On certain categories of modules for twisted affine Lie algebras

0.21 MB·English

by  Yongcun Gao

On certain categories of modules for twisted affine
Lie algebras
Yongcun Gao1 and Jiayuan Fu2
School of Science, Communication University of China, 100024
Abstract
9 Inthispaper,usinggeneratingfunctionswestudytwocategoriesE andC ofmod-
0 ules for twisted affineLie algebras ˆg[σ], which were firstly introduced and studied in
0
[Li]foruntwistedaffineLiealgebras. Weclassify integrableirreducibleˆg[σ]-modules
2
in categories E and C, where E is proved to contain the well known evaluation mod-
n
a ules and C to unify highest weight modules, evaluation modules and their tensor
J
product modules. We determine the isomorphism classes of those irreducible mod-

4
ules.

]
A
R 1 Introduction
.
h
t For affine Lie algebras, a very important class of modules is the class of highest weight
a
m modules in the well known category O, where highest weight integrable modules (of
[
nonnegative integral levels) (cf. [K]) have been the main focus. In [C], Chari proved that

1 every irreducible integrable g˜-module of a positive level with finite-dimensional weight
v
7 subspaces must beahighest weight moduleandthatevery irreducible integrablegˆ-module
7
of level zero with finite-dimensional weight subspaces must be finite-dimensional.
3
0 Another important class of modules is so called “evaluation modules” (of level zero)
.
1 associated with a finite number of g-modules and with the same number of nonzero
0
9 complex numbers, studied by Chari and Pressley in [CP2] (cf. [CP1], [CP3]). It was
0
proved in [CP2] (together with [C]) that every finite-dimensional irreducible (integrable)
:
v
gˆ-module is isomorphic to an evaluation module. Furthermore, Chari and Pressley in
i
X
[CP2] studied the first time the tensor product module of an integrable highest weight gˆ-
r
a modulewitha(finite-dimensional) evaluationgˆ-moduleassociatedwithfinite-dimensional
irreducible g-modules and distinct nonzero complex numbers. A result, proved in [CP2],
is that such a tensor product module is also irreducible. These irreducible (integrable) gˆ-
modules are greatly different from highest weight modules and finite-dimensional modules
in many aspects. In this way, a new family of irreducible integrable gˆ-modules were
constructed.
With this new family of irreducible integrable gˆ-modules having been constructed,
important problems are to determine the isomorphism classes and to find a canonical
characterization using internal structures, instead of presenting them as tensor product
1
Partially supported by a grant from Communication University of China
2
Partially supported by the National Science Foundation of China (10726057)
1
modules, and then to unify the new family with the family of highest weight irreducible
integrable modules toward a classification of all the irreducible integrable gˆ-modules.
In the paper [Li], all the above mentioned problems have been completely solved for
untwisted affineLiealgebrasbyexploiting generatingfunctionsandformalcalculus, which
have played an essential role in the theory of vertex operator algebras.
In [Li] two category E and C of modules for untwisted affine Lie algebras gˆ are de-
fined and studied. A key result is a factorization which states that every irreducible
representation of gˆ in the category C can be factorized canonically as the product of
two representations such that the first representation defines a restricted module and the
second one defines a module in the category E.
We generalize all the results of [Li] to twisted affine Lie algebras gˆ[σ]. We classify
integrable irreducible ˆg[σ]-modules in categories E and C, where E is proved to contain
the well known evaluation modules and C to unify highest weight modules, evaluation
modules and their tensor product modules. We determine the isomorphism classes of
those irreducible modules.
This paper is organized as follows: In Section 2, we review the notions of restricted
module and integrable module and we prove a complete reducibility theorem about the
restricted integrable gˆ[σ]-modules. We also collect and restate certain results on modules
for tensor product algebras of two associative algebras in [Li] . In Section 3, we study
the category E, and in Section 4, we study the category C. In Section 5, we classify the
irreducible integrable modules in the category C.
2 Category R of restricted gˆ[σ]-modules
First let us fix some formal variable notations (see [FLM], [FHL], [LL]). Throughout this
paper, t,x,x ,x ,... are independent mutually commuting formal variables. We shall
1 2
typically use z,z ,z ,... for complex numbers. For a vector space U, U[[x±1,...,x±1]]
1 2 1 n
denotes the space of all formal (possibly doubly infinite) series in x ,...,x with coef-
1 n
ficients in U, U((x ,...,x )) denotes the space of all formal (lower truncated) Laurent
1 n
series inx ,...,x with coefficients inU andU[[x ,...,x ]]denotes the space ofallformal
1 n 1 n
(nonnegative) powers series in x ,...,x with coefficients in U.
1 n
We shall use the traditional binomial expansion convention: For m ∈ Z,
m
(x ±x )m = (±1)ixm−ixi ∈ C[x ,x−1][[x ]]. (2.1)
1 2 i 1 2 1 1 2
i≥0 (cid:18) (cid:19)
X
Recall from [FLM] the formal delta function
δ(x) = xn ∈ C[[x,x−1]]. (2.2)
n∈Z
X
2
Its fundamental property is that
f(x)δ(x) = f(1)δ(x) for f(x) ∈ C[x,x−1]. (2.3)
For any nonzero complex number z,
z
δ = znx−n ∈ C[[x,x−1]] (2.4)
x
(cid:16) (cid:17) Xn∈Z
and we have
z z
f(x)δ = f(z)δ for f(x) ∈ C[x,x−1]. (2.5)
x x
(cid:16) (cid:17) (cid:16) (cid:17)
In particular,
z
(x−z)δ = 0. (2.6)
x
(cid:16) (cid:17)
Let g be a simple finite-dimensional Lie algebra equipped with a nondegenerate sym-
metric invariant bilinear form h·,·i so that the squared length of the longest roots is 2,
and let σ be an automorphism of g of order N, ε = exp2Nπi. Then we have
N−1
g = g , where g = {a ∈ g|σ(a) = εia}.
i i
i=0
M
Let gˆ[σ] be the corresponding twisted affine Lie algebra, i.e.,
N−1
gˆ[σ] = gi ⊗tNi C[t,t−1]⊕Ck (2.7)
i=0
M
with the defining commutator relations
i
[a⊗tm+Ni ,b⊗tn+Nj ] = [a,b]⊗tm+n+iN+j +(m+ N)ha,biδm+Ni ,−n−Nj k, (2.8)
for a ∈ g ,b ∈ g , m,n ∈ Z, and with k as a nonzero central element. A gˆ[σ]-module W
i j
is said to be of level ℓ in C if the central element k acts on W as the scalar ℓ.
For a ∈ g , form the generating function
i
a(x) = (a⊗tn+Ni )x−n−Ni −1 ∈ gˆ[σ][[xN1 ,x−N1 ]]. (2.9)
n∈Z
X
In terms of generating functions the defining relations (2.8) exactly amount to
i i
x N x ∂ x N x
[a(x ),b(x )] = [a,b](x )x−1 2 δ 2 +ha,bi x−1 2 δ 2 k.(2.10)
1 2 2 1 x x ∂x 1 x x
(cid:18) 1(cid:19) (cid:18) 1(cid:19) 2 " (cid:18) 1(cid:19) (cid:18) 1(cid:19)#
3
Following the tradition (cf. [FLM], [LL]), for a ∈ g , n ∈ Z we shall use a(n) for the
i
corresponding operator associated to a ⊗ tn+Ni on ˆg[σ]-modules. We have the category
R of the so-called restricted modules for the affine Lie algebra ˆg[σ]. A gˆ[σ]-module W is
said to be restricted (cf. [K1]) if for any w ∈ W,
a(n)w = 0 for n sufficiently large. (2.11)
Notice that in terms of generating functions, the condition (2.11) amounts to that
1
a(x)w ∈ W((xN)) for w ∈ W. (2.12)
That is, a gˆ[σ]-module W is restricted if and only if
1
a(x) ∈ Hom(W,W((xN))) for a ∈ g. (2.13)
Let U be a g-module and let ℓ be any complex number. Let k act on U as the scalar
ℓ and let
ˆg[σ]+ = gi ⊗tn+Ni
n∈N,i=0,···X,N−1,n+Ni>0
act trivially, making U a (g ⊕ gˆ[σ] ⊕ Ck)-module. Form the following induced ˆg[σ]-
0 +
module
Mˆg(ℓ,U) = U(gˆ[σ])⊗g0⊕ˆg[σ]+⊕Ck U. (2.14)
It is clear that M (ℓ,U) is a restricted ˆg[σ]-module. We have also that the category R
ˆg
contains all the highest weight modules.
Lemma 2.1. There are homogeneous elements {a1,...,ar} of g = N−1g such that
i=0 i
g = span{a1,a2,...,ar}, [ak(m),ak(n)] = 0 for 1 ≤ k ≤ r, Lm,n ∈ Z (2.15)
and such that for 1 ≤ k ≤ r and for any n ∈ Z, ak(n) acts locally nilpotently on all
integrable gˆ[σ]-modules.
It is similar to untwisted case, we have the following result:
Theorem 2.2. Let g be a finite-dimensional simple Lie algebra equipped with the nor-
malized Killing form. Every nonzero restricted integrable gˆ[σ]-module is a direct sum of
(irreducible) highest weight integrable modules. In particular, every irreducible integrable
gˆ[σ]-module W is a highest weight integrable module.
4
Proof. In view of the complete reducibility theorem in [K1] we only need to show that
every nonzero restricted integrable gˆ[σ]-module W contains a highest weight integrable
(irreducible) submodule.
Claim 1: There exists a nonzero u ∈ W such that gˆ[σ] u = 0.
+
For any nonzero u ∈ W, since W is restricted, ˆg[σ] u is finite-dimensional. For any
+
u ∈ W, we define d(u) = dimgˆ[σ] u. We need to prove that there is a 0 6= u ∈ W such
+
that d(u) = 0.
Suppose that d(u) > 0 for any 0 6= u ∈ W. Take 0 6= u ∈ W such that d(u) is
minimal. By Lemma 2.1, There are {a1,...,ar} of g such that ak(n) locally nilpotently
act on W for k = 1,...,r, n ∈ Z. Let l+ j be the positive number such that g (l)u 6= 0
N j
and g (n)u = 0 whenever n + i > l + j . By the definition of l, ak(l)u 6= 0 for some
i N N
ak ∈ g ,1 ≤ k ≤ r.
j
Notice that ak(l)su = 0 for some nonnegative integer s. Let m be the nonnegative
integer such that ak(l)mu 6= 0 and ak(l)m+1u = 0. Set v = a (l)mu. We will obtain a
k
contradiction by showing that d(v) < d(u). First we prove that if a(n)u = 0 for some
a ∈ g , n+ i > 0, then a(n)v = 0. In the following we will show by induction on m that
i N
a(n)a (l)mu = 0 for any a ∈ g and m ≥ 0. If m = 0 this is immediate. Now assume that
k i
the result holds for m. Since [a,ak](l+n)u = 0 (from the definition of l) and a(n)u = 0,
by the induction assumption that a(n)ak(l)mu = 0 we have:
[a,ak](l+n)ak(l)mu = 0, a(n)ak(l)mu = 0. (2.16)
Thus
a(n)ak(l)m+1u = [a(n),ak(l)]ak(l)mu+ak(l)a(n)ak(l)mu
= [a,ak](l+n)ak(l)mu+ak(l)a(n)ak(l)mu
= 0, (2.17)
as required. In particular, we see that a(n)v = a(n)ak(l)mu = 0. Therefore, d(v) ≤ d(u).
Since ak(l)v = 0 and ak(l)u 6= 0, we have d(v) < d(u), a contradiction.
Claim 2: W contains an irreducible highest weight integrable submodule. Set
Ω(W) = {u ∈ W |gˆ[σ] u = 0}. (2.18)
+
Then Ω(W) is a g -submodule of W and it is nonzero by Claim 1. Since ak(0) for
0
k = 1,...,r act locally nilpotently on Ω(W), it follows from the PBW theorem that for
any u ∈ Ω(W), U(g )u is finite-dimensional, so that U(g )u is a direct sum of finite-
0 0
dimensional irreducible g -modules. Let u ∈ Ω(W) be a highest weight vector for g . It is
0 0
clear that u is a singular vector forgˆ[σ]. It follows from [K] that u generates an irreducible
gˆ-module.
5
We recall from [Li] the following result:
Lemma 2.3. Let A and A be associative algebras (with identity) and let U and U
1 2 1 2
be irreducible modules for A and A , respectively. If either End U = C, or A is of
1 2 A1 1 1
countable dimension, then U ⊗U is an irreducible A ⊗A -module.
1 2 1 2
Lemma 2.4. Let A and A be associative algebras (with identity) and let U be an irre-
1 2
ducible A ⊗A -module. Suppose that U as an A -module has an irreducible submodule
1 2 1
U and assume that either A is of countable dimension or End U = C. Then U is
1 1 A1 1
isomorphic to an A ⊗A -module of the form U ⊗U as in Lemma 2.3.
1 2 1 2
Lemma 2.5. Let A and A be associative algebras (with identity) and let W be an A ⊗
1 2 1
A -module. Assume that A is of countable dimension and assume that W is a completely
2 1
reducible A -module and a completely reducible A -module. Then W is isomorphic to
1 2
a direct sum of irreducible A ⊗ A -modules of the form U ⊗ V with U an irreducible
1 2
A -module and V an irreducible A -module.
1 2
3 Category E of gˆ[σ]-modules
In this section, we study the category E of gˆ[σ]-modules, which is shown to include the
well known evaluation modules (of level zero). We show that every irreducible integrable
gˆ[σ]-module in the category E is isomorphic to a finite-dimensional evaluation module.
Definition 3.1. For the twisted affine Lie algebra gˆ[σ], the category E is defined to
consists of gˆ[σ]-modules W for which there exists a nonzero polynomial p(x) ∈ C[x],
depending on W, such that
p(x)a(x)w = 0 for a ∈ g, w ∈ W. (3.1)
Lemma 3.2. The central element k of gˆ[σ] acts as zero on any gˆ[σ]-module in the
category E.
Proof. Let W be a ˆg[σ]-module in the category E with a nonzero polynomial p(x) such
that p(x)a(x) = 0 on W for a ∈ g. If p(x) is a (nonzero) constant, we have a(x) = 0 for
all a ∈ g, i.e., a(n) = 0 for a ∈ g, n ∈ Z. In view of the commutator relation (2.8) we
see that k must be zero on W. Assume that p(x) is of positive degree, so that p′(x) 6= 0.
Pick a,b ∈ g such that ha,bi = 1. (Notice that h·,·i is nondegenerate on g .) Using the
0 0
commutator relations (2.10) we get
∂ x
0 = p(x )p(x )[a(x ),b(x )] = kp(x )p(x )x−1 δ 2 . (3.2)
1 2 1 2 1 2 1 ∂x x
2 (cid:18) 1(cid:19)
6
∂ x x
Res p(x )p(x )x−1 δ 2 = −Res p(x )p′(x )x−1δ 2
x2 1 2 1 ∂x x x2 1 2 1 x
2 (cid:18) 1(cid:19) (cid:18) 1(cid:19)
= −p(x )p′(x ), (3.3)
1 1
we get kp(x )p′(x ) = 0, which implies that k = 0 on W.
1 1
Next, we give some examples of ˆg[σ]-modules in E. Let U be a g-module and let z be
1
a nonzero complex number. With a fixed zN, define an action of gˆ[σ] on U by
N−1 N−1
a(n)·u = zn+Ni (aiu) for a = ai ∈ g, n ∈ Z, (3.4)
i=0 i=0
X X
k·U = 0. (3.5)
Then U equipped with the defined action is a gˆ[σ]-module (of level zero), which is denoted
byU(z). IfU isanirreducible g-module, itisclearthatU(z) isanirreducible gˆ[σ]-module.
Moregenerally,letU ,...,U beg-modulesandletz ,...,z benonzerocomplexnumbers.
1 r 1 r
Then the tensor product ˆg[σ]-module ⊗r U (z ) is called an evaluation module.
k=1 k k
We now show that the evaluation module ⊗r U (z ) is in the category E. For a ∈
k=1 k k
g , u ∈ U (z ) = U , we have
i k k k k
r
a(x)(u1 ⊗···⊗ur) = zkn+Ni x−n−Ni −1(u1 ⊗···⊗auk ⊗···⊗ur)
n∈Z k=1
XX
r z i z
= x−1 k N δ k (u ⊗···⊗au ⊗···⊗u ). (3.6)
1 k r
x x
Xk=1 (cid:16) (cid:17) (cid:16) (cid:17)
Since (x−z )δ zk = 0 for k = 1,...,r, we get (x−z )···(x−z )a(x)(u ⊗···⊗u ) = 0.
k x 1 r 1 r
Thus we have proved:
(cid:0) (cid:1)
Lemma 3.3. Let U ,...,U be g-modules and let z ,...,z be nonzero complex numbers.
1 r 1 r
Then on the tensor product gˆ[σ]-module U (z )⊗···⊗U (z ),
1 1 r r
(x−z )···(x−z )a(x) = 0 for a ∈ g. (3.7)
1 r
In particular, the evaluation gˆ[σ]-module U (z )⊗···⊗U (z ) is in the category E.
1 1 r r
Proposition 3.4. Let gˆ[σ] be a twisted affine Lie algebra. Then for any irreducible g-
modules U ,...,U and for any distinct nonzero complex numbers z ,...,z , the tensor
1 r 1 r
product gˆ[σ]-module U (z )⊗···⊗U (z ) is irreducible.
1 1 r r
7
Proof. Notice that the universal enveloping algebra U(gˆ[σ]) is of countable dimension. It
follows fromLemma 2.3 (andinduction) that U (z )⊗···⊗U (z ) is an irreducible module
1 1 r r
for the product Lie algebra gˆ[σ]⊕···⊕ˆg[σ] (r copies). Denote by π the representation
homomorphism map. For 1 ≤ j ≤ r, denote by ψ the j-th embedding of gˆ[σ] into
j
gˆ[σ]⊕···⊕gˆ[σ] (r copies) and denote by ψ the diagonal map from gˆ[σ] to ˆg[σ]⊕···⊕gˆ[σ]
(r copies). Then ψ = ψ +···+ψ . We also extend the linear maps ψ and ψ ,...,ψ on
1 r 1 r
gˆ[σ][[x,x−1]] canonically.
For 1 ≤ j ≤ r, set p (x) = (x−z )/(z −z ). Then
j k6=j k j k
Qz z z
k k k
p (x)δ = p (z )δ = δ δ (3.8)
j j k j,k
x x x
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
for j,k = 1,...,r. Using (3.6) We have that on U (z )⊗···⊗U (z ),
1 1 r r
p (x)πψ (a(x)) = δ πψ (a(x)) for 1 ≤ j,k ≤ r, a ∈ g. (3.9)
j k j,k k
Thus on U (z )⊗···⊗U (z ),
1 1 r r
p (x)πψ(a(x)) = πψ (a(x)) for 1 ≤ j ≤ r, a ∈ g, (3.10)
j j
which implies that
πψ (gˆ[σ]) ⊂ πψ(gˆ[σ]) for j = 1,...,r. (3.11)
j
From this we have
πψ(gˆ[σ]) = πψ (gˆ[σ])+···+πψ (gˆ[σ]). (3.12)
1 r
It follows that U (z )⊗···⊗U (z ) is an irreducible gˆ[σ]-module.
1 1 r r
Proposition 3.5. Let U ,...,U , V ,...,V be nontrivial irreducible g-modules and let
1 r 1 s
z ,...,z and ξ ,...,ξ be two groups of distinct nonzero complex numbers. Then the
1 r 1 s
gˆ[σ]-module U (z )⊗···⊗U (z ) is isomorphic to V (ξ )⊗···⊗V (ξ ) if and only if r = s,
1 1 r r 1 1 s s
∼
z = ξ and U = V up to a permutation.
j j j j
Proof. We only need to prove the “only if” part. Let U be any gˆ[σ]-module in category
E. There exists a (unique nonzero) monic polynomial p(x) of least degree such that
p(x)a(x)U = 0 for a ∈ g. Clearly, isomorphic ˆg[σ]-modules in category E have the same
monic polynomial. If U = U (z ) ⊗ ··· ⊗ U (z ), we are going to show that p(x) =
1 1 r r
(x−z )···(x−z ) is the associated monic polynomial. First, by Lemma 3.3 we have that
1 r
p(x)a(x) = 0 on U (z ) ⊗ ···⊗ U (z ) for a ∈ g. Let q(x) be any polynomial such that
1 1 r r
q(x)a(x) = 0 on U (z )⊗···⊗U (z ) for a ∈ g. Set p (x) = (x−z )/(z −z ) for
1 1 r r j k6=j k j k
Q
8
j = 1,...,r as in the proof of Proposition 3.4. For a ∈ g , u ∈ U with j = 1,...,r, we
i j j
have
0 = q(x)p (x)a(x)(u ⊗···⊗u )
j 1 r
z i z
= q(x)x−1 j N δ j (u ⊗···⊗au ⊗···⊗u )
1 j r
x x
(cid:16) z(cid:17) i (cid:16) z(cid:17)
= q(z )x−1 j N δ j (u ⊗···⊗au ⊗···⊗u ).
j 1 j r
x x
(cid:16) (cid:17) (cid:16) (cid:17)
Since each U is a nontrivial g-module, we must have q(z ) = 0 for j = 1,...,r. Thus
j j
p(x) divides q(x). This proves that p(x) is the associated monic polynomial.
Assume that U (z ) ⊗ ··· ⊗ U (z ) is isomorphic to V (ξ ) ⊗ ··· ⊗ V (ξ ) with Φ a
1 1 r r 1 1 s s
gˆ[σ]-module isomorphism map. Then the two tensor product modules must have the
same associated monic polynomial. That is, (x− z )···(x− z ) = (x− ξ )···(x −ξ ).
1 r 1 s
Thus r = s and up to a permutation z = ξ for j = 1,...,r. Assume that z = ξ for
j j j j
j = 1,...,r. For 1 ≤ j ≤ r, a ∈ g and for u ∈ U , v ∈ V with k = 1,...,r, we have
i k k k k
z i z
p (x)a(x)(u ⊗···⊗u ) = x−1 j N δ j (u ⊗···⊗au ⊗···⊗u ),(3.13)
j 1 r 1 j r
x x
(cid:16)z (cid:17)i (cid:16)z (cid:17)
p (x)a(x)(v ⊗···⊗v ) = x−1 j N δ j (v ⊗···⊗av ⊗···⊗v ). (3.14)
j 1 r 1 j r
x x
(cid:16) (cid:17) (cid:16) (cid:17)
Then
z
Φ(u ⊗···⊗au ⊗···⊗u ) = Res x−1δ j Φ(u ⊗···⊗au ⊗···⊗u )
1 j r x 1 j r
x
z (cid:16)− i (cid:17)
j N
= Res Φ(p (x)a(x)(u ⊗···⊗u ))
x j 1 r
x
(cid:16)z (cid:17)− i
j N
= Res p (x)a(x)Φ(u ⊗···⊗u )
x j 1 r
x
= σ (a)Φ(cid:16)(u(cid:17)⊗···⊗u ), (3.15)
j 1 r
where for a ∈ g , v ∈ V ,...,v ∈ V ,
i 1 1 r r
σ (a)(v ⊗···⊗v ) = (v ⊗···⊗av ⊗···⊗v ).
j 1 r 1 j r
It follows that U is isomorphic to V . For example, consider j = 1. Pick up nonzero
j j
vectors u ∈ U for 2 ≤ k ≤ r. Using these vectors we get an embedding
k k
Θ : U → U ⊗···⊗U ; u 7→ u⊗u ⊗···⊗u .
1 1 r 2 r
On the other hand, for any linear functions f ∈ V∗ for k = 2,...,r, we have a linear
k k
map
Ψ : V ⊗···⊗V → V ; v ⊗v ⊗···⊗v 7→ f (v )···f (v )v .
f2,...,fr 1 r 1 1 2 r 2 2 r r 1
9
Since 0 6= ΦΘ(U ) ⊂ V ⊗···⊗V , there exist linear functions f ∈ V∗ for k = 2,...,r
1 1 r k k
such that
Ψ ΦΘ(U ) 6= 0.
f2,...,fr 1
By (3.15), Ψ ΦΘ is a nonzero g-homomorphism from U to V . It is an isomorphism
f2,...,fr 1 1
because both U and V are irreducible.
1 1
We next classify finite-dimensional irreducible gˆ[σ]-modules in category E. For a ∈ g ,
i i
we have
i
ai(x) = (ai ⊗tn+Ni )x−n−Ni −1 = ai ⊗x−1δ t t N . (3.16)
x x
n∈Z (cid:18) (cid:19)(cid:18) (cid:19)
X
For f(x) ∈ C[x], m ∈ Z, a ∈ g , we have
i i
i
xm+Ni f(x)ai(x) = ai ⊗xm+Ni f(x)x−1δ t t N = ai ⊗tm+Ni f(t)x−1δ t , (3.17)
x x x
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
so that
Resxxm+Ni f(x)ai(x) = ai ⊗tm+Ni f(t). (3.18)
It follows immediately that for any gˆ[σ]-module W, f(x)a (x)W = 0 if and only if (a ⊗
i i
tNi f(t)C[t,t−1])W = 0.
For a nonzero polynomial p(x), we define a subcategory E of E, consisting of ˆg[σ]-
p(x)
modules W such that
p(x)a(x)w = 0 for a ∈ g, w ∈ W. (3.19)
NoticingLemma3.2,thenagˆ[σ]-moduleinthecategoryE exactlyamountstoamodule
p(x)
for the Lie algebra iN=−01gi ⊗tNi C[t,t−1]/ iN=−01gi ⊗tNi p(t)C[t,t−1].
Lemma 3.6. Let pP(x) = (x − z )···(x −Pz ) with z ,...,z distinct nonzero complex
1 r 1 r
numbers and with k ∈ N. Then any finite-dimensional irreducible ˆg[σ]-module W in
the category E is isomorphic to a gˆ[σ]-module U (z ) ⊗ ··· ⊗ U (z ) for some finite-
p(x) 1 1 r r
dimensional irreducible g-modules U ,...,U .
1 r
Proof. We have
N−1 N−1
gi ⊗tNi C[t,t−1]/ gi ⊗tNi p(t)C[t,t−1]
i=0 i=0
X X
r N−1 N−1
= gi ⊗tNi C[t,t−1]/ gi ⊗tNi (t−zl)C[t,t−1] .
!
l=1 i=0 i=0
M X X
10