Table Of ContentA link between two elliptic quantum groups
8
9
9
1 Pavel Etingof and Olivier Schiffmann
n
a February 8, 2008
J
3
2 Abstract
] We consider the category CB of meromorphic finite-dimensional rep-
A
resentations of the quantum elliptic algebra B constructed via Belavin’s
Q R-matrix, and the category CF of meromorphic finite-dimensional rep-
h. cres∈enCta,twioensusoefaFevlderesri’osnelolifptthiceqVuearntteuxm-IRgFrocuoprrEeτsp,γ2o(ngdlnen).ceFtoor caonnystfirxuecdt
t
a two families of (generically) fully faithful functors Hcx : CB → DB and
m Fxc : CF → DB where DB is a certain category of infinite-dimensional
representations of B by difference operators. We use this to construct
[
an equivalence between the abelian subcategory of CB generated by ten-
1 sor products of vector representations and theabelian subcategory of CF
v generated bytensor products of vector representations.
8
0
1 1 Categories of meromorphic representations
1
0
Inthissection,werecallthedefinitionsofvariouscategoriesofrepresentations
8
9 of quantum elliptic algebras.
/
h
at Notations: letusfixτ ∈C,Im(τ)>0,γ ∈R\Qandn≥2. Denoteby(vi)ni=1
the canonical basis of Cn and by (E )n the canonical basis of End(Cn), i.e
m ij i,j=1
E v = δ v . Let h = { λ E | λ = 0} be the space of diagonal
: ij k jk i i i ii i i
v traceless matrices. We havePa natural idPentification h∗ = { iλiEi∗i | Iλi =
Xi 0}. In particular, the weight of vi is ωi =Ei∗i− n1 kEk∗k. P P
P
r
a Classical theta functions: the theta functionθκ,κ′(t;τ) withcharacteristics
κ,κ′ ∈R is defined by the formula
θκ,κ′(t;τ)= eiπ(m+κ)((m+κ)τ+2(t+κ′)).
mX∈Z
It is an entire function whose zeros are simple and form the (shifted) lattice
{1 −κ+(1 −κ′)τ}+Z+τZ.
2 2
Theta functions satisfy (and are characterized up to renormalizationby) the
following fundamental monodromy relations
θκ,κ′(t+1;τ)=e2iπκθκ,κ′(t;τ), (1)
θκ,κ′(t+τ;τ)=e−iπτ−2iπ(t+κ′)θκ,κ′(t;τ). (2)
1
Theta functions withdifferentcharacteristicsare relatedto eachother by shifts
of t:
θκ1+κ2,κ′1+κ′2(t;τ)=eiπκ22τ+2iπκ2(t+κ′1+κ′2)θκ1,κ′1(t+κ2τ +κ′2;τ). (3)
In particular, we set θ(t)=θ (t;τ).
1,1
2 2
1.1 Meromorphic representations of the Belavin quantum
elliptic algebra
Consider the two n×n matrices
0 1 ... 0
1 0 ... 0
0 ξ ... 0 ... ... ... ...
A=0... 0... ...... ξn...−1 B =01 00 ...... 01
where ξ = e2iπ/n. We have An = Bn = Id, BA = ξAB, i.e A,B generate
the Heisenberg group. Belavin ([2]) introduced the matrix RB(z)∈End(Cn)⊗
End(Cn), uniquely determined by the following properties:
1. Unitarity: RB(z)RB(−z)=1,
21
2. RB(z) is meromorphic, with simple poles at z =γ+Z+τZ,
3. RB(0)=P :x⊗y 7→y⊗x for x,y ∈Cn (permutation),
4. Lattice translation properties:
RB(z+1)=A RB(z)A−1 =A−1RB(z)A ,
1 1 2 2
RB(z+τ)=e−2iπn−n1γB1RB(z)B1−1 =e−2iπn−n1γB2−1RB(z)B2.
In particular, RB(z) commutes with A⊗A and B ⊗ B. The matrix RB(z)
satisfies the quantum Yang-Baxter equation with spectral parameters:
RB(z−w)RB(z)RB(w)=RB(w)RB(z)RB(z−w).
12 13 23 23 13 12
The category C : followingFaddeev, Reshetikhin, TakhtajanandSemenov-
B
Tian-Shansky,onecandefineanalgebraBfromRB(z),usingtheRLLformalism-
see[3],[10]. However,wewillonlyneedtoconsideracertaincategoryofmodules
over this algebra, defined as follows.
LetC bethecategorywhoseobjectsarepairs(V,L(z))whereV isafinitedi-
B
mensionalvectorspaceandL(z)∈End(Cn)⊗End(V)isaninvertiblemeromor-
phicfunction(theL-operator)suchthatL(z+n)=L(z)andL(z+nτ)=L(z),
satisfying the following relation in the space End(Cn)⊗End(V)⊗End(V):
RB(z−w)L (z)L (w)=L (w)L (z)RB(z−w) (4)
12 13 23 23 13 12
(as meromorphic functions of z and w); morphisms (V,L(z))→(V′,L′(z)) are
linear maps ϕ : V → V′ such that (1⊗ϕ)L(z) = L′(z)(1⊗ϕ) in the space
Hom(Cn⊗V,Cn⊗V′). ThequantumYang-BaxterrelationforRB implies that
2
(Cn,χ(z)RB(z−w))∈Ob(C ) for all w ∈C, where we set χ(z)= θ(z−(1−n1)γ).
B θ(z)
This object is called the vector representation and will be denoted simply by
V (w).
B
The category C is naturally a tensor category with tensor product
B
(V,L(z))⊗(V′,L′(z))=(V ⊗V′,L (z)L′ (z)) (5)
12 13
at the level of objects and with the usual tensor product at the level of mor-
phisms.
ThereisanotionofadualrepresentationinthecategoryC : the(right)dual
B
of (V,L(z)) is (V∗,L∗(z)) where L∗(z) = L−1(z)t2 (first apply inversion, then
apply the transposition in the second component t ). If V,W ∈ Ob(C ) and
2 B
ϕ∈Hom (V,W) then ϕt ∈Hom (W∗,V∗).
CB CB
We will also need an extended category Cx defined as follows: objects of Cx
B B
are objects of C but we set
B
HomCBx(V,V′)=HomCB(V,V′)⊗MC
whereMCisthefieldofmeromorphicfunctionsofacomplexvariablex. Inother
words,morphismsinCx aremeromorphic1-parameterfamiliesofmorphismsin
B
C .
B
The category D : We now define a difference-operator variant of the cate-
B
gories CB,CBx. Let us denote by Mh∗ the field of (nωi)-periodic meromorphic
functions h∗ →C and by Dh∗ the C-algebra generatedby Mh∗ and shift opera-
tors Tµ : Mh∗ →Mh∗,f(λ)7→f(λ+µ) for µ∈h∗. If V is a finite-dimensional
vector space, we set Vh∗ = Mh∗ ⊗CV, and D(V)=Dh∗ ⊗End(V). Let DB be
the category whose objects are pairs (V,L(z)) where V is a finite-dimensional
C-vector space and L(z) ∈ End(Cn) ⊗ D(V) is an invertible operator with
meromorphiccoefficients satisfying (4)in End(Cn)⊗D(V)⊗D(V); morphisms
(V,L(z)) → (V′,L′(z)) are (nω )-periodic meromorphic functions ϕ : h∗ →
i
Hom(V,V′) such that (1⊗ϕ)L(z)=L(z)(1⊗ϕ) in HomC(Cn⊗Vh∗,Cn⊗Vh′∗)
(i.e morphisms are Mh∗-linear).
ThecategoryD isaright-modulecategoryoverC ,i.ewehavea(bi)functor
B B
⊗: D ×C →D defined by (5).
B B B
ThecategoryDx isdefinedinananalogousway: objectsarepairs(V,L(z,x))
B
as in D but the L-operator is now a meromorphic function of z and x, and
B
morphisms (V,L(z,x)) → (V′,L′(z,x)) are meromorphic maps ϕ(λ,x) : h∗ ×
C→HomC(V,V′) satisfying (1⊗ϕ)L(z,x)=L(z,x)(1⊗ϕ).
1.2 Meromorphic representations of the elliptic quantum
group Eτ,γ/2(gln)
Felder’s dynamical R-matrix: letusconsiderthefunctionsoftwocomplex
variables
θ(l+γ)θ(z) θ(z−l)θ(γ)
α(z,l)= , β(z,l)= .
θ(l)θ(z−γ) θ(l)θ(z−γ)
3
As functions of z, α and β have simple poles at z =γ+Z+τZ and satisfy
α(z+1,l)=α(z,l), α(z+τ,l)=e−2iπγα(z,l),
β(z+1,l)=β(z,l), β(z+τ,l)=e−2iπ(γ−l)β(z,l).
Felder introducedin [4] the matrixRF(z,λ):C×h∗ →End(Cn)⊗End(Cn):
RF(z,λ)= E ⊗E + α(z,λ −λ )E ⊗E + β(z,λ −λ )E ⊗E
ii ii i j ii jj i j ji ij
Xi Xi6=j Xi6=j
where λ= λ E∗ ∈h∗.
i i ii
This matPrix is a solution of the quantum dynamical Yang-Baxter equation
with spectral parameters
RF (z−w,λ−γh )RF (z,λ)RF (w,λ−γh )
12 3 13 23 1
=RF (w,λ)RF (z,λ−γh )RF (z−w,λ)
23 13 2 12
wherewehaveusedthefollowingconvention: ifV arediagonalizableh-modules
i
with weight decomposition V = Vµ and a(λ)∈End( V ) then
i µ i i i
L N
a(λ−γhl)| Vµi =a(λ−γµl)
i i
N
As usual, indices indicate the components of the tensor product on which the
operators act.
In addition, R(z,λ) satisfies the following two conditions:
1. Unitarity: R (z,λ)R (−z,λ)=Id,
12 21
2. Weight zero: ∀h∈h, [h(1)+h(2),R(z,λ)]=0.
The category C : It is possible to use R(z,λ) to define an algebra by the
F
RLL-formalism (see [4]): the elliptic quantum group E (gl (C)). However,
τ,γ/2 n
we will only need the following category of its representations C , introduced
F
by Felder in [4] and studied by Felder and Varchenko in [5]: objects are pairs
(V,L(z,λ))whereV isafinite-dimensionaldiagonalizableh-moduleandL(z,λ):
C×h∗ → End(Cn)⊗End(V) is an invertible meromorphic function which is
(nω )-periodic in λ and which satisfies the following two conditions:
i
[h +h ,L(z,λ)]=0,
1 2
R (z−w,λ−γh )L (z,λ)L (w,λ−γh )
12 3 13 23 1
=L (w,λ)L (z,λ−γh )R (z−w,λ) (6)
23 13 2 12
Morphisms (V,L(z,λ))→(V′,L′(z,λ)) are (nω )-periodic meromorphic weight
i
zeromapsϕ(λ):V →V′ suchthatL′(z,λ)(1⊗ϕ(λ−γh ))=(1⊗ϕ(λ))L(z,λ).
1
The dynamical quantum Yang-Baxter relation for RF(z,λ) implies that (Cn,
RF(z−w,λ))∈Ob(C ) for all w ∈C. This is the vector representation and it
F
will be denoted by V (w).
F
The category C is naturally equipped with a tensor structure: it is defined
F
on objects by
(V,L(z,λ))⊗(V′,L′(z,λ))=(V ⊗V′,L (z,λ−γh )L′ (z,λ)),
12 3 13
4
and if ϕ∈Hom (V,W),ϕ′ ∈Hom (V′,W′) then
CF CF
(ϕ⊗ϕ′)(λ)=ϕ(λ−γh )⊗ϕ′(λ)∈Hom (V ⊗V′,W ⊗W′).
2 CF
There is a notion of a dual representation in the category C : the (right)
F
dual of (V,L(z,λ)) is (V∗,L∗(z,λ)) where L∗(z,λ) = L−1(z,λ+γh2)t2 (apply
inversion, shifting and then apply the transposition in the second component
t ). If V,W ∈ Ob(C ) and ϕ(λ) ∈ Hom (V,W) then ϕ∗(λ) := ϕ(λ+γh )t ∈
2 B CB 1
Hom (W∗,V∗).
CB
The extended category Cx is defined by Ob(Cx)=Ob(C ) and
F F F
HomCFx(V,V′)=HomCF(V,V′)⊗MC
i.emorphismsinCx aremeromorphic1-parameterfamilies ofmorphismsinC .
F F
2 The functor Fc : C → D
x F B
In this section, we define a family of functors from meromorphic (finite-
dimensional) representations of Eτ,γ(gln(C)) to infinite-dimensional represen-
2
tations of the quantum elliptic algebra B.
2.1 Twists by difference operators:
For any finite-dimensional diagonalizable h-module V, let eγD ∈ End(Vh∗)
denotetheshiftoperator: eγD. f (λ)v = f(λ+γµ)v ,v ∈V . Nowlet
µ µ µ µ µ µ µ
(V,L(z,λ))∈CF, and let S(z,λP),S′(z,λ):C×Ph∗ →End(Cn) be meromorphic
and nondegenerate. Define the difference-twist of (V,L(z,λ)) to be the pair
(V,LS,S′(z)) where
LS,S′(z)=S (z,λ−γh )L(z,λ)e−γD1S′(z,λ)−1 ∈End(Cn)⊗D(V). (7)
1 2 1
This is a difference operator acting on Cn⊗Vh∗.
Lemma 1 The difference operator LS(z,λ) satisfies the following relation in
End(Cn)⊗D(V)⊗D(V):
T (z,w,λ−γh )LS,S′(z)LS,S′(w)=LS,S′(w)LS,S′(z)T′ (z,w,λ)
12 3 13 23 23 13 12
where
T(z,w,λ)=S (w,λ)S (z,λ−γh )RF (z−w,λ)S (w,λ−γh )−1S (z,λ)−1
2 1 2 12 2 1 1
(8)
T′(z,w,λ)=S′(z,λ)S′(w,λ+γh )RF (z−w,λ)S′(z,λ+γh )−1S′(w,λ)−1
1 2 1 12 1 1 2
(9)
Proof: theproofisstraightforward,usingrelation(6)forL(z,λ)andtheweight
zero property of RF(u,λ) and L(u,λ).(cid:3)
5
2.2 The Vertex-IRF transform
Let φl(u) = e2iπ(l2nτ+lnu)θ0,0(u + lτ;nτ) for l = 1,...n. Then the vector
Φ(u) = (φ (u),... ,φ (u)) is, up to renormalization, the unique holomorphic
1 n
vector in Cn satisfying the following monodromy relations:
Φ(u+1)=AΦ(u), (10)
Φ(u+τ)=e−iπnτ−2iπnuBΦ(u) (11)
NowletS(z,λ):C×h∗ →End(Cn)bethematrixwhosecolumnsare(Φ (z,λ),
1
... ,Φ (z,λ)) where Φ (z,λ) = Φ(z −nλ ). Using (10)-(11), it is easy to see
n j j
that we have det(S(z,λ)) = Const(λ)θ(z) and hence that S(z,λ) is invertible
for z 6=0 and generic λ.
Lemma 2 We have
RB(z−w)S (z,λ)S (w,λ−γh )=S (w,λ)S (z,λ−γh )RF(z−w,λ)
1 2 1 2 1 2
RB(z−w)S (w,λ)S (z,λ+γh )=S (z,λ)S (w,λ+γh )RF(z−w,λ)
2 1 2 1 2 1
Proof: thefirstrelationisequivalenttothefollowingidentitiesfori,j =1,...n:
RB(z−w)Φ (z,λ)⊗Φ (w,λ−γω )=Φ (z,λ−γω )⊗Φ (w,λ)
i i i i i i
RB(z−w)Φ (z,λ)⊗Φ (w,λ−γω )=α(z−w,λ −λ )Φ (z,λ−γω )⊗Φ (w,λ)
i j i i j i j j
+β(z−w,λ −λ )Φ (z,λ−γω )⊗Φ (w,λ)
i j j i i
These identities are proved by comparing poles and transformation properties
under lattice translations as functions of z and w, and using the uniqueness of
Φ. The second relation of the lemma is proved in the same way. These identi-
tiesareessentiallytheVertex/Interaction-Round-a-Facetransformofstatistical
mechanics (see [8],[9] and [6] for the case n=2).(cid:3)
2.3 Construction of the functor Fc : C → D
x F B
Let us fix some c ∈ C. We can now define the family of functors Fc :
x
C → C indexed by x ∈ C: for (V,L(z,λ)) ∈ C , set Fc((V,L(z,λ))) =
F B F x
(V,LSx,Sx+c(z)) with Su(z,λ) = S(z −u,λ) as above and let Fxc be trivial at
the level of morphisms.
Proposition 1 Fc :C →D is a functor.
x F B
Proof: itfollowsfromLemma2that(V,LSx,Sx+c(z))∈Ob(DB). Furthermore,
if ϕ(λ)∈Hom ((V,L(z,λ)),(V′,L′(z,λ))) then by definition we have L′(z,λ)
CF
(1⊗ϕ(λ−γh ))=(1⊗ϕ(λ))L(z,λ), so that
1
S (z−x,λ−γh )L′(z,λ)e−γD1S (z−x−c,λ)−1(1⊗ϕ(λ))
1 2 1
=S (z−x,λ−γh )L′(z,λ)(1⊗ϕ(λ−γh ))e−γD1S (z−x−c,λ)−1
1 2 1 1
=S (z−x,λ−γh )(1⊗ϕ(λ))L′(z,λ)e−γD1S (z−x−c,λ)−1
1 2 1
=(1⊗ϕ(λ))S (z−x,λ−γh )L′(z,λ)e−γD1S (z−x−c,λ)−1
1 2 1
since ϕ(λ) is of weight zero. Thus Fc(ϕ(λ)) is an intertwiner in the category
x
D .(cid:3)
B
6
We can also think of the family of functors Fc as a single functor Fc :Cx →
x F
Dx.
B
Remark: wecanthinkofthedifference-twistandtherelationsinLemma2as
adynamicalanalogueofthenotionofequivalenceofR-matricesduetoDrinfeld
and Belavin-see [1].
3 The image of the trivial representation and
the functor Hc : C → D
x B B
Applying the functor Fc to the trivial representation(C,Id)∈Ob(C ) yields
x F
Fc((C,Id))=(C,S(z−x,λ)e−γD1S(z−x−c,λ)−1).
x
We will denote this object by Ic. For instance, when n = 2, we obtain a
x
representation of the Belavin quantum elliptic algebra as difference operators
acting on the space of periodic meromorphic functions in one variable λ, i.e
given by an L-operator
a(z) b(z)
L(z)=
(cid:18)c(z) d(z)(cid:19)
where a(z),b(z),c(z),d(z) are operators of the form f(z)T +g(z) where T
−γ −γ
is the shift by −γ.
SuchrepresentationsofBbydifferenceoperatorsalreadyappearedinthework
of Krichever, Zabrodin ([9]) (for n = 2) and Hasegawa ([7],[8])(for the general
case), where they were also derived by some Vertex-IRF correspondence.
Definition: Let c ∈ C and let Hc : C → D be the functor defined by the
x B B
assignment V → Ic ⊗V and which is trivial at the level of morphisms. The
x
family of functors Hc gives rise to a functor Hc :Cx →Dx.
x B B
4 Full Faithfulness of the functor Hc : C → D
x B B
In this section, we prove the following result
Proposition 2 Let V,V′ ∈ Ob(C ). Then for all but finitely many values of
B
xmodZ+Zτ, the map
Hc :Hom (V,V′)→∼ Hom (Hc(V),Hc(V′))
x CB DB x x
is an isomorphism.
Proof: since Hom (V,V′)≃Hom (C,V′⊗V∗), Hom (Ic⊗V,Ic⊗V′)≃
CB CB DB x x
Hom (Ic,Ic⊗V′⊗V∗),itisenoughtoshowthatthemapHc :Hom (C,W)→
DB x x x CB
Hom (Ic,Ic⊗W) is an isomorphism for all W ∈Ob(C ). Since Hc is trivial
DB x x B x
at the level of morphisms, this map is injective. Now let W ∈ Ob(C ) and
B
7
let ϕ(λ) ∈ Hom (Ic,Ic ⊗W), that is, ϕ(λ) is a (nω )-periodic meromorphic
DB x x i
function h∗ →W satisfying the equation
ϕ (λ)S (z−x,λ)e−γD1S (z−x−c,λ)−1
2 1 1
=S (z−x,λ)e−γD1S (z−x−c,λ)−1L (z)ϕ (λ)
1 1 12 2
where L(z) is the L-operator of W. This is equivalent to
L (z)ϕ (λ)=S (z−x−c,λ)ϕ (λ+γh )S (z−x−c,λ)−1 (12)
12 2 1 2 1 1
Now L(z) is an elliptic function (of periods n and nτ) so it is either constant
orithasa pole. Restricting W tothe subrepresentationSpan(ϕ(λ),λ∈h∗), we
see thatthe latter caseisimpossible for genericxasthe RHSof(12)has apole
at z = x+c only; hence L(z) is constant. Furthermore, from (12) we see that
the matrix
M(λ)=S (z−x−c,λ)−1L S (z−x−c,λ)
1 12 1
isindependentofz. Inparticular,settingz 7→z+1andusingthetransformation
properties (10) of S(z,λ), we obtain [A ,L ] = 0. This implies that L =
1 12
E ⊗D for some D ∈End(W).
i ii i i
P
Lemma 3 Let U be a finite dimensional vector space, let T ∈ End(Cn) ⊗
End(U) be an invertible solution of the equation
RB(z)T T =T T RB(z) (13)
12 13 23 23 13 12
such that T = E ⊗D for some D ∈ End(U). Then [D ,D ] = 0 for all
i ii i i i j
i,j and there ePxists X ∈ End(U) such that Xn = 1 and Di+1 = XDi for all
i=1,...n.
Proof: let us write RB(z) = R (z)E ⊗ E . Then equation
p,q,r,s p,q,r,s pq rs
(13) is equivalent to Rp,q,r,s(z)DPpDq = Rp,q,r,s(z)DsDr for all p,q,r,s. But it
follows from the general formula for RB(z) that R (z) 6= 0 if and only if
p,q,r,s
p+q ≡r+s(modn). Thus we have [D ,D ]=0 for all i,j and X :=D D−1
i j i i+1
is independent of i, and satisfies Xn =1.(cid:3)
By the above lemma, there exists X ∈ End(W) such that Xn = 1 and
D =XD . SupposethatX 6=1andchoosee∈W suchthatX(e)=ξkewith
i+1 i
ξk 6=1. Nowweapplythetransformationz 7→z+τ tothematrixM(λ). Noting
that, by (11), S(z−x−c+τ,λ) = e−iπτ/2−2iπ(z−x−c)/nBS(z−x−c,λ)F(λ)
where F(λ)=diag(e−2iπλ,...e−2iπλn), we obtain the equality
F(λ)−1S (z−x−c,λ)−1B−1L B S (z−x−c,λ)F(λ)
1 1 12 1 1
=S (z−x−c,λ)−1L S (z−x−c,λ)
1 12 1
Applying this to the vector e yields AdF(λ)(M(λ))(e) = ξ−kM(λ)(e). This is
possible for all λ only if k ≡ 0 (mod n). Hence X = 1 and (12) reduces to the
equation Dϕ (λ)=ϕ (λ+γh ). In particular ϕ(λ) is γ(ω −ω )-periodic. But
2 2 1 i j
byourassumption,ϕ(λ)is(nω )-periodicandγ isrealandirrational. Therefore
i
ϕ(λ) is constant and it is a morphism in the category C .(cid:3)
B
Now, considering x as a parameter, we obtain:
Corollary 1 The functor Hc :Cx →Dx is fully faithful.
B B
8
Remark: equation (12) shows that Hom (Ic,Ic ⊗V) = Hom (V∗,I0 ).
DB x x CB x+c
Thus the above proposition states that for any finite-dimensional represen-
tation V ∈ Ob(C ) and for all but finitely many x mod Z + τZ, we have
F
Hom (V∗,I0)=Hom (V∗,C), wherethe somorphismis induced bythe em-
DB x CB
bedding C ⊂ I0 (constant functions). However, for finitely many values of
x
xmod
Z+τZ, this may not be true: see [9] and [8] where some finite-dimensional
subrepresentations of I0 are considered.
x
5 Full faithfullness of the functor Fc : C → D
x F B
In this section, we prove the following result:
Proposition 3 The functor Fc :C →D is fully faithful.
x F B
Proof: we have to show that for any two objects V,V′ in C there is an iso-
F
morphism Fc : Hom (V,V′) → Hom (Fc(V),Fc(V′)). Since Fc is trivial
x CF DB x x x
at the level of morphisms, this map is injective. Now let V,W ∈ Ob(C ) and
F
let ϕ(λ) ∈ Hom (Fc(V),Fc(W)). By definition, ϕ(λ) : V → W satisfies the
DB x x
relation
ϕ (λ)S (z−x,λ−γh )LV (z,λ)e−γD1S (z−x−c,λ)−1
2 1 2 12 1
=S (z−x,λ−γh )LW(z,λ)e−γD1S (z−x−c,λ)−1ϕ (λ)
1 2 12 1 2
where LV(z,λ) (resp. LW(z,λ)) is the L-operator of V (resp. W). This is
equivalent to
ϕ (λ)S (z−x,λ−γh )LV (z,λ)=S (z−x,λ−γh )LW(z,λ)ϕ (λ−γh )
2 1 2 12 1 2 12 2 1
(14)
Introduce the following notations: write W = W , V = V , ϕ(λ) =
ξ ξ µ µ
νϕν(λ) for the weight decompositions (so thatLϕν : Vξ → WLξ+ν). Also let
PS(z−x,λ)= Sij(z−x,λ)E , LV (z,λ)= E ⊗Lij(z,λ)and use the
i,j ij 12 i,j ij V
same notationPfor LW(z,λ). Applying (14) to vP⊗ζ for some i and ζ ∈ V
i µ µ µ
yields
Skj(z−x,λ−γ(µ+ω −ω ))v ⊗ϕ (λ)(Lji(z,λ)ζ )
i j k ν V µ
jX,k,ν
= Skl(z−x,λ−γ(µ+ω −ω +σ))v ⊗Lli (z,λ)ϕ (λ−γω )ζ
i l k W σ i µ
lX,k,σ
(15)
where we usedthe weight-zeropropertyofLV(z,λ) andLW(z,λ). Applying v∗
k
to (15) and projecting on the weight space W gives the relation
µ+ωi+ξ
Skj(z−x,λ−γ(µ+ω −ω ))ϕ (λ)(Lji(z,λ)ζ )
i j ν V µ
X
ν,j
ν−ωj=ξ
= Skl(z−x,λ−γ(µ+ω −ω +σ))Lli (z,λ)(ϕ (λ−γω )ζ )
i j W σ i µ
X
σ,l
σ−ωl=ξ
(16)
9
for any i,k,ξ and ζ ∈ V . Now let A = {χ | ϕ (λ) 6= 0}. Fix some j and
µ µ χ
let β ∈ A be an extremal weight in the direction −ω (i.e β−ω +ω 6∈ A for
j j k
k 6=j). Then (16) for ξ =β−ω reduces to
j
Skj(z−x,λ−γ(µ+ω −ω ))ϕ (λ)(Lji(z,λ)ζ )
i j β V µ
=Skj(z−x,λ−γ(µ+ω −ω +β))Lji(z,λ)ϕ (λ−γω )ζ
i j W β i µ
(17)
Claim: thereexistsi∈{1,...n},µandζ ∈V suchthatϕ (λ)(Lji(z,λ)ζ )6=
µ µ β V µ
0 for generic z and λ.
Proof: recall the central element Qdet(z,λ) ∈ Eτ,γ(gln). By definition, its
2
action on V is invertible. Expanding Qdet(z,λ) along the jth-line, we have
Qdet(z,λ) = Lji(z,λ)P (z,λ) for some operators P (z,λ) ∈ End(V). In
i V i i
particular, PImLji(z,λ)=V, and the claim follows.
i
Thus,the raPtioSkj(z−x,λ−γ(µ+ωi−ωj+β))/Skj(z−x,λ−γ(µ+ωi−ωj))
is independent of k. This is possible only if β ∈ CE∗ . Applying this to
r6=j rr
j = 1,...n, we see that A = {0}. Hence ϕ(λ) isPan h-module map. But then
relation (14) reduces to ϕ (λ)LV (z,λ)=LW(z,λ)ϕ (λ−γh ), and ϕ(λ) is an
2 12 12 2 1
intertwiner in the category C .(cid:3)
F
Corollary 2 The functor Fc :Cx →Dx is fully faithful.
F B
6 The image of the vector representation
LetusdenoteV˜ (w)=(Cn,χ(w)RF(w,λ)). ItisanobjectofC whichequals
F F
the tensor product of the vector representation V (w) by the one-dimensional
F
representation (C,χ(z)).
Proposition 4 For any x,w, x+c6≡w (modZ+τZ), we have Fc(V (w))≃
x F
Hc(V (w)).
x B
Proof: by definition, we have
Fc(V˜ (w))=(Cn,χ(z)S (z−x,λ−γh )RF(z−w,λ)e−γD1×S (z−x−c,λ)−1),
x F 1 2 1
Ic⊗V (w)=(Cn,χ(z)S (z−x,λ)e−γD1S (z−x−c,λ)RB(z−w))
x B 1 1
We claim that the map ϕ(λ) = e−γD(S(w−x−c,λ)−1)eγD ∈ End(Cn) is an
intertwiner Hc(V (w))≃Ic⊗V (w)→∼ Fc(V˜ (w)). Indeed, we have
x B x B x F
S (z−x,λ−γh )RF(z−w,λ)e−γD1S (z−x−c,λ)−1(1⊗ϕ(λ))
1 2 1
=e−γD2S (z−x,λ)eγD2RF(z−w,λ)e−γ(D1+D2)S (z−x−c,λ+γh )−1S (w−x−c,λ)−1eγD2
1 1 2 2
=e−γD2S (z−x,λ)e−γD1RF(z−w,λ)S (z−x−c,λ+γh )−1S (w−x−c,λ)−1eγD2
1 1 2 2
=e−γD2S (z−x,λ)e−γD1S (w−x−c,λ+γh )−1S (z−x−c,λ)−1RB(z−w)eγD2
1 2 1 1
=e−γD2S (z−x,λ)S (w−x−c,λ)e−γD1S (z−x−c,λ)−1RB(z−w)eγD2
1 2 1
=(1⊗ϕ(λ))S (z−x,λ)e−γD1S (z−x−c,λ)−1RB(z−w)
1 1
where we used Lemma 2 and the zero-weightproperty of RF(u,λ).(cid:3)
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