Let Gbeaconnected simply-connected simple complex algebraic
group and g the corresponding simple Lie algebra. In the first half of the
present paper, we study the relation between the positive part Uq(n+) of the
quantizedenvelopingalgebraUq(g)andthespecificirreduciblerepresentations
of the quantized function algebra Qq[G], taking into account the right Uq(g)-
algebra structure of Qq[G]. This work is motivated by Kuniba, Okado and
5 Yamada’sresult([KOY])togetherwithTanisakiandSaito’sresults([T],[S2]).
1
Inthelatterhalf,wecalculatethetransitionmatricesfromthecanonicalbasis
0
2 to thePBW bases of Uq(n+) usingtheaboverelation. Consequently,weshow
that the constants arising from our calculation are described by the structure

ul constants for the comultiplication of Uq(g). In particular, when g is of type
ADE, this result implies the positivity of the transition matrices, which was
J
originally proved by Lusztig ([L1]) in the case when the PBW bases are as-
3
sociated with the adapted reduced words of the longest element of the Weyl

] group, and by Kato ([Kato]) in arbitrary cases. In fact, the constants in our
A calculation coincide with ones arising from the calculation using the bilinear
Q form on Uq(n±). Weexplain this coincidence in Appendix.
.
h
t
a
m
[

3
v Contents
6
1 1. Introduction 2
4
2. Preliminaries I : the quantized enveloping algebras and the quantized
1
function algebras 5
0
. 2.1. Definitions of U (g) and Q [G] 5
1 q q
0 2.2. A review of the representation theory of the quantized function
5 algebras 8
1
2.3. Kuniba-Okado-Yamada’s theorem 10
:
v 3. The isomorphism Q [G/N+] ≃ F(λ)K 11
q λ
i
X λM∈P+
4. Preliminaries II : the canonical and dual canonical bases 16
r
a 4.1. The category of crystals 16
4.2. The canonical bases of U (n−) 17
q
4.3. The canonical and dual canonical bases of V(λ) 17
4.4. Some properties of the canonical and dual canonical bases 19
5. The transition matrices from Canonical bases to PBW bases 24
5.1. The main theorem 25
5.2. Calculations 25
A. Appendix 39
A.1. A simpler proof of the positivity 40
A.2. Comparison with Section 5 42
References 45
1
2 HIRONORIOYA
1. Introduction
Notation 1.1. Denote the set of non-negative integers(= Z ) by N.
≧0
Let G be a connected simply-connected simple complex algebraic group and
g the corresponding simple Lie algebra. Then, we can define two Hopf algebras
over thecomplexnumberfieldC,thequantizedenvelopingalgebraU (g) andthe
q C
quantized function algebra C [G]. (In this case, q is a complex number which is
q
neither0norarootofunity.) TheHopfalgebrasU (g) andC [G]areq-analogues
q C q
of the universal enveloping algebra U(g) of g and the coordinate algebra C[G] of
G respectively. We can also define these Hopf algebras as the Hopf algebras over
the rational function field Q(q) in the same way, and denote these by U (g) and
q
Q [G]. We will deal with U (g) and Q [G]. The algebra Q [G] has the natural
q q q q
left and right U (g)-algebra structure.
q
In the first half of this paper, we study the relation between the positive part
U (n+) of U (g) and the specific irreducible representations of Q [G], which has
q q q
been pointed outby Kuniba,Okadoand Yamada [KOY]. To explain this relation,
we recall the “tensor product construction” of the irreducible representations of
Q [G] due to Soibelman [Soi].
q
Let I be the index set of simple roots of g and W the Weyl group of g. It
is known that, for each i ∈ I, there exists an infinite dimensional irreducible
representation π of Q [G], whose representation space V has a natural basis
i q si
|mi indexed by the non-negative integers ([VS], [Soi]). Take an element
i m∈Z
≧0
w ∈ W and fix its reduced expression w =s ···s .
(cid:8) (cid:9) i1 il
Fact 1.2([Soi,5.4]). The representation π := π ⊗···⊗π ofQ [G]isirreducible
w i1 il q
and its isomorphism class does not depend on the choice of the reduced expressions
of w.
See for instance [KS, Chapter 3], [Jos, Chapter 10] for more details.
We explain Kuniba, Okado and Yamada’s result [KOY].Let w be the longest
0
element of W and N its length.It follows from the construction that the represen-
tation space V of π has a basis |m i ⊗···⊗|m i (=:|(m)i ) for
w0 w0 1 i1 l iN i m∈(Z )N
≧0
each reduced word i = (i ,...,i ) of w (which means that w = s ···s is a
1 N (cid:8) 0 0 (cid:9)i1 iN
reducedexpressionofw ). AnintertwinerΘ fromπ ⊗···⊗π toπ ⊗···⊗π
0 j,i i1 iN j1 jN
(i,j are reduced words of w ) is given by
0
|(0)i 7→ |(0)i .
i j
We regard the map Θ as the identity map on V .
j,i w0
On the other hand, we have a basis {Ec} , called the PBW basis, of
i c∈(Z )N
≧0
U (n+)foreachreducedwordiofw . SeeDefinition2.19fortheprecisedefinition.
q 0
Set the Q(q)-linear isomorphism Φ :U (n+)→ V by
i q w0
Ec 7→ |(c)i (c ∈ (Z )N).
i i ≧0
Then, Kuniba, Okado and Yamada’s result is the following:
Fact 1.3 ([KOY, Theorem 5]). For any reduced words i,j of w , we have
0
Φ = Θ ◦Φ .
j j,i i
This fact says that the definition of the map Φ does not depend on the choice
i
of i, so we denote this map by Φ .
KOY
We also use the conjecture [KOY, Conjecture 1], recently proved by Saito [S2]
and Tanisaki [T]. Let cλ denote the element of U (g)∗ determined by
fλ,vw0λ q
x 7→ hf ,x.v i,
λ w0λ
REPRESENTATIONS OF QUANTIZED FUNCTION ALGEBRAS 3
where v is a lowest weight vector of the integrable highest weight U (g)-
w0λ q
module V(λ) with highest weight λ, and f is a highest weight vector of the
λ
right U (g)-module V(λ)∗. See Notation 2.10 for their normalization. We set
q
S := cλ , where P is the set of the dominant weights. Then, S is
fλ,vw0λ λ∈P+ +
a (left and right) Ore multiplicative set in Q [G]. We consider the quotient ring
(cid:8) (cid:9) q
Q [G] and set
q S
ξ := (1−q2)−1(c̟i .E )(c̟i )−1 ∈ Q [G] ,
i i f̟i,vw0̟i i f̟i,vw0̟i q S
where ̟ is the fundamental weight and E is the positive Chevalley generator
i i
associated with i ∈ I. Then, the action of Q [G] on V can be extended to the
q w0
action of Q [G] , and
q S
Fact 1.4 ([T, Proposition 7.6], [S2, Corollary 4.3.3]).
Φ (E x) = ξ .Φ (x),
KOY i i KOY
for all x ∈ U (n+), where ξ . is the action of ξ on V .
q i i w0
We calculate the action of the elements of Q [G] on V other than ξ ’s. More
q S w0 i
precisely, taking into account the right U (g)-action, we investigate the action of
q
the elements of Q [G/N−], which is a right U (g)-subalgebra of Q [G] defined as
q q q
the set of invariants of Q [G] with respectto the left action of U (n−). We mainly
q q
use the method appearing in the reference [Jos, Chapter 9].
The set S is also a (left and right) Ore multiplicative set in Q [G/N−]. The
q
quotient ring is denoted by Q [G/N−] . We construct a right U (g)-algebra C
q S q
with the following properties:
• The right U (g)-algebra C is isomorphic to Q [G/N−] (the multiplicative
q q
set in C corresponding to S is denoted by S′).
• As a Q(q)-algebra, the quotient ring CS′ is naturally regarded as the non-
negative(=Borel) part Uˇ≧0 of (a variant of) the quantized enveloping al-
gebra whose basis {K } of its Cartan part is indexed by the elements
λ λ∈P
of the weight lattice P.
TherightU (g)-algebraC canberegardedasasubalgebraofUˇ≧0withacertain
q
right Uq(g)-algebra structure. Let {Gi}i∈I be the set of elements in CS′(= Uˇ≧0)
corresponding to the positive Chevalley generators. Then, we have the following
theorem:
Theorem 1. Let Υ be the isomorphism Uˇ≧0 → Q [G/N−] . (Note that Υ| :
q S C
C → Q [G/N−].) Then,
q
Υ(K )= cλ and Υ(G ) = ξ (λ ∈ P , i∈ I).
λ fλ,vw0λ i i +
Theorem 1 will be stated as Theorem 3.9 below. Combining this theorem with
Fact 1.4, we can compute the action of the element of the form cλ .x (.x
fλ,vw0λ
means the right action of x ∈U (g)) on V by computing the image of K under
q w0 λ
the right action of x in C.
In the latter half of this paper, we also deal with the canonical basis of U (n+),
q
which is defined by Lusztig [L1] and subsequently by Kashiwara [K1] under the
different methods. The canonical basis of U (n+) is a basis of U (n+) which is
q q
different from the PBW bases and does not depend on the choice of the reduced
expressions of w . The elements of this basis have many nice properties, but it is
0
difficult to calculate the explicit forms of them unlike the PBW bases, in general.
For an element G+ of the canonical basis of U (n+), we can write
q
G+ = ζG+Ec with ζG+ ∈ Z[q±1].
i c i i c
c
X
4 HIRONORIOYA
Our aim is to investigate the coefficients ζG+ by using the representation V of
i c w0
Q [G].
q
Let us explain our method briefly. Fix an arbitrary reduced word i of w .
0
Take the sufficiently “large” dominant integral weight λ associated with G+ and
0
set c := cλ0 . ∗ (G+), where ∗ is a certain Q(q)-algebra anti-involution
G+ fλ0,vw0λ0
of U (g). (Definition 2.4). By Theorem 1 (and Fact 1.4), we can show that the
q
element Φ−1 (c .|(0)i ) of U (n+) is equal to G+ modulo the terms of the form
KOY G+ i q
(the sufficiently large powers of q)·Ec when we express this element using the
i
PBW basis.
On the other hand, it is possible in principle to compute the element c .|(0)i
G+ i
according tothedefinitionof thetensor productmoduleV . Infact, itis difficult
w0
to obtain the explicit result, but we can compute it modulo the terms of the form
(the sufficiently large powers of q)·|(c)i . Taking the identification via Φ
i KOY
into consideration, we obtain the desired coefficients ζG+ and show that they
i c
are described by the structure constants for the comultiplication of U (g). In
q
particular, when the Lie algebra g is of type ADE, these structure constants are
theelements of N[q±1]by Lusztig’s theorem ([L2, Theorem 11.5]). Usingthis fact,
we obtain the following theorem:
Theorem 2. Assume that the Lie algebra g is of type ADE. Take an arbitrary
reduced word i of w and an element G+ of the canonical basis of U (n+). Then,
0 q
we have
G+ = ζG+Ec with ζG+ ∈ N[q±1].
i c i i c
c
X
Theorem 2 will be stated as Theorem 5.2 below. We remark that Theorem 5.2
is actually the negative counterpart of Theorem 2. However, the statement in the
negative side can be immediately translated into that for the positive side and
vice versa.
In fact, this theorem itself is not a new result. It was originally proved by
Lusztig in his original paper of the canonical bases ([L1, Corollary 10.7]) in the
case when PBW bases are associated with the “adapted” reduced words of w
0
(See [L1, 4.7].), through his geometric realization of the elements of the canon-
ical bases and PBW bases. Recently, this fact for arbitrary PBW bases was
proved by Kato ([Kato, Theorem 4.17]), through the categorification of PBW
bases by using the Khovanov-Lauda-Rouquier algebras. Beyond type ADE, by
the way, McNamara also established the categorification of PBW bases via the
Khovanov-Lauda-Rouquier algebras for arbitrary finite types ([M1, Theorem 3.1],
thedualPBWbases) andsymmetricaffinetypes ([M2, Theorem 24.4]). Hence, in
fact, such positivity also holds for symmetric affine types ([M2, Theorem 24.10])
though we do not deal with them in this paper. (For nonsymmetric finite types,
the “canonical basis” arising from the Khovanov-Lauda-Rouquier categorification
does not coincide with the above-mentioned canonical basis and the positivity
theorem fails in general.)
Whenweperformourcalculation, weheavilyusethepropertiesofthecanonical
basis of U (n±), the canonical and dual canonical bases of the highest weight
q
integrable modules of U (g), since the coproduct of c is described by these
q G+
objects. One of the important properties, “Similarity of the structure constants”,
will be proved in Section 4. (Proposition 4.17)
Afterthispaperwassubmittedtothepreprintserver,YoshiyukiKimurapointed
out to the author the existence of a much simpler proof of the positivity of the
transition matrices from canonical bases to PBW bases. We explain the proof of
this which is obtained from his comments in the first half of Appendix. Moreover,
REPRESENTATIONS OF QUANTIZED FUNCTION ALGEBRAS 5
this method provides, in fact, the same constants as in Section 5 even when g
is of nonsymmetric (finite) type. We check this point in the latter half. Since
the calculation in Section 5 is slightly complicated, it might be helpful to read
Subsection A.1 before reading Section 5. At last, we also state the corollaries of
this comparison.
Acknowledgements. TheauthorisgreatlyindebtedtoYoshiyukiKimura,Yoshi-
hisa Saito and Toshiyuki Tanisaki for many helpful discussions and comments.
The appendix in this paper is attached essentially motivated by Kimura’s com-
ments. IwouldalsoliketothankRyoSatoformanyhelpfuldiscussions. Thiswork
was supported by the Program for Leading Graduate Schools, MEXT, Japan.
2. Preliminaries I : the quantized enveloping algebras and the
quantized function algebras
2.1. Definitions of U (g) and Q [G]. Firstly, we review the definition of the
q q
quantized enveloping algebras and the quantized function algebras.
Notation 2.1. Let g be a finite dimensional complex simple Lie algebra, ∆ the
root system of g with respect to a fixed Cartan subalgebra h, Q(⊂ h∗) the root
lattice, Q∨(⊂ h) the coroot lattice. Fix a set of simple roots Π = {α } (⊂ ∆ ⊂
i i∈I
h∗) and denote the set of simple coroots by Π∨ := {α∨} (⊂ h). (hα∨,α i)
i i∈I i j i,j∈I
is called the Cartan matrix of g. We say that the Lie algebra g is of type ADE if
the Cartan matrix of g is symmetric. Let W be the Weyl group, e the unit of W,
s (∈ W) the simple reflection corresponding to α . (i.e. s (λ) = λ−hλ,α∨iα for
i i i i i
λ ∈h∗.), w the longest element of W and l(w) the length of an element w of W.
0
Definethe symmetricbilinear form (, )on h∗ by 2(α ,α )/(α ,α ) = hα∨,α iand
i j i i i j
(α ,α ) = 2 for all short simple root α . Define Q := Z α , Q := −Q ,
i i i + i∈I ≧0 i − +
∆+ := ∆∩Q+ (the set of positive roots), P := {λ ∈ h∗P| hλ,α∨i i ∈ Z for all i∈ I }
(the integral weight lattice), and P+ := λ ∈ h∗ hλ,α∨i i ∈ Z≧0 for all i ∈I (the
set of dominant integral weights). ̟ denotes the fundamental weight such that
i (cid:8) (cid:12) (cid:9)
h̟ ,α∨i = δ . We define the partial order ≤ o(cid:12)n Q by α ≤ β ⇔ β −α ∈ Q .
i j i,j +
For an element α := m α ∈ Q (m ∈ Z), we set htα := m (called the
i∈I i i i i∈I i
height of α).
P P
We set
(αi,αi)
qi := q 2 ,
qn−q−n
[n]:= for n ∈ Z,
q−q−1
[n][n−1]···[n−k+1]
n if n ∈ Z,k ∈ Z>0,
:= [k][k−1]···[1]
k 
(cid:20) (cid:21) 1 if n ∈ Z,k = 0,
[n]! := [n][n−1]···[1] for n ∈ Z ,[0]! := 1.
>0

n n [n]!
Note that [n], ∈ Z[q±1] and = if n ≧ k ≧ 0. For a
k k [k]![n−k]!
(cid:20) (cid:21) (cid:20) (cid:21)
rational function X ∈ Q(q), we define X by the rational function obtained from
i
X by substituting q by q (i ∈ I).
i
Definition 2.2. The quantized enveloping algebra U (g) is the unital associative
q
Q(q)-algebra defined by the generators
E , F (i ∈ I), K (h ∈ Q∨),
i i h
6 HIRONORIOYA
and the relations (i)-(vi) below.
(i) K0 = 1, KhKh′ = Kh+h′ for all h,h′ ∈ Q∨,
(ii) K E = qhh,αiiE K for all h ∈ Q∨,i ∈ I,
h i i h
(iii) K F = q−hh,αiiF K for all h ∈ Q∨,i ∈ I,
h i i h
K −K
i −i
(iv) [E ,F ] = δ for all i,j ∈ I,
i j ij q −q−1
i i
where K := K and K := K ,
i (αi,αi)α∨ −i −(αi,αi)α∨
2 i 2 i
(v) 1−hα∨i,αji(−1)kE(k)E E(1−hα∨i,αji−k) = 0 for all i,j ∈ I with i 6= j,
i j i
k=0
(vi) 1−hXα∨i,αji(−1)kF(k)F F(1−hα∨i,αji−k) = 0 for all i,j ∈ I with i 6= j
i j i
k=0
X Xn
where X(n) := i .
i [n] !
i
For α = m α ∈ Q (m ∈ Z), we set K := K . Note that
i i i α P mi(αi,αi)α∨
i∈I 2 i
i∈I
X
K = K .
±αi ±i
The subalgebra of Uq(g) generated by {Ei}i∈I (resp. {Fi}i∈I, {Kα∨}i∈I ) is
i
denoted by U (n+) (resp. U (n−), U0 ).
q q
For α ∈ Q, we set
Uq(g)α := u∈ Uq(g) KhuK−h = qhα,hiu for all h∈ Q∨ .
The elements of Uq(g)(cid:8)α are called(cid:12)homogeneous and said to have wei(cid:9)ghts α. For a
(cid:12)
homogeneous element u∈ U (g) , we set wtu= α. U (n+) , U (n−) are defined
q α q α q α
similarly.
There exists a Q(q)-linear isomorphism U (n−)⊗U0⊗U (n+) → U (g), a⊗b⊗
q q q
c 7→ abc. This is called the triangular decomposition of U (g).
q
The algebra U (g) has a Hopf algebra structure given by the following comul-
q
tiplication ∆, counit ε and antipode S:
∆(E )= E ⊗1+K ⊗E , ε(E ) = 0, S(E ) = −K E ,
i i i i i i −i i
∆(F )= F ⊗K +1⊗F , ε(F ) = 0, S(F ) = −F K ,
i i −i i i i i i
∆(Kα∨)= Kα∨ ⊗Kα∨, ε(Kα∨) = 1, S(Kα∨) = K−α∨,
i i i i i i
for all i∈ I.
Remark 2.3. This U (g) is called of simply-connected type in [L4].
q
Definition2.4. WedefinetheQ(q)-algebra,anti-coalgebrainvolutionω :U (g) →
q
U (g) by
q
ω(Ei)= Fi, ω(Fi) = Ei, ω(Kα∨)= K−α∨,
i i
for i ∈ I. We define the Q(q)-algebra anti-involutions ∗,ϕ,ω′ : U (g) → U (g) by
q q
∗(Ei) = Ei, ∗(Fi) = Fi, ∗(Kα∨)= K−α∨,
i i
ϕ(Ei) = Fi, ϕ(Fi) = Ei, ϕ(Kα∨)= Kα∨,
i i
ω′(Ei) = KiFi, ω′(Fi) = EiK−i, ω′(Kα∨) = Kα∨,
i i
for i ∈ I. Note that ω′ is a Q(q)-coalgebra involution and ω = ∗◦ϕ= ϕ◦∗.
We define the Q-algebra involution (·) : U (g)→ U (g) by
q q
Ei = Ei, Fi = Fi, Kα∨ = K−α∨, q = q−1,
i i
for i ∈ I.
REPRESENTATIONS OF QUANTIZED FUNCTION ALGEBRAS 7
Definition 2.5. For i ∈ I, define the Q(q)-linear maps r, r : U (n+) → U (n+)
i i q q
by
hwtx,α∨i
r(xy) = r(x)y+q i x r(y) and r(E ) = δ ,
i i i i i j ij
hwty,α∨i
r (xy) = q i r (x)y+xr (y) and r (E )= δ ,
i i i i i j ij
for j ∈ I and homogeneous elements x,y ∈U (n+).
q
Moreover,weset e′ := (·)◦ω◦r ◦ω◦(·) ,e′ := (·)◦ω◦ r◦ω◦(·) :
i i Uq(n−) i i Uq(n−)
U (n−)→ U (n−). (cid:12) (cid:12)
q q (cid:12) (cid:12)
We have ∗◦ r◦∗| = r and ∗◦ e′◦(cid:12)∗| = e′ for all i ∈ I. (cid:12)
i Uq(n+) i i Uq(n−) i
For any homogeneous element x ∈ U (n+),y ∈ U (n−) and p ∈Z , we have
q q ≧0
∆(x)= Ei(p)Kx−pαi ⊗qi−12p(p−1)(ir)p(x)+ x′Kwtx′′ ⊗x′′,
x′∈Uq(n+)X:homogeneous,
wtx′6=pαi
∆(x)= qi−21p(p−1)(ri)p(x)Kpαi ⊗Ei(p) + x′Kwtx′′ ⊗x′′,
x′′∈Uq(n+)X:homogeneous,
wtx′′6=pαi
∆(y)= Fi(p)⊗qi12p(p−1)K−pαi(e′i)p(y)+ y′⊗Kwty′y′′,
y′∈Uq(n−)X:homogeneous,
wty′6=−pαi
∆(y)= qi12p(p−1)(ie′)p(y)⊗Kwty+pαiFi(p)+ y′⊗Kwty′y′′.
y′′∈Uq(n−)X:homogeneous,
wty′′6=−pαi
For a homogeneous element x ∈ U (n+) (resp. U (n−)) with wtx 6= 0, we have
q q
(1) if r(x) = 0 (resp. e′(x) = 0) for all i∈ I, then x = 0, and
i i
(2) if r (x) = 0 (resp. e′(x) = 0) for all i∈ I, then x = 0.
i i
For a homogeneous element x ∈ U (n+) (resp. U (n−)), we have
q q
r (x) = qhwtx−αi,α∨ii r(x) (2.1)
i i i
(resp. e′(x) = qhwtx+αi,α∨ii e′(x)) (2.2)
i i i
For x ∈U (n+) and y ∈ U (n−), we have
q q
K r(x)−r (x)K
−ii i i
• F x−xF = in U (g), and
i i q −q−1 q
i i
e′(y)K −K e′(y)
• E y−yE = i i −i i in U (g).
i i q −q−1 q
i i
See [L4, Chapter 1, 3.1.5] for details.
Definition 2.6. Let M be a left (resp. right) U (g)-module. For any λ ∈ P, we
q
set
Mλ := m ∈ M Kh.m = qhh,λim (resp. m.Kh = qhh,λim) for all h ∈ Q∨ .
We say that M is integrable if M satisfies
(cid:8) (cid:12) (cid:9)
(cid:12)
• M = M ,
λ
λ∈P
M
• The action of E and F are locally nilpotent on M for all i ∈I.
i i
8 HIRONORIOYA
Let O (g) (resp. O (gopp)) be the category of finite dimensional integrable
int int
left (resp. right) U (g)-modules. The category O (g) (resp. O (gopp)) is a
q int int
semisimplecategory anditsirreducibleobjectsareisomorphictoexactly oneV(λ)
(resp. Vr(λ)) for some λ ∈ P , where V(λ) (resp. Vr(λ)) is the irreducible inte-
+
grable left (resp. right) U (g)-module with highest weight λ. (See. [Jan, Chapter
q
5])
Remark 2.7. A highest weight vector of a right U (g)-module is a vector which
q
vanishes by the action of F ’s (i ∈ I).
i
The dual space V(λ)∗ of V(λ) has a natural right U (g)-module structure, and
q
V(λ)∗ is isomorphic to Vr(λ) as a right U (g)-module.
q
Definition 2.8. The dual space U (g)∗ of U (g) has a natural U (g)-bimodule
q q q
structure. Hence, we can define the subspace Q [G] of U (g)∗ by
q q
{f ∈ Uq(g)∗ |Uq(g).f belongs to Oint(g) and f.Uq(g) belongs to Oint(gopp)},
Then, Q [G] has a Hopf algebra structure ([Jan, Chapter 7]) induced from one of
q
U (g), andaleftandrightU (g)-algebrastructure. NotethatU (g)∗ hasanatural
q q q
Q(q)-algebra structure but does not have the bialgebra (hence, the Hopf algebra)
structure. This Hopf algebra Q [G] is called the quantized function algebra. By
q
abuse of notation, the coproduct (resp. the counit, the antipode) of Q [G] is also
q
denoted by ∆ (resp. ε, S).
The Hopf algebra Q [G] is a quantum analogue of the algebra of regular func-
q
tions on G where G is the connected simply-connected simple complex alge-
braic group whose Lie algebra is g. There is the q-analogue of the Peter-Weyl
theorem([K2, Proposition 7.2.2]):
Proposition 2.9. For λ ∈ P , we define the U (g)-bimodule homomorphism
+ q
Ψ : V(λ)∗ ⊗V(λ) → Q [G] by f ⊗v 7→ (u7→ hf,u.vi). Then,
λ q
Ψ : V(λ)∗ ⊗V(λ) → Q [G]
λ q
λM∈P+ λM∈P+
is an isomorphism of U (g)-bimodules.
q
Notation 2.10. For f ∈ V(λ)∗ and v ∈ V(λ), we set
cλ := Ψ (f ⊗v).
f,v λ
For each λ ∈ P , we fix a highest weight vector of V(λ) (resp. V∗(λ)), denoted by
+
v (resp. f ). We make an assumption that hf ,v i = 1. For w = s ···s ∈
λ λ λ λ il(w) i1
W, we set
v := F(hsil(w)−1···si1λ,α∨il(w)i)···F(hsi1λ,α∨i2i)F(hλ,α∨i1i).v ,
wλ il(w) i2 i1 λ
f := f .E(hλ,α∨i1i)E(hsi1λ,α∨i2i)···E(hsil(w)−1···si1λ,α∨il(w)i).
wλ λ i1 i2 il(w)
It is well known that v and f depend only on the weight wλ (i.e. not on the
wλ wλ
choice of w and its reduced expression. See [L4, Proposition 39.3.7]). Moreover,
hf ,v i = 1.
wλ wλ
2.2. A review of the representation theory of the quantized function al-
gebras. Letusrecallsomebasicfacts onsimpleQ [G]-modulesduetoSoibelman
q
et al.
Definition 2.11 (The Q [G]-modules V ). First, we construct a certain simple
q si
Qq[SL2]-module. The algebra Qq[SL2] is generated by cij := Fδ2,j.c̟f̟,v̟.Eδ2,i
(i,j ∈ {1,2}). (Since I = {∗}, E , F , ̟ (i ∈ I) are simply denoted by E, F,
i i i
REPRESENTATIONS OF QUANTIZED FUNCTION ALGEBRAS 9
̟, respectively.) Moreover, c ’s satisfies the following relations and, in fact, the
ij
relations of c ’s are exhausted by them:
ij
c c = qc c , c c = qc c ,
11 12 12 11 21 22 22 21
c c = qc c , c c = qc c ,
11 21 21 11 12 22 22 12
[c ,c ]= 0, [c ,c ] = (q−q−1)c c ,
12 21 11 22 12 21
c c −qc c = 1.
11 22 12 21
Let V := Q(q)|mi bean infinitedimensionalQ(q)-vector spacewith abasis
m∈Z
M≧0
indexed by non-negative integers. We define a Q [SL ]-module structure on V by
q 2
0 if m = 0,
c .: |mi 7−→
11 (|m−1i if m ∈ Z>0,
c .: |mi 7−→ qm|mi for m ∈ Z ,
12 ≧0
c .: |mi 7−→ −qm+1|mi for m ∈ Z ,
21 ≧0
c .: |mi 7−→ (1−q2(m+1))|m+1i for m ∈ Z .
22 ≧0
By the construction, it is easy to see that this is a simple Q [SL ]-module. The
q 2
algebra homomorphism Q [SL ] → End (V) corresponding to this infinite di-
q 2 Q(q)
mensional Q [SL ]-module is denoted by π.
q 2
Next, we construct simple representations of Q [G] using this representation π.
q
For i∈ I, thereexists theinjective Hopfalgebra homomorphismφ : U (sl ) →
i qi 2
Uq(g) defined by E 7→ Ei, F 7→ Fi and Kα(= Kα∨) 7→ Ki. The image of φi is
denoted by U (sl ). Hence, we have the dual surjective Hopf algebra homomor-
qi 2,i
phism φ∗ : Q [G] → Q [SL ], f 7→ f| ◦φ .
i q qi 2 Uqi(sl2,i) i
Therefore,weobtaintherepresentations π| ◦φ ofQ [G](i ∈ I). Thesimple
q=qi i q
Q [G]-module corresponding to this representation will be denoted by V (= V
q si
as a vector space) and its natural basis will be denoted by |mi . The
i m∈Z
≧0
trivial Q [G]-module will be denoted by V .
q e (cid:8) (cid:9)
The following theorem is known as the tensor product theorem.
Theorem 2.12 ([Soi, 5.4]). Let w ∈ W. Then, for any reduced expression w =
s ···s , the Q [G]-module V ⊗··· ⊗V is simple and its isomorphism
i1 il(w) q si1 sil(w)
class does not depend on the choice of the reduced expressions of w. Moreover,
for any two reduced expression w = s ···s = s ···s , we have the Q [G]-
i1 il(w) j1 jl(w) q
module isomorphism Θ : V ⊗ ··· ⊗ V → V ⊗ ··· ⊗ V given by
j,i si1 sil(w) sj1 sjl(w)
|0i ⊗ ··· ⊗ |0i 7→ |0i ⊗ ··· ⊗ |0i , where i = (i ,...,i ) and j =
i1 il(w) j1 jl(w) 1 l(w)
(j ,...,j ).
1 l(w)
Hence, we denote this module by V .
w
ThemoduleV is a “highest weight module” in some sense. Let us explain this
w
point.
Definition 2.13. We define the subspaces Q [G/N+], Q [G/N−] of Q [G] by
q q q
Q [G/N+] := span cλ f ∈V(λ)∗,v ∈ V(λ) ,
q Q(q) f,v λ
n (cid:12) o
Q [G/N−] := span cλ (cid:12) f ∈V(λ)∗,v ∈ V(λ) .
q Q(q) f,v (cid:12) w0λ
n (cid:12) o
Then, Qq[G/N+] and Qq[G/N−] are th(cid:12)e right Uq(g)-subalgebras of Qq[G].
(cid:12)
The following lemma is the “triangular decomposition” of Q [G]. ([KS, Chap-
q
ter3])
10 HIRONORIOYA
Lemma 2.14. The multiplication Q [G/N−]⊗ Q [G/N+]→ Q [G], c⊗c′ 7→
q Q(q) q q
cc′ is surjective.
We have the following proposition. ([Jos, Proposition 10.1.1, Lemma 10.1.7,
Theorem 10.1.18])
Proposition 2.15. Let w ∈ W and w = s ···s be its reduced expression.
i1 il(w)
Set |(0)i := |0i ⊗···⊗|0i ∈ V ⊗···⊗V .
w i1 il(w) si1 sil(w)
Then,
(i) Q [G/N+].|(0)i = Q(q)|(0)i , and
q w w
(ii) if cλ .|(0)i 6= 0 for some weight vector f ∈ V(λ)∗, then wtf = wλ.
f,vλ w
Moreover,
(I) if cλ .V 6= 0 for some λ ∈ P and a weight vector f ∈ V(λ)∗, then
f,vλ w +
wtf ≥ wλ,
(II) if cλ .V 6= 0 for some λ ∈ P and a weight vector f ∈ V(λ)∗, then
f,vw0λ w +
wtf ≤ ww λ.
0
2.3. Kuniba-Okado-Yamada’s theorem. Kuniba-Okado-Yamada’s theorem
points out the relation between the Q(q)-vector space U (n+) and the simple
q
Q [G]-module V . To explain this, we introducethe PBW bases of U (n+). ([L4,
q w0 q
Chapter 37,40])
Definition 2.16 (Symmetries T′ , T′′ ). For i ∈ I and ǫ ∈ {±1}, there exist
i,ǫ i,ǫ
Q(q)-algebra automorphisms T′ , T′′ :U (g) → U (g) defined by
i,ǫ i,ǫ q q
T′ (E )= −K F ,T′ (F ) = −E K ,
i,ǫ i ǫi i i,ǫ i i −ǫi
T′ (E ) = (−1)rqǫrE(r)E E(s)for j 6= i,
i,ǫ j i i j i
r+s=X−hα∨i,αji
T′ (F ) = (−1)rq−ǫrF(s)F F(r)for j 6= i,
i,ǫ j i i j i
r+s=X−hα∨i,αji
Ti′,ǫ(Kh) = Kh−hαi,hiα∨i ,
T′′(E )= −F K ,T′′ (F ) = −K E ,
i,ǫ i i ǫi i,ǫ i −ǫi i
T′′(E ) = (−1)rq−ǫrE(s)E E(r)for j 6= i,
i,ǫ j i i j i
r+s=X−hα∨i,αji
T′′(F ) = (−1)rqǫrF(r)F F(s)for j 6= i,
i,ǫ j i i j i
r+s=X−hα∨i,αji
Ti′,′ǫ(Kh) = Kh−hαi,hiα∨i .
Note that
T′ = (T′′ )−1 ω◦T′ ◦ω = T′′ ,
i,ǫ i,−ǫ i,ǫ i,ǫ
∗◦T′ ◦∗ = T′′ , (·)◦T# ◦(·) = T#
i,ǫ i,−ǫ i,ǫ i,−ǫ
where # ∈ { ′, ′′}.
Definition 2.17 (PBW bases). Let ǫ ∈ {±1} and i = (i ,i ,...,i ) be a
1 2 l(w0)
reduced word of the longest element w of W. (i.e. w = s s ···s .)
0 0 i1 i2 il(w0)
Then, the vectors
E(c1)T′ (E(c2))···T′ T′ ···T′ (E(cl(w0))) (c ,c ,...,c ) ∈ (Z )l(w0)
i1 i1,ǫ i2 i1,ǫ i2,ǫ il(w0)−1,ǫ il(w0) 1 2 l(w0) ≧0
fnorm a basis of Uq(n+). Moreover, the vectors (cid:12)(cid:12) o
(cid:12)
E(c1)T′′ (E(c2))···T′′ T′′ ···T′′ (E(cl(w0))) (c ,c ,...,c ) ∈ (Z )l(w0)
i1 i1,ǫ i2 i1,ǫ i2,ǫ il(w0)−1,ǫ il(w0) 1 2 l(w0) ≧0
anlso form a basis of Uq(n+). They are called the PBW(cid:12)(cid:12) bases of Uq(n+). (See, for o
(cid:12)
instance, [Jan, Chapter 8] for details.)