Quantum Affine Algebras at Roots of Unity

arXiv:q-alg/9609031v2 22 Apr 1997 Quantum Affine Algebras at Roots of Unity Vyjayanthi Chari1 and Andrew Pressley2 ABSTRACT Let Uq (ĝ) be the quantized universal enveloping algebra of the affine Lie algebra ĝ associated to a finite-dimensional complex simple Lie algebra g, and let Uqres (ĝ) be the C[q, q −1 ]-subalgebra of Uq (ĝ) generated by the q-divided powers of the Chevalley generators. Let Uǫres (ĝ) be the Hopf algebra obtained from Uqres (ĝ) by specialising q to a non-zero complex number ǫ of odd order. We classify the finite-dimensional irreducible representations of Uǫres (ĝ) in terms of highest weights. We also give a ‘factorisation’ theorem for such representations: namely, any finite-dimensional irreducible representation of Uǫres (ĝ) is isomorphic to a tensor product of two representations, one factor being the pull-back of a representation of ĝ by Lusztig’s Frobenius homomorphism F̂rǫ : Uǫres (ĝ) → U(ĝ), the other factor being an irreducible representation of the Frobenius kernel. Finally, we give a conc! ! ! rete construction of all of the finite-dimensional irreducible ˆ 2 ). The proofs make use of several interesting new identities in Uq (ĝ). representations of Uǫres (sl 1 Partially supported by NATO and the ESPRC (GR/K65812) 2 0. Introduction Let Uq (g) be the Drinfel’d–Jimbo quantum group associated to a symmetrizable Kac–Moody algebra g. Thus, Uq (g) is a Hopf algebra over the field C(q) of rational functions of an indeterminate q, and is defined by certain generators and relations (which are written down in Proposition 1.1 below for the cases of interest in this paper). Roughly speaking, one thinks of Uq (g) as a family of Hopf algebras depending on a ‘parameter’ q. To make this precise, one constructs a ‘specialisation’ Uǫ (g) of Uq (g), for a non-zero complex number ǫ, as follows. If ǫ ∈ C is transcendental, define Uǫ (g) = Uq (g)⊗C(q) C, via the algebra homomorphism C(q) → C that takes q to ǫ. If, on the other hand, ǫ is algebraic, the latter homomorphism does not exist, and one proceeds by first constructing a C[q, q −1 ]-form of Uq (g), i.e. a !!!C[q, q −1 ](Hopf) subalgebra Ũq (g) of Uq (g) such that Uq (g) = Ũq (g) ⊗C[q,q −1 ] C(q). Then one defines Uǫ (g) = Ũq (g) ⊗C[q,q −1 ] C, via the algebra homomorphism C[q, q −1 ] → C that takes q to ǫ. Two such C[q, q −1 ]-forms have been studied in the literature. They lead to the same algebra Uǫ (g) when ǫ is not a root of unity, but different algebras, with very different representation theories, when ǫ is a root of unity. In the ‘non-restricted’ form, one takes Ũq (g) to be the C[q, q −1 ]-subalgebra of Uq (g) generated by the Chevalley generators ei , fi of Uq (g). The finite-dimensional representations of the non-restricted specialisation Uǫ (g) have been studied by De Concini, Kac and Procesi [10], [11], when g is finite-dimensional, and by Beck and Kac [2], when g is (untwisted) affine. In the ‘restricted’ form, one takes Ũq (g) to be the C[q, q −1 ]-subalgebra of Uq (g) generated by the divided powers eri /[r]q ! and fir /[r]q !, for all r > 0, where [r]q ! denotes a q-factorial. When g is finite-dimensional and ǫ is a root of unity, the structure and representation theory of the restricted specialisation Uǫres (g) was worked out by Lusztig (see [5], [15] and the references there). It is the purpose of the present work to study the finite-dimensional representations of Uǫres (g) when ǫ is a root of unity and g is (untwisted) affine. Part of our work may be regarded as the quantum analogue of Garland’s paper [13]. A crucial role is played in [13] by a certain identity (Lemma 7.5) that is needed to prove a suitable triangular decomposition of the restricted form of U (ĝ) (analogous to Uqres (ĝ)). The proof of the analogue of Garland’s lemma in the quantum case (Lemma 5.1 below) is, however, more difficult than in the classical situation because the generators of the ‘positive part’ of Uǫres (ĝ) do not commute, whereas their classical analogues do commute. Moreover, Garland makes use of a natural derivation of U (ĝ) which turns out to have no straightforward analogue in the quantum situation. Lemma 5.1 is one of several interesting new identities in Uqres (ĝ) that we obtain below. One of these (equation (19) below) shows an unexˆ 2 ) and Young diagrams. pected (and as yet unexplained) connection between Uqres (sl (This relation is invisible i! ! ! n the classical situation considered by Garland.) Once the triangular decomposition of Uǫres (ĝ) is available, we are able to give, in Theorem 8.2, an abstract highest weight description of its finite-dimensional irreducible representations. It turns out that there is a natural one-to-one correspondence between the finite-dimensional irreducible representations of Uǫres (ĝ) when ǫ is a root of unity, and those of Uǫ (ĝ) when ǫ is not a root of unity, although 3 tations of Uǫ (ĝ) when ǫ is not a root of unity were classified in [4], [5] and [6]). This is exactly parallel to the situation for Uǫres (g) when dim(g) < ∞, where the finite-dimensional irreducible representations are parametrised by dominant integral weights whether or not ǫ is a root of unity (see [5], Chapter 11, and [15]). One of the most important results about the finite-dimensional irreducible representations V of Uǫres (g) when dim(g) < ∞ and ǫ is a root of unity of order ℓ asserts that V factorises into the tensor product of a representation whose highest weight is divisible by ℓ and one whose highest weight is ‘less than ℓ’, in the sense that its value on every simple coroot of g is less than ℓ (see [5] and [15]). In Section 9, we prove an analogue of this result for Uǫres (ĝ) (Theorem 9.1). It is well known that there is a close relationship between finite-dimensional representations of quantum affine algebras and affine Toda theories (see [7] and the references there). The value of ǫ is determined by the ‘coupling constant’ of the associated theory. Since the representation theory of Uǫres (ĝ) depends crucially on whether ǫ is a root of unity or not, one would expect that the structure of affine Toda theories will be different at certain special values of the coupling constant. This does indeed appear to be the case (T. J. Hollowood, private communication), but we shall leave further discussion of this point to another place. 1. Preliminaries In this section and the next, we recall certain facts about g and ĝ and their associated quantum groups that will be needed later. See [1], [5] and [15] for further details. Let (aij )i,j∈I be the Cartan matrix of the finite-dimensional complex simple Lie ` algebra g, let Iˆ = I {0}, and let (aij )i,j∈Iˆ be the generalised Cartan matrix of the untwisted affine Lie algebra ĝ of g. Let (di )i∈Iˆ be the coprime positive integers such that the matrix (di aij )i,j∈Iˆ is symmetric. P Let P̌ be the lattice over Z with basis (λ̌i )i∈I , let α̌j = i∈I aji λ̌i (j ∈ I), and P let Q̌ = i∈I Zα̌i ⊆ P̌ . The root lattice Q = HomZ (P̌ , Z) has basis the simple roots (αi )i∈I of g, where αi (λ̌j ) = δij . Similarly, the weight lattice P = HomZ (Q̌, Z) has basis the fundamental weights (λi )i∈I of g, where λi (α̌j ) = δij . Let P + = P {λ ∈ P | λ(α̌i ) ≥ 0 for all i ∈ I}, and let Q+ = i∈I N.αi . For i ∈ I, define si : P̌ → P̌ by si (x) = x − αi (x)α̌i , and let W be the group of automorphisms of P̌ generated by the si . Then, W acts on Q by si (ξ) = ξ −ξ(α̌i )αi , for S ξ ∈ Q. TheSroot and coroot systems of g are given, respectively, by R = i∈I W αi , Ř = i∈I W α̌i . There is a partial order on R such that α ≤ β if and only if β − α is a linear combination of the αi (i ∈ I) with non-negative integer coefficients. The correspondence αi ↔ α̌i extends to a W -invariant correspondence R ↔ Ř, written α ↔ α̌, such that α(α̌) = 2 for all α ∈ R. Define sα : P̌ → P̌ by sα (x) = x − α(x)α̌, for all α ∈ R. ˜ P̌ be the semi-direct product group defined using the W action on Let Ŵ = W × P̌ . For s ∈ W , write s for (s, 0) ∈ Ŵ , and for x ∈ P̌ write x for (1, x) ∈ Ŵ , where 1 is the identity element of W . Let θ be the highest root of g with respect to ≤, and write s0 for (sθ , θ̌) ∈ Ŵ . Let W̃ be the (normal) subgroup of Ŵ generated by the 4 of the group of diagram automorphisms of ĝ, i.e. the bijections τ : Iˆ → Iˆ such ˆ Moreover, there is an isomorphism of groups that aτ (i)τ (j) = aij for all i, j ∈ I. ˜ W̃ , where the semi-direct product is defined using the action of T in W̃ Ŵ ∼ =T× given by τ.si = sτ (i) (see [1]). If w ∈ Ŵ , a reduced expression for w is an expression ! ! ! w = τ si1 si2 . . . sin with τ ∈ T , i1 , i2 , . . . , in ∈ Iˆ and n minimal. The universal enveloping algebra U (g) (resp. U (ĝ)) is the associative algebra ˆ over C with generators e± i , hi for i ∈ I (resp. i ∈ I) and defining relations ± ± hi e± j − ej hi = ±aij ej , hi hj = hj hi , − − + e+ i ej − ej ei = δij hi ,   1−aij X 1 − aij r r ± ± 1−aij −r (e± (−1) =0 i ) ej (ei ) r r=0 if i 6= j, ˆ where i, j ∈ I (resp. i, j ∈ I). Let q be an indeterminate, let C(q) be the field of rational functions of q with complex coefficients, and let C[q, q −1 ] be the ring of Laurent polynomials. For r, n ∈ N, n ≥ r, define q n − q −n , q − q −1 [n]q ! = [n]q [n − 1]q . . . [2]q [1]q , hni [n]q ! = . r q [r]q ![n − r]q ! [n]q = Then n r q ∈ C[q, q −1 ] for all n ≥ r ≥ 0. One has the identity n hni X =0 (−1)r q r(n−1) r q r=0 (1) ˆ for all n ≥ 0. Let qi = q di for i ∈ I. Proposition 1.1. There is a Hopf algebra Uq (ĝ) over C(q) which is generated as ±1 ˆ with the following defining relations: (i ∈ I), an algebra by elements e± i , ki ki ki−1 = ki−1 ki = 1, ki k j = kj k i , ±aij ± ej , −1 ki e± = qi j ki − [e+ i , ej ] 1−aij X (−1) r=0 r  1 − aij r  ki − ki−1 , = δij qi − qi−1 r ± ± 1−aij −r (e± =0 i ) ej (ei ) qi The comultiplication of Uq (ĝ) is given on generators by + + + − − −1 − if i 6= j. 5 ˆ for i ∈ I. Restricting the indices i, j to I, one obtains a Hopf algebra Uq (g). ± 0 Define Uq (g) (resp. Uq (g) ) to be the C(q)-subalgebra of Uq (g) generated by ±1 the e± i (resp. by the ki ) for all i ∈ I. There is a natural injective homomorphism of C(q)-Hopf algebras Uq (g) → Uq (ĝ) ± that takes e± i to ei and ki to ki , for all i ∈ I. In particular, any representation of Uq (ĝ) can be regarded as a representation of Uq (g). It is convenient to use the following notation: (r) (e± i ) (e± )r = i . [r]qi ! ˆ be the C(q)-algebra automorphisms of Uq (ĝ) defined by Lusztig Let Ti (i ∈ I) [15] (our notational conventions differ slightly from his): −aij − − −1 + Ti (e+ i ) = −ei ki , Ti (ei ) = −ki ei , Ti (kj ) = ki kj , −aij Ti (e+ j ) = X (−aij −r) + + (r) if i 6= j, ej (ei ) (−1)r−aij qi−r (e+ i ) X (r) − − (−aij −r) (−1)r−aij qir (e− ej (ei ) if i 6= j. i ) r=0 −aij Ti (e− j ) = r=0 If i ∈ I, Ti induces an automorphism of the subalgebra Uq (g) of Uq (ĝ), also denoted by Ti . ˆ given by the above There is, of course, a classical analogue T i of Ti for all i ∈ I, formulas with ei replaced by ei and ki replaced by 1, and with T i (hj ) = hj − aij hi . The T i are Hopf algebra automorphisms of U (ĝ). The finite group T acts as C(q)-Hopf algebra automorphisms of Uq (ĝ) by ± τ.e± i = eτ (i) , τ.ki = kτ (i) , ˆ for all i ∈ I. Similarly, T acts as Hopf algebra automorphisms of U (ĝ) by ± τ.e± i = eτ (i) , τ.hi = hτ (i) , ˆ for all i ∈ I. If w ∈ W has a reduced expression w = τ si1 . . . sin , let Tw be the C(q)-algebra automorphism of Uq (ĝ) given by Tw = τ Ti1 . . . Tin . Then, Tw depends only on w, and not on the choice of its reduced expression [1]. In particular, for any i ∈ I, we have a well defined C(q)-algebra automorphism Tλ̌i of Uq (ĝ). Similarly, one has Hopf algebra automorphisms T λ̌i of U (ĝ). Let Uqres (g) (resp. Uqres (ĝ)) be the C[q, q −1 ]-subalgebra of Uq (g) (resp. Uq (ĝ)) (r) ˆ r ∈ N). for all i ∈ I, r ∈ N (resp. for all i ∈ I, generated by the ki±1 and the (e± i ) res res −1 Then, Uq (g) (resp. Uq (ĝ)) is a C[q, q ]-Hopf subalgebra of Uq (g) (resp. Uq (ĝ)). 6 ˆ In fact [15], if i, j ∈ I, ˆ n ∈ N, (resp. for all i ∈ I). −n(n−1) (n) Ti ((e+ ) = (−1)n qi i ) (n) n (n) (n) (e− ki , Ti ((e− ) = (−1)n q n(n−1) ki−n (e+ , i ) i ) i ) −naij (n) ) Ti ((e+ j ) = X (−naij −r) + (n) + (r) if i 6= j, (ej ) (ei ) (−1)r−naij qi−r (e+ i ) X (r) − (n) − (−naij −r) (−1)r−naij qir (e− (ej ) (ei ) if i 6= j. i ) r=0 −naij (n) Ti ((e− )= j ) r=0 The action of T obviously preserves Uqres (ĝ). For r ∈ N, n ∈ Z, i ∈ I, define   Y r ki qin−s+1 − ki−1 qi−n+s−1 ki ; n . = s − q −s r q i i s=1 The importance of these elements stems from the identity min(r,s) (2) (r) − (s) (ei ) (e+ i ) = X (s−t) (e− i ) t=0   ki ; 2t − r − s (r−t) , (e+ i ) t h ki ;n r i for all r, s ∈ N, i ∈ I. It follows from this identity that ∈ Uqres (g) for all r ∈ N, n ∈ Z, i ∈ I. (r) for all i ∈ I, Let Uqres (g)± be the subalgebra of Uqres (g) generated by the (e± i ) h i ki ;n ±1 res 0 r ∈ N, and let Uq (g) be the subalgebra generated by the ki and the , r for all r ∈ N, n ∈ Z, i ∈ I. Multiplication defines an isomorphism of C(q)-vector spaces Uqres (g)− ⊗ Uqres (g)0 ⊗ Uqres (g)+ → Uqres (g). The appropriate definitions of Uqres (ĝ)± and Uqres (ĝ)0 will be given at the end of Section 3. Let ℓ be an odd integer with ℓ ≥ 3, and assume that ℓ is not divisible by 3 if g is of type G2 . Let ǫ ∈ C be a primitive ℓth root of unity, and set Uǫres (g) = Uqres (g) ⊗C[q,q −1 ] C, Uǫres (ĝ) = Uqres (ĝ) ⊗C[q,q −1 ] C, via the algebra homomorphism C[q, q −1 ] → C that takes q to ǫ. If x is any element of Uqres (g) (resp. Uqres (ĝ)), we denote the corresponding element of Uǫres (g) (resp. Uǫres (ĝ)) also by x. res 0 Define Uǫres (g)± and Uǫresh(g)0 in i the obvious way. It is known that Uǫ (g) is generated by the ki±1 and the of C-vector spaces ki ;0 ℓ , for i ∈ I. Multiplication defines an isomorphism Uǫres (g)− ⊗ Uǫres (g)0 ⊗ Uǫres (g)+ → Uǫres (g). ± 7 We shall make use of the characteristic zero Frobenius homomorphisms defined by Lusztig (see [15], Chapter 35). Namely, there exist homomorphisms of Hopf algebras Frǫ : Uǫres (g) → U (g), F̂rǫ : Uǫres (ĝ) → U (ĝ) such that, for i ∈ I, Frǫ (ki ) = 1, (r) Frǫ ((e± ) i ) = ( = ( r/ℓ (e± i ) (r/ℓ)! if ℓ divides r, 0 otherwise. r/ℓ (e± i ) (r/ℓ)! if ℓ divides r, 0 otherwise. ˆ and for i ∈ I, F̂rǫ (ki ) = 1, (r) F̂rǫ ((e± ) i ) To discuss finite-dimensional representations of Uqres (ĝ), or of its specialization Uǫres (ĝ), it is advantageous to use another realization of Uq (ĝ), due to Drinfel’d [12], Beck [1] and Jing [14]. Theorem 1.2. There is an isomorphism of C(q)-Hopf algebras from Uq (ĝ) to the ±1 (i ∈ I), hi,r (i ∈ I, r ∈ Z\{0}) and algebra with generators x± i,r (i ∈ I, r ∈ Z), ki ±1/2 c , and the following defining relations: c±1/2 are central, ki ki−1 = ki−1 ki = 1, c1/2 c−1/2 = c−1/2 c1/2 = 1, ki kj = kj ki , ki hj,r = hj,r ki , ±aij ± xj,r , ki xj,r ki−1 = qi 1 cr − c−r [hi,r , hj,s ] = δr,−s [raij ]qi , r qj − qj−1 1 ∓|r|/2 ± [hi,r , x± xj,r+s , j,s ] = ± [raij ]qi c r ±a ±aij ± ± ± ± ± ± xj,s xi,r+1 = qi ij x± x± i,r xj,s+1 − xj,s+1 xi,r , i,r+1 xj,s − qi (3) − [x+ i,r , xj,s ] m X X π∈Σm k=0 (−1)k hmi k qi = δij − + − c−(r−s)/2 ψi,r+s c(r−s)/2 ψi,r+s qi − qi−1 , ± ± ± ± x± i,rπ(1) . . . xi,rπ(k) xj,s xi,rπ(k+1) . . . xi,rπ(m) = 0, if i 6= j, for all sequences of integers r1 , . . . , rm , where m = 1 − aij , Σm is the symmetric ± group on m letters, and the ψi,r are determined by equating powers of u in the formal power series ! ∞ ∞ X X ± hi,±s u±s . u±r = ki±1 exp ±(qi − qi−1 ) ψi,±r s=1 r=0 The isomorphism is given by ± r ∓r ± 8 where o : I → {±1} is a map such that o(i) = −o(j) whenever aij < 0 (it is clear that there are exactly two possible choices for o). The classical analogues of the generators x± i,r and hi,r are given by ∓r ± r x± i,r = o(i) T λ̌i (ei ), − hi,r = [x+ i,r , xi,0 ]. The latter elements, together with a central element c, generate U (ĝ) and the h defining relations are obtained from those in 1.2 by interpreting c as q c , ki as qi i,0 , + − ψi,r (resp. ψi,r ) as (qi − qi−1 )hi,r if r > 0 (resp. if r < 0), and letting q tend to one. For each i ∈ I, there is an injective homomorphism of C(q)-algebras ϕi : ˆ 2 ) → Uq (ĝ) that takes x± to x± and ψ ± to ψ ± , for all r ∈ Z (as usual, Uqi (sl r r i,r i,r ˆ 2 )) (see we drop the (unique) Dynkin diagram subscript when dealing with Uq (sl res ˆ [1], Proposition 3.8). It follows immediately from 1.2 that Uqi (sl2 ) is gener(r) ated as a C[q, q −1 ]-algebra by c±1/2 , k ±1 and the (x± for n ∈ Z, r ∈ N (for n) − + 1 −1 res e+ = x and e = c kx ). Since the T preserve U (ĝ), it is also clear that 0 q 0 1 −1 λ̌i ± (r) res ˆ 2 )) ⊆ U res (ĝ), (xi,n ) ∈ Uq (ĝ) for all i ∈ I, n ∈ Z, r ∈ N. It follows that ϕi (Uqres (sl q i res ˆ and hence that ϕi induces a C-algebra homomorphism ϕi : Uǫi (sl2 ) → Uǫres (ĝ). ˆ 2 )) pointwise if i, j ∈ I, i 6= j ([1], Corollary 3.2). We have that Tλ̌j fixes ϕi (Uqi (sl ˆ 2 ) → U (ĝ). We denote the obvious classical analogue of ϕi by ϕi : U (sl Finally, we shall need the following automorphisms of Uq (ĝ): Proposition 1.3. (a) There exists a C(q)-algebra anti-automorphism Φ of Uq (ĝ) such that, for all i ∈ I, r ∈ Z, ± ± ∓ 1/2 , Φ(hi,r ) = hi,r . ) = ψi,r ) = c−1/2 , Φ(ψi,r Φ(x± i,r ) = xi,r , Φ(c (b) There exists a C-algebra and coalgebra anti-automorphism Ω of Uq (ĝ) such that Ω(q) = q −1 and, for all i ∈ I, r ∈ Z, ∓ ± ∓ 1/2 Ω(x± ) = c−1/2 . i,r ) = xi,−r , Ω(ψi,r ) = ψi,−r , Ω(hi,r ) = hi,−r , Ω(c Moreover, Ω and Φ each preserve Uqres (ĝ), commute with T , and we have, for all i ∈ I, ΩTi = Ti Ω, ΦTi = Ti−1 Φ, ΦΩ = ΩΦ. Proof. The existence of Φ and Ω can be verified directly by using 1.2. It is obvious that Φ and Ω commute with T . The relations between Ω, Φ and the Ti are checked ±1 ˆ of Uq (ĝ), on which they by verifying agreement on the generators e± (j ∈ I) j , kj are given by ∓ Φ(e± j ) = ej , Φ(kj ) = kj , ∓ Ω(e± j ) = ej , Ω(kj ) = kj−1 . These formulas obviously imply that Φ and Ω preserve Uqres (ĝ). The fact that Ω ±1 is a coalgebra anti-automorphism can also be checked on the generators e± j , kj 9 2. Representation Theory of Uq A representation V of Uq (g) is said to be of type I if, for all i ∈ I, ki acts semisimply on V with eigenvalues in qiZ . Any finite-dimensional irreducible representation of Uq (g) can be obtained from a type I representation by tensoring with a onedimensional representation that takes each e± i to zero and each ki to ±1. If λ ∈ P , the weight space Vλ of a representation V of Uq (g) is defined by  Vλ = v ∈ V ki .v = qini v , where ni = λ(α̌i ). We have e± i .Vλ ⊆ Vλ±αi . A vector v in a type I representation V of Uq (g) is said to be a highest weight vector if there exists λ ∈ P such that v ∈ Vλ and e+ i .v = 0 for all r ∈ N, i ∈ I. If, in addition, V = Uq (g).v, then V is said to be a highest weight representation with highest weight λ. For any λ ∈ P , there exists, up to isomorphism, a unique irreducible representation V (λ) of Uq (g) with highest weight λ. We have V (λ) = M V (λ)µ . µ≤λ Every finite-dimensional irreducible type I representation of Uq (g) is isomorphic to V (λ) for some (unique) λ ∈ P + . The character of V (λ), and in particular its dimension, is the same as that of the irreducible representation V (λ) of g with the same highest weight λ. Turning now to Uq (ĝ), we say that a representation V of Uq (ĝ) is type I if it is type I as a representation of Uq (g) and c1/2 acts as one on V . A vector v in a type I representation V of Uq (ĝ) is said to be a highest weight vector if v is ± annihilated by x+ i,r , and is an eigenvector of ψi,r , for all i ∈ I, r ∈ Z. If, in addition, V = Uq (ĝ).v, then V is said to be a highest weight representation of Uq (ĝ). One shows by the usual Verma module arguments that, for any set of scalars Ψ± i,r such + − that Ψi,0 Ψi,0 = 1, there is an irreducible highest weight representation V of Uq (ĝ) ± with highest weight vector v such that ψi,r .v = Ψ± i,r v for all i ∈ I, r ∈ Z (compare the proof of 7.3 below). Moreover, V is uniquely determined by the scalars Ψ± i,r , up to isomorphism. Theorem 2.1. The irreducible highest weight representation of Uq (ĝ) determined by the scalars Ψ± i,r is finite-dimensional if and only if there exist polynomials Pi ∈ C(q)[u], for i ∈ I, such that Pi (0) 6= 0 and ∞ X r Ψ+ i,r u = −2 deg(Pi ) Pi (qi u) qi = ∞ X −r , Ψ− i,−r u 10 in the sense that the left- and right-hand sides are the Laurent expansions of the middle term about u = 0 and u = ∞, respectively. This is proved in [4], [5], [6] and [9]. From now on, we denote by Πq the set of polynomials P ∈ C(q)[u] such that P (0) 6= 0, and by ΠIq the set of I-tuples of such polynomials. If P = (Pi )i∈I ∈ ΠIq , we denote by V (P) the finite-dimensional irreducible representation of Uq (ĝ) determined by P as in 2.1. If P = (Pi )i∈I , Q = (Qi )i∈I ∈ ΠIq , define P ⊗ Q = (Pi Qi )i∈I . The following result is proved in [9] (see 7.4 for the proof of an analogous result): Proposition 2.2. Let P, Q ∈ ΠIq . If V (P)⊗V (Q) is irreducible as a representation of Uq (ĝ), then it is isomorphic to V (P ⊗ Q). We shall also need the classical analogue of 2.1. Let Π the set of polynomials P ∈ C[u] such that P (0) 6= 0, and let ΠI be the set of I-tuples of such polynomials. Then, the classical analogue of 2.1 states that every finite-dimensional irreducible representation V of U (ĝ) is generated by a vector v that is annihilated by c and the x± i,r , and such that hi,r .v = Hi,r v, where the scalars Hi,r ∈ C satisfy ∞ X ∞ Pi′ (u) X P ′ (u) Hi,r u = deg(Pi ) − u , Hi,−r u−r = u i , Pi (u) r=0 Pi (u) r=0 r for some P = (Pi )i∈I ∈ ΠI . We write V as V (P). The obvious classical analogue of 2.2 holds. Finite-dimensional irreducible representations of Uq (ĝ) can be constructed explicitly when g = sl2 , by means of the following result. We take Iˆ = {0, 1} when g = sl2 . Proposition 2.3. For any non-zero a ∈ C(q), there is an algebra homomorphism ˆ 2 ) → Uq (sl2 ) such that eva : Uq (sl (4) ±1 ±1 ∓ ± eva (e± a e , eva (k0 ) = k −1 , eva (e± 0 )=q 1 ) = e , eva (k1 ) = k. Moreover, we have, for all r ∈ Z, (5) −r r r + −r r − r eva (x+ a k e , eva (x− a e k . r ) =q r )=q See [4], Proposition 5.1, for the proof. If V is a representation of Uq (sl2 ), we ˆ 2 ) obtained by pulling back V denote by Va the ‘evaluation representation’ of Uq (sl by eva . Let V (n) be the (n + 1)-dimensional irreducible type I representation of Uq (sl2 ). Proposition 2.4. Let r, n1 , . . . , nr ∈ N, and let a1 , . . . , ar ∈ C(q) be non-zero. Then, the tensor product V (n1 )a1 ⊗ · · · ⊗ V (nr )ar ˆ 2 ) if and only if, for all 1 ≤ s 6= t ≤ r, is irreducible as a representation of Uq (sl ±(n +n −2p) 11 This is proved by exactly the same argument as Theorem 5.8 in [4]. It is easy to compute that V (n)a ∼ = V (Pn,a ), where (6) Pn,a (u) = n Y (1 − aq n−2m+1 u) m=1 (see [4], Corollary 4.2). It follows from 2.2 that, when the conditions in 2.4 are satisfied, V (n1 )a1 ⊗ · · · ⊗ V (nr )ar ∼ = V (P ), Qr where P = s=1 Pns ,as . The classical analogues of the homomorphisms eva exist for all g. They are the homomorphisms eva : U(ĝ) → U (g), defined for any non-zero a ∈ C, such that ± ±1 ∓ eva (e± eθ , eva (e± 0) =a i ) = ei , eva (h0 ) = −θ̌, ev a (hi ) = hi , for i ∈ I. Here, e± θ are root vectors corresponding to ±θ, normalised so that + − [eθ , eθ ] = θ̌. Evaluation representations V a of U (ĝ) are defined in the obvious way. Using this construction, one can describe explicitly the representation V (P) for all P ∈ ΠI . −1 −1 Theorem 2.5. Let P = (Pi )i∈I ∈ ΠI , and let {a−1 1 , a2 , . . . , ar } be the union of the set of roots of Pi for all i ∈ I. Write Pi (u) = r Y (1 − as u)ni,s s=1 for some integers ni,s ≥ 0, and let µs = P i∈I ni,s λi ∈ P + . Then, V (P) ∼ = V (µ1 )a1 ⊗ V (µ2 )a2 ⊗ · · · ⊗ V (µr )ar . Conversely, if µ1 , µ2 , . . . , µr ∈ P + and a1 , a2 , . . . , ar ∈ C are non-zero, the tensor product V (µ1 )a1 ⊗ V (µ2 )a2 ⊗ · · · ⊗ V (µr )ar is irreducible as a representation of ĝ if and only if a1 , a2 , . . . , ar are distinct. Proof. By using the methods in [3], one shows that every finite-dimensional irreducible representation of U (ĝ) is isomorphic to a tensor product V (µ1 )a1 ⊗ V (µ2 )a2 ⊗ · · · ⊗ V (µr )ar for some µ1 , µ2 , . . . , µr ∈ P + and non-zero a1 , a2 , . . . , ar ∈ C, and that the condition for irreducibility of such a tensor product is as stated in 2.5. By the classical analogue of 2.2, it therefore suffices to prove that the I-tuple of polynomials (Pi )i∈I associated to V (µs )as is given by Pi (u) = (1 − as u)ni,s . This is a straightforward computation.  3. Some Identities ˆ 2 ) that will be needed later. In this section we establish certain identities in Uq (sl ˆ 2 ) → Uq (ĝ) defined in Section 1, one By applying the homomorphisms ϕi : Uqi (sl obtains corresponding identities in Uq (ĝ). As usual, we drop the subscript i ∈ I 12 ˆ 2 )0 by induction on n ∈ N using Definition 3.1. Set P0 = 1 and define Pn ∈ Uq (sl the formula n −k −1 X + Pn = ψ Pn−r . 1 − q −2n r=1 r Define P−n = Ω(Pn ). Definition 3.1 can be conveniently reformulated by introducing the following ˆ 2 )[[u]] of formal power series in an indeterminate u elements of the algebra Uq (sl ˆ 2 ): with coefficients in Uq (sl Ψ± (u) = ∞ X ± un , ψ±n P ± (u) = ∞ X P±n un . n=0 n=0 Then, 3.1 is equivalent to ± (7) Ψ (u) = k ±1 P ± (q ∓2 u) , P ± (u) as one sees in the case of the upper sign by multiplying both sides of (7) by P + (u) and equating coefficients of un (and in the case of the lower sign by applying Ω). Lemma 3.2. For all n ∈ N, we have n Pn = − 1 X rq r hr Pn−r . n r=1 [r]q Proof. We first prove that (8) ∞ Y  n  q hn un P (u) = exp − . [n]q n=1 + For this, it suffices to prove that if we substitute this formula for P + into the right-hand side of (7), then (7) is satisfied:  −n n  Q∞ hn u exp − q [n] + −2 n=1 P (q u) q  n n =k Q k + ∞ P (u) exp − q hn u n=1 ∞ Y [n]q  (q n − q −n )hn un =k exp [n]q n=1 ! ∞ X = k exp (q − q −1 ) hn un ,  n=1 which equals Ψ+ (u) by 1.2. Differentiating both sides of the equation + log P (u) = − ∞ X q n hn un 13 with respect to u now gives u ∞ X nq n hn un (P + )′ (u) = − . P + (u) [n] q n=1 Multiplying both sides of this equation by P + (u) and then equating coefficients of un gives the identity in 3.2.  Remark The identity (8) is equivalent to  k1  k2  k3 X h1 h2 h3 1 n − − − ··· (9) Pn = q k1 !k2 !k3 ! · · · [1]q [2]q [3]q k ,k ,k ...≥0 1P2 3 rkr =n for n ≥ 0. We now study certain commutation formulas between the Pn and the x± r . Lemma 3.3. Let n ∈ N, r ∈ Z. We have + + 2 2 + P n x+ r = xr Pn − (q + 1)xr+1 Pn−1 + q xr+2 Pn−2 (if n = 1, the last term on the right-hand side is omitted). Remark One can obtain a similar formula for n ≤ 0 by applying ΦΩ to both sides. Proof of 3.3. We proceed by induction on n. If n = 0 there is nothing to prove, and if n = 1 the identity follows from P1 = −qh1 , + [h1 , x+ r ] = [2]q xr+1 . Assume then that n ≥ 2 and that the result is known for smaller values of n. Then we have, by 3.2, [Pn , x+ r ] n−1 1 X mq m =− [hm Pn−m , x+ r ] n m=1 [m]q n−1 1 X mq m [2m] + x Pn−m =− n m=1 [m]q m r+m n−1  1 X mq m 2 + − hm −(q 2 + 1)x+ r+1 Pn−m−1 + q xr+2 Pn−m−2 n m=1 [m]q by the induction hypothesis. So [Pn , x+ r ] n−1 1 X m [2m]q + =− q x Pn−m n m=1 [m]q r+m + n−1 1 X m [2m]q  2 2 + q (q + 1)x+ r+m+1 Pn−m−1 − q xr+m+2 Pn−m−2 n m=1 [m]q n−1 X mq m 1 2 x+ hm Pn−m−1 + (q + 1) n [m]q r+1 m=1 − n−1 1 X mq m+2 x+ r+2 hm Pn−m−2 . 14 If m ≥ 3, the expression which right-multiplies x+ r+m on the right-hand side of the last equation is 1 − n   [2m − 2]q m [2m]q m−1 2 m [2m − 4]q q −q (q + 1) +q Pn−m = 0, [m]q [m − 1]q [m − 2]q while the expression which right-multiplies x+ r+1 is, by 3.2, n−1 X mq m 1 2 1 2 − (q + 1)Pn−1 + (q + 1) hm Pn−m−1 n n [m] q m=1 (n − 1) 2 1 (q + 1)Pn−1 = −(q 2 + 1)Pn−1 , = − (q 2 + 1)Pn−1 − n n and that which right-multiplies x+ r+2 is n−2 1 2 [4]q 1 X mq m+2 1 − q Pn−2 − hm Pn−m−2 + (q 2 + 1)2 Pn−2 n [2]q n m=1 [m]q n = (n − 2) 2 2 2 q Pn−2 + q Pn−2 = q 2 Pn−2 . n n This completes the inductive step, and the proof of 3.3.  Lemma 3.4. Let r ∈ Z, n ∈ N. We have x+ r Pn = n X q m [m + 1]q Pn−m x+ r+m . m=0 The proof is similar to that of 3.3. We omit the details. We shall also need the following result, which is easily deduced from the r = 0 case of 3.4: Lemma 3.5. Let n, r ∈ N. Then, r (x+ 0 ) Pn = X + + q m1 +m2 +···+mr [m1 +1]q [m2 +1]q . . . [mr +1]q Pn−m1 −m2 −···−mr x+ m1 xm2 . . . xmr , where the sum is over those non-negative integers m1 , m2 , . . . , mr such that m1 + m2 + · · · + mr ≤ n. (n) for Let Uqres (ĝ)± be the C[q, q −1 ]-subalgebras of Uq (ĝ) generated by the (x± i,r ) res 0 −1 all i ∈ I, r ∈ Z, n ∈ N, and let Uq (ĝ) be the C[q, q ]-subalgebra of Uq (ĝ) generated by Uqres (ĝ)0 and the Pi,r = ϕi (Pr ) for all i ∈ I, r ∈ Z. Note that, by (9), Uqres (ĝ)0 is also generated by Uqres (g)0 and the hi,r /[r]qi for all i ∈ I, r ∈ Z\{0}. (n) (n) , Since, by 1.2, (x± is equal, up to a sign, to a power of Tλ̌i applied to (e± i,r ) i ) 0 res res res ± res it is obvious that Uq (ĝ) ⊂ Uq (ĝ). That Uq (ĝ) ⊂ Uq (ĝ) will be proved in Section 5 (see 5.2). 15 Let ξ be an indeterminate, and form the polynomial algebra C(q)[ξ] over C(q). ˆ 2 )[[u]]: Define the following elements of the algebra Uq (sl ∞ ∞ X X n + n − + x− xn u , X (u) = X (u) = n+1 u . n=0 n=0 Let ˆ 2 )[[u]] D ± (u) : C(q)[ξ] → Uq (sl be the unique homomorphisms of C(q)-algebras such that D ± (u)(ξ) = X ± (u). Writing D ± (u) = ∞ X Dn± un , n=0 ± the homomorphism property of D (u) is equivalent to n X ± ± ± Dm (f )Dn−m (g) Dn (f g) = m=0 ˆ 2 )± are uniquely for all f, g ∈ C(q)[ξ]. The C(q)-linear maps Dn± : C(q)[ξ] → Uq (sl determined by this relation, together with Dn+ (ξ) = x+ n, Dn− (ξ) = x− n+1 , Dn± (1) = δn,0 . Proposition 4.1. Let r, n ∈ N. Writing ξ (r) for ξ r /[r]q !, we have: (a) q n+r−1 [n]q Dn+ (ξ (r) ) = n X + (r−1) q t [t]q x+ ). t Dn−t (ξ t=0 (b) [n + r]q Dn− (ξ (r) ) = n X q −t [n − t + 1]q Dt− (ξ (r−1) )x− n−t+1 . t=0 ˆ 2 ) defined in Section 1; we Let T be the C(q)-algebra automorphism Tλ̌1 of Uq (sl have ± T (x± T (ψn± ) = ψn± , T (c1/2 ) = c1/2 n ) = −xn∓1 , for all n ∈ N. Proposition 4.2. Let r, n ≥ 1. Then, r−1 X + + (r) (s) + (r−s) (ξ (r) ), Dn (ξ ) = (−1)s+1 q s(r−1) (x+ Dn (ξ ) + q r(r−1) T Dn−r 0) s=1 the last term being present if and only if r ≤ n. ˆ 2 )++ be The main tool for proving these identities is the next lemma. Let Uq (sl ˆ 2 ) generated by the x+ for all n ≥ 0. It follows from the C(q)-subalgebra of Uq (sl n ˆ 2 ) given in [1], Proposition 6.1, that the Poincaré–Birkhoff–Witt basis of Uq (sl + + s0 sn sn−1 s1 {(x+ . . . (x+ n ) (xn−1 ) 1 ) (x0 ) }n≥0,s0 ,s1 ,... ,sn−1 ,sn ≥0 ˆ 2 )++ . An element x ∈ Uq (sl ˆ 2 )++ is said to have degree r ≥ 1 is a C(q)-basis of Uq (sl + + s0 sn sn−1 s1 if x is a linear combination of monomials (x+ . . . (x+ with n ) (xn−1 ) 1 ) (x0 ) 16 ˆ 2 )++ have degree r. Suppose that x acts as Lemma 4.3. Let r ≥ 1 and let x ∈ Uq (sl zero on v1⊗r ∈ V (1)a1 ⊗ V (1)a2 ⊗ · · · ⊗ V (1)ar for all non-zero a1 , a2 , . . . , ar ∈ C(q). Then, x = 0. Proof of 4.3. We proceed by induction on r. If r = 1, then x= ∞ X λn x+ n, n=0 with all but finitely many coefficients λn ∈ C(q) being zero. Using the formulas in Section 2 we have, in V (1)a , ! ∞ X x.v1 = λn an v0 . n=0 This can vanish for all non-zero a ∈ C(q) only if λn = 0 for all n. ˆ 2 )++ of degree < r, and consider Now assume the result for x ∈ Uq (sl X sn s0 (10) x= λ(s0 , s1 , . . . , sn )(x+ . . . (x+ n) 0) , P where the sum is over those s0 , s1 , . . . ≥ 0 such that m sm = r, and all but finitely many λ(s0 , s1 , . . . ) ∈ C(q) are zero. Let N be the maximum value of n that occurs in (10), i.e. N = max{n | there exists s0 , . . . , sn with λ(s0 , . . . , sn ) 6= 0}, and consider the action of x on V (1)a1 ⊗ V , where V = V (1)a2 ⊗ · · · ⊗ V (1)ar . Clearly, x.v1⊗r = F (a1 , a2 , . . . , ar )v0⊗r , where F (a1 , a2 , . . . , ar ) is a polynomial in a1 , a2 , . . . , ar with coefficients in C(q) whose degree in a1 is ≤ N . We are going to identify the terms in F (a1 , a2 , . . . , ar ) ⊗r N that involve aN 1 . Since x acts as zero on v1 , the coefficient of a1 must actually be zero. ˆ 2 ) spanned (over C(q)) by the x± for all Let X ± be the linear subspace of Uq (sl n n ∈ Z. By Proposition 5.4 in [4], if n ≥ 0 we have ∆(x+ n) ≡ n X + + + 2 − ˆ ˆ x+ m ⊗ ψn−m + 1 ⊗ xn modulo Uq (sl2 )(X ) ⊗ Uq (sl2 )X . m=0 Since (X + )2 annihilates V (1)a1 for all a1 , it follows that x+ n acts on V (1)a1 ⊗ V as + s x+ ⊗ k + 1 ⊗ x and hence, by an easy induction on s ≥ 1, that (x+ n n n ) acts as (11) (x+ n ⊗k+1⊗ s x+ n) = s X t=0 q t(s−t) hsi t q t + s−t t k. (x+ n ) ⊗ (xn ) ⊗r−1 It is now clear that terms involving aN ) can arise only from the 1 in x.(v1 ⊗ v1 part X sN −1 s0 sN . . . (x+ ∆((x+ (12) λ(s0 , . . . , sN )∆(x+ 0) ) N−1 ) N) 17 of ∆(x), and in (12) only from the terms in which x+ N occurs in the first factor in the tensor product (note that if a square or higher power of x+ N occurs in the first factor of the tensor product, the corresponding term acts as zero on v1 ⊗ v1⊗r−1 ). ⊗r By (11), the coefficient of aN 1 in x.v1 is therefore X s0 ⊗r−1 sN −1 + (xN−1 )sN −1 . . . (x+ λ(s0 , . . . , sN )q sN −1 [sN ]q q r−2sN +1 v1 ⊗ (x+ 0 ) .v1 N) s0 ,... ,sN = q r v1 ⊗ X sN −1 + s0 ⊗r−1 q −sN [sN ]q λ(s0 , . . . , sN )(x+ (xN−1 )sN −1 . . . (x+ . 0 ) .v1 N) s0 ,... ,sN Thus, if x acts as zero on v1⊗r , this last expression must vanish, so by the induction hypothesis, X sN −1 + s0 q −sN [sN ]q λ(s0 , . . . , sN )(x+ (xN−1 )sN −1 . . . (x+ = 0. 0) N) s0 ,... ,sN By [1], Proposition 6.1, this forces all the coefficients λ(s0 , . . . , sN ) to vanish. But this contradicts the assumption that xN does occur on the right-hand side of (10).  We shall also need the computations contained in the next two lemmas. Lemma 4.4. Let r ≥ s ≥ 1, and let a1 , . . . , ar ∈ C(q) be non-zero. Then, in V (1)a1 ⊗ · · · ⊗V (1)ar , we have (13) (s) (x+ (X + (u))(r−s) .v1⊗r = fs,r (a1 , . . . , ar ; u)v0⊗r , 0) where   s X 1 t r−t q t(r−s) Et (a1 , . . . , ar )ut fs,r (a1 , . . . , ar ; u) = (−1) s−t q (1 − a1 u) · · · (1 − ar u) t=0 and Et (a1 , . . . , ar ) is the tth elementary symmetric function of a1 , . . . , ar , i.e. X Et (a1 , . . . , ar ) = a r1 a r2 . . . a rt . 1≤r1 <r2 <···<rt ≤r Proof. It is clear that (13) holds with some scalars fs,r (a1 , . . . , ar ; u) that are formal power series in u with coefficients in C(q). We can derive an inductive formula for them as follows. Working in V (1)a1 ⊗ V (1)a2 ⊗ · · · ⊗ V (1)ar we have, by (11) and the fact that V (1)a1 is annihilated by (X + )2 , + (s−1) (s) + r−s ⊗r (s) (x+ .v1 = (1 ⊗ (x+ + q s−1 x+ k)X + (u)r−s .(v1 ⊗ v1⊗r−1 ). 0 ) X (u) 0) 0 ⊗ (x0 ) Arguing as in the proof of 4.3, we see that X + (u) acts on V (1)a1 ⊗ V as X + (u) ⊗ Ψ+ (u) + 1 ⊗ X + (u). Hence, (s) (s) + r−s ⊗r .v1 = X + (u).v1 ⊗ (x+ (x+ 0) 0 ) X (u) +q −s+1 v0 ⊗ r−s X X + (u)t Ψ+ (u)X + (u)r−s−t−1 t=0 (s−1) + X (u)r−s .v1⊗r−1 k(x+ 0) ! .v1⊗r−1 18 say. Clearly, B = q r−s fs−1,r−1 (a2 , . . . , ar ; u)v0⊗r . To evaluate A, we first show that, for r ≥ 1, r X + r+1 X + (u)t Ψ+ (u)X + (u)r−t ≡ q r [r + 1]q k −1 X + (u)r − (q − q −1 )x− 0 X (u) t=0 ˆ 2 )X − [[u]]. modulo Uq (sl (15) The proof is by induction on r. If r = 1, we have [X + (u), x− 0]= Ψ+ (u) − k −1 , q − q −1 so + −1 − + Ψ+ (u)X + (u) = (q − q −1 )X + (u)x− )x0 X (u)2 + k −1 X + (u) 0 X (u) − (q − q   Ψ+ (u) − k −1 − −1 + + = (q − q )X (u) X (u)x0 − q − q −1 + 2 −1 + − (q − q −1 )x− X (u). 0 X (u) + k Hence, + 2 −1 Ψ+ (u)X + (u)+X + (u)Ψ+ (u) = q[2]q k −1 X + (u)−(q−q −1 )x− )X + (u)2 x− 0 X (u) +(q−q 0, proving (15) when r = 1. Assume now that (15) is known for some r ≥ 1. Then, r+1 X X + (u)t Ψ+ (u)X + (u)r−t+1 t=0 + = X (u) r X X + (u)t Ψ+ (u)X + (u)r−t + Ψ+ (u)X + (u)r+1 t=0 r + r+1 ≡ X (u)(q [r + 1]q k −1 X + (u)r − (q − q −1 )x− ) + Ψ+ (u)X + (u)r+1 0 X (u) ˆ 2 )X − [[u]] modulo Uq (sl + + + −1 ≡ q r+2 [r + 1]q k −1 X + (u)r+1 − ((q − q −1 )x− )X + (u)r+1 0 X (u) + Ψ (u) − k ˆ 2 )X − [[u]] + Ψ+ (u)X + (u)r+1 modulo Uq (sl + r+2 ˆ 2 )X − [[u]], ≡ q r+1 [r + 2]q k −1 X + (u)r+1 − (q − q −1 )x− modulo Uq (sl 0 X (u) completing the inductive step. Thus,
⊗r−1 (s) + r−s A = X + (u).v1 ⊗(x+ q r−s−1 [r − s]q k −1 X + (u)r−s−1 − (q − q −1 )x− .v1 . 0) 0 X (u) Now, by the formulas in 2.3, we have, in V (1)a1 , + X (u).v1 = ∞ X un x+ n .v1 = ∞ X an1 un v0 = 1 v0 . 19 Moreover, an easy induction on s shows that (s) −s+1 (s−1) (q − q −1 )[(x+ , x− k − q s−1 k −1 )(x+ . 0) 0 ] = (q 0) Using these results, we get  (s) + (1 − a1 u)A = v0 ⊗ q r−s−1+2(r−s−1)−2(r−1) [r − s]q (x+ X (u)r−s−1 0)  (s−1) + r−s−1 .v1⊗r−1 ) X (u) −(q −s+1+r−1 − q s−1−r+1 )(x+ 0
= q s [r − s]q fs,r−1 (a2 , . . . , ar ; u) − (q r−s − q −r+s )fs−1,r−1 (a2 , . . . , ar ; u) v0⊗r . Inserting these values of A and B into (14) gives fs,r (a1 , . . . , ar ; u) = (16)   q −r+s − q r−s a1 u fs−1,r−1 (a2 , . . . , ar ; u) 1 − a1 u + q s [r − s]q fs,r−1 (a2 , . . . , ar ; u). This identity clearly determines the fs,r , by induction on r, in terms of f0,1 and f1,1 . It therefore suffices to prove that the formula for fs,r in the statement of the lemma satisfies (16) and is correct when r = 1. Correctness for r = 1 is trivially checked. To verify (16), we must show that s X t t(r−s) (−1) q t=0  r−s t−s = (q  Et (a1 , . . . , ar )ut − q s q s X (−1)t q t(r−s−1) Et (a2 , . . . , ar )ut t=0 −r+s −q r−s a1 u) s−1 X t t(r−s) (−1) q t=0  r−t−1 s−t−1  Et (a2 , . . . , ar )ut . q Thus, equating coefficients of ut , we are reduced to proving that  r−t s−t   r−t−1 Et (a2 , . . . , ar ) Et (a1 , . . . , ar ) − q s−t q q     r−t −r+s r − t − 1 a1 Et−1 (a2 , . . . , ar ). Et (a2 , . . . , ar ) + =q s−t q s−t−1 q s−t  Since Et (a1 , . . . , ar ) = a1 Et−1 (a2 , . . . , ar ) + Et (a2 , . . . , ar ), this reduces to  r−t s−t  −q s−t q This identity is easily checked.  r−t−1 s−t   =q q −r+s  r−t−1 s−t−1  . q 20 Lemma 4.5. Let r ≥ 1 and let a1 , . . . , ar ∈ C(q) be non-zero. Then, in V (1)a1 ⊗ · · · ⊗ V (1)ar , we have T (D + (u)(ξ (r) )).v1⊗r = a1 . . . ar D + (ξ (r) ).v1⊗r . Proof. Assume that, for all 1 ≤ s 6= t ≤ r, as /at ∈ / q Z . Then, V = V (1)a1 ⊗ · · · ⊗ V (1)ar ˆ 2 ) (see [9]). Let ρ : Uq (sl ˆ 2) → is an irreducible highest weight representation of Uq (sl End(V ) denote the action. Then, ρ ◦ T is another irreducible representation of ˆ 2 ), and since T takes Uq (sl ˆ 2 )+ to itself and fixes Uq (sl ˆ 2 )0 pointwise, ρ ◦ T is Uq (sl highest weight with the same highest weight as ρ. Hence, ρ ◦ T is equivalent to ρ, i.e. there exists F ∈ GL(V ) such that ˆ 2 ), v ∈ V . F (x.v) = T (x).F (v) for all x ∈ Uq (sl We might as well assume that F (v0⊗r ) = v0⊗r . Then, F (v1⊗r ) = cv1⊗r for some non-zero c ∈ C(q). We proceed to compute c. Using (11), we have + + r ⊗r r ⊗r (x+ 0 ) .v1 = (x0 ⊗ k + 1 ⊗ x0 ) .v1 + r−1 = q r−1 [r]q (x+ k).(v1 ⊗ v1⊗r−1 ) 0 ⊗ (x0 ) r−1 ⊗r−1 = [r]q v1 ⊗ (x+ .v1 , 0) and hence by iteration, (r) ⊗r .v1 = v0⊗r . (x+ 0) Similarly, (r) ⊗r (x− .v0 = v1⊗r . 0) Hence, (r) ⊗r (r) (r) ⊗r .v0 ). .F (v0⊗r ) = F ((x− .v0 = T (x− v1⊗r = (x− 1) 1) 0) Obviously, (r) ⊗r .v0 = c′ v1⊗r , (x− 1) (r) to both sides of the last equation for some c′ ∈ C(q). To compute c′ , apply (x+ 0) and use Proposition 3.5 in [4]: 2 c′ v0⊗r = (−1)r q −r Pr k r .v0⊗r = (−1)r × (coefficient of ur in (1 − a1 u) . . . (1 − ar u))v0⊗r = (a1 . . . ar )v0⊗r , where the second equality used (6). Hence, c′ = a1 . . . ar and v1⊗r = (a1 . . . ar )F (v1⊗r ) = (a1 . . . ar )cv1⊗r , 21 Finally, T (D + (u)(ξ (r) )).v1⊗r = T (X + (u)(r) ).v1⊗r = c−1 T (X + (u)(r) ).F (v1⊗r ) = (a1 . . . ar )F (X + (u)(r) .v1⊗r ) = (a1 . . . ar )F (D + (u)(ξ (r) ).v1⊗r ). But D + (u)(ξ (r)).v1⊗r is obviously a scalar multiple of v0⊗r , and so is fixed by F . Hence, T (D + (u)(ξ (r) )).v1⊗r = (a1 . . . ar )D + (u)(ξ (r) ).v1⊗r , as claimed. Although we have proved this relation only under the assumption that as /at ∈ / qZ for all s 6= t, it is clear that both sides are polynomials in a1 , . . . , ar , so it must hold identically.  We are finally in a position to prove 4.1 and 4.2. Proof of 4.1. By equating coefficients of un on both sides, the identity in part (a) is easily seen to be equivalent to [r]q (X + (q 2 u) − X + (u))D + (u)(ξ r−1 ) = q r−1 (D + (q 2 u)(ξ r ) − D + (u)(ξ r )), or, since D + (u)(ξ r ) = X + (u)r , to [r]q X + (q 2 u)X + (u)r−1 − q −1 [r − 1]q X + (u)r = q r−1 X + (q 2 u)r . (17) Now (17) holds trivially if r = 1, and we claim that it follows for all r if it holds for r = 2. Indeed, assuming that r ≥ 3 and that the result is known for r − 1, we get q r−1 X + (q 2 u)r = qX + (q 2 u)([r − 1]q X + (q 2 u)X + (u)r−2 − q −1 [r − 2]q X + (u)r−1 ) = q[r − 1]q X + (q 2 u)2 X + (u)r−2 − [r − 2]q X + (q 2 u)X + (u)r−1 = [r − 1]q ([2]q X + (q 2 u)X + (u) − q −1 X + (u)2 )X + (u)r−2 − [r − 2]q X + (q 2 u)X + (u)r−1 = [r]q X + (q 2 u)X + (u)r−1 − q −1 [r − 1]q X + (u)r , where the penultimate equality is from the r = 2 case. Thus, it suffices to prove that [2]q X + (q 2 u)X + (u) − q −1 X + (u)2 = qX + (q 2 u)2 . (18) By 4.3, it is enough to prove that both sides of (18) act in the same way on v1 ⊗v1 ∈ V (1)a1 ⊗V (1)a2 , for all non-zero a1 , a2 ∈ C(q). Recalling that ˆ 2 )(X + )2 ⊗Uq (sl ˆ 2 )X − )[[u]], ∆(X + (u)) = X + (u)⊗Ψ+ (u) + 1⊗X + (u) modulo (Uq (sl and noting that, in V (1)a , + 1 + q − q −1 au + q −1 − qau 22 by the formulas in 2.3, we have [2]q X + (q 2 u)X + (u).(v1 ⊗v1 )
= [2]q X + (q 2 u) ⊗ Ψ+ (q 2 u)X + (u) + X + (u)⊗X + (q 2 u)Ψ+ (u) .(v1 ⊗v1 )   1 1 q − qa2 u 1 q −1 − qa2 u 1 = [2]q v1 ⊗v1 + 1 − q 2 a1 u 1 − q 2 a2 u 1 − a2 u 1 − a2 u 1 − a2 u 1 − q 2 a2 u   q −1 q v1 ⊗v1 , + = [2]q (1 − q 2 a1 u)(1 − q 2 a2 u) (1 − a1 u)(1 − a2 u)
q −1 X + (u)2 .(v1 ⊗v1 ) = q −1 X + (u)⊗Ψ+ (u)X + (u) + X + (u)⊗X + (u)Ψ+ (u) .(v1 ⊗v1 )   1 1 q − q −1 a2 u 1 1 q −1 − qa2 u −1 v1 ⊗v1 =q + 1 − a1 u 1 − a2 u 1 − a2 u 1 − a1 u 1 − a2 u 1 − a2 u q −1 [2]q v1 ⊗v1 , = (1 − a1 u)(1 − a2 u) and hence qX + (q 2 u)2 .(v1 ⊗v1 ) = (1 − q2 a q[2]q v1 ⊗v1 . 2 1 u)(1 − q a2 u) It is now clear that the two sides of (18) agree in their action on v1 ⊗v1 , for all a1 , a2 . This proves part (a). The identity in part (b) can be converted into an equivalent identity for Dn+ by using Dn− = −T ◦ Φ ◦ Dn+ , and this identity can then be proved by an argument similar to that used for part (a). We omit the details.  Proof of 4.2. Multiplying both sides by un and summing from n = 0 to ∞, we see that the identity to be proved is equivalent to + X (u) (r) r−1   X + (r−s) (s) + (r−s) (r) = (−1)s+1 q s(r−1) (x+ ) X (u) − (x ) +q r(r−1) ur T (X + (u)(r) )+(x+ . 0 0 0) s=1 As in the proof of 4.1, we compute the action of both sides on v1⊗r ∈ V (1)a1 ⊗ · · · ⊗V (1)ar , for arbitrary non-zero a1 , . . . , ar ∈ C(q). Using 4.4 and 4.5, we see that it suffices to prove that   r−1 X s X 1 1 s+t+1 s(r−1)+t(r−s) r − t Et (a1 , . . . , ar )ut (−1) q = s−t q (1 − a1 u) . . . (1 − ar u) (1 − a1 u) . . . (1 − ar u) s=1 t=0 + r−1 X (−1)s q s(r−1) s=1 hri s q + q r(r−1) a1 . . . ar ur + 1, (1 − a1 u) . . . (1 − ar u) i.e., after using the identity (1), 0 = q r(r−1) a1 . . . ar ur − (−1)r q r(r−1) (1 − a1 u) . . . (1 − ar u) − 1   r−1 X s X s+t+1 s(r−1)+t(r−s) r − t Et (a1 , . . . , ar )ut . + (−1) q 23 The constant term on the right-hand side vanishes by identity (1) again, and the coefficient of ur obviously vanishes. If 0 < t < r, the coefficient of ut on the right-hand side is (−1)r+t+1 q r(r−1) + r−1 X (−1)s+t+1 q s(r−1)+t(r−s) s=t  r−t s−t  ! Et (a1 , . . . , ar ) q   r−t X p p(r−s−1) r − t Et (a1 , . . . , ar ), = (−1) q p q p=0 which vanishes by identity (1) once again.  The following corollary of 4.2 will be needed later. Corollary 4.5. Let r, n ≥ 1. Then, Dn+ (ξ (r) ) = X (s0 ) + (s1 ) µ(s0 , s1 , . . . )(x+ (x1 ) ... , 0) P wherePthe sum is over those non-negative integers s0 , s1 , . . . such that t st = r and t tst = n, and the coefficients µ(s0 , s1 , . . . ) ∈ C[q, q −1 ]. In particular, the (r) + (ξ (r) ) is q nr(r−1) . in Dnr coefficient of (x+ n) Proof. The first part is immediate from 4.2. The second part follows by induction (r) can arise only from the second term on r, noting that the term involving (x+ n) + on the right-hand side of the formula for Dnr (ξ (r) ) given in 4.2.  Remark It is possible to deduce an exact formula for Dn+ (ξ (r) ) from 4.2, namely (19) Dn+ (ξ (r) ) = q −n X (r−l(π)) + (r1 ) + (r2 ) q f (π)+rl(π) (x+ (x1 ) (x2 ) ... . 0) π Here, the sum is over all sequences π : r1 , r2 , . . . of non-negative integers such that r1 + 2r2 + 3r3 + · · · = n and r1 + r2 + · · · ≤ r. To such a sequence we associate a Young diagram which has r1 rows of length 1, r2 rows of length 2, etc., so that l(π) = r1 + r2 + · · · is the total number of rows, and define f (π) to be the sum of the products of the lengths of adjacent columns. For example, if π is the sequence 2, 1, 3, 1, 0, 0, . . . , whose Young diagram is we have l(π) = 7, f (π) = 7 × 5 + 5 × 4 + 4 × 1 = 59. To prove the formula (19), one observes that 4.2 clearly determines Dn+ (ξ (r) ), by induction on n for fixed r, and then by induction on r. Thus, it suffices to prove 24 the right-hand side of (19), then the identity in 4.2 is satisfied. We omit the details as we shall not make use of (19) in the sequel. We conclude this section with the following result, which will be needed in Section 6. Proposition 4.6. For all r ∈ N, n ∈ Z, we have 1 + r−1 + + + r r + res ˆ + ((x+ xn x0 + · · · + x+ n (x0 ) ) ∈ Uq (sl2 ) . 0 ) xn + (x0 ) [r + 1]q ! ˆ 2 ) inductively by Proof. For r ∈ N, n ∈ Z, define elements Ar,n ∈ Uq (sl (20) 2r + Ar,n = Ar−1,n x+ 0 − q x0 Ar−1,n , A0,n = x+ n, and let Br,n = r X s + + r−s (x+ . 0 ) xn (x0 ) s=0 We shall prove the following identities: r (21) X Ar,n (s) = (−1)s q r(r−s+1) (x+ 1) T [r + 1]q ! s=0 r (22)  Br−s,n−r−1 [r − s + 1]q !  , X Br,n (r−s) As,n = q (r−s)(s+1) (x+ . 0) [r + 1]q ! s=0 [s + 1]q ! The statement in the proposition for n ≥ 0 follows from these identities by induction (r+1) on n. Indeed, the result is trivial if n = 0, since Br,0 /[r+1]q ! = (x+ . Assuming 0) res ˆ + that Br,m /[r + 1]q ! ∈ Uq (sl2 ) for all r ∈ N, m < n, it follows from (21) that ˆ 2 )+ for all r ∈ N, and then from (22) that Br,n /[r + 1]q ! ∈ Ar,n /[r + 1]q ! ∈ Uqres (sl ˆ 2 )+ . Uqres (sl Note that (21) can be written in the form Ar,n = r X + r−s s + cr,s (x+ , 1 ) xn−r (x1 ) s=0 where cr,s = s X t=0 t r(r−t+1) (−1) q  r+1 t  . q + r−s s + Comparing coefficients of (x+ on both sides of the second equation 1 ) xn−r (x1 ) in (20), we see using (3) that (21) follows from cr,s = q 2r cr−1,s − cr−1,s−1 . This is an easy consequence of the identity     h i r r+1 t r t−r+1 . =q +q 25 To prove (22), we proceed by induction on r. If r = 0, the result is trivial. Assume now that (22) holds for all smaller values of r and all n. Then, + (x+ )r x+ Br,n n + Br−1,n x0 = 0 [r + 1]q ! [r + 1]q ! 1 = [r + 1]q ! = 1 [r + 1]q ! r + (x+ 0 ) xn + [r]q ! r + (x+ 0 ) xn + r−1 X r−1 X q (r−s−1)(s+1) (r−s−1) (x+ 0) As,n [s + 1]q ! ! x+ 0 ! s=0 r−1 X + (r−s−1) As,n [r]q ! q (r−s−1)(s+1)+2s+2 x+ 0 (x0 ) [s + 1]q ! s=0 (r−s−1) As+1,n +[r]q ! q (r−s−1)(s+1) (x+ 0) [s + 1]q ! s=0 ! , using the inductive relation 2s+2 + As,n x+ x0 As,n = As+1,n . 0 −q A s,n (r−s) Comparing coefficients of (x+ 0) [s+1]q ! , we see that to prove (22) it is enough to show that q (r−s)(s+1) [r + 1]q = q (r−s−1)(s+1)+2s+2 [r − s]q + q (r−s)s [s + 1]q . This identity is easily checked. This completes the proof of 4.6 for n ≥ 0. The statement for n ≤ 0 follows from this by applying ΦΩ.  5. The Basic Lemma In this section, we prove the following result which is central to the whole paper. ˆ 2 ). We work in Uq (sl ˆ 2 ), let Lx : Uq (sl ˆ 2 ) → Uq (sl ˆ 2 ) be left multiplication by x. Define If x ∈ Uq (sl Dn = LPn D0+ + LPn−1 D1+ + · · · + LP0 Dn+ . ˆ 2 ): Lemma 5.1. Let r, s ∈ N. The following identity holds in Uq (sl min(r,s) (r) − (s) (x+ (x1 ) 0) = X t=0 X − (s−t) t (−1)t q −t(r+s−t) Dm (ξ )k Dn (ξ (r−t) ). m+n=t m,n≥0 Remarks 1. By applying Ω and/or powers of T to both sides, analogous identities (r) − (s) (xn ) whenever m + n = ±1. (The case m + n = 0 can be obtained for (x+ m) 26 2. Lemma 5.1 is the quantum analogue of Lemma 7.5 in [9]. The classical result is much easier, however, because there the x+ n commute among themselves, as do the x− . In addition, one has the relation n Dn± = (D1± )n n! classically, whereas in the quantum case this equation does not even make sense. Before proving 5.1, we note the following important corollary. Corollary 5.2. For all n ∈ N, ˆ 2 ) for all r ∈ N; (i)n Dn± (ξ (r) ) ∈ Uqres (sl ˆ 2 ). (ii)n Pn ∈ Uqres (sl ˆ 2 ) for all n < 0. Of course, it follows from part (ii), 1.3 and 3.1 that Pn ∈ Uqres (sl Proof. We prove both statements simultaneously by induction on n, according to the following scheme: n n n+1 (i) =⇒ (ii) =⇒ (i) , n n where (i) is the statement that (i)m holds for all m ≤ n, and similarly for (ii) . n Assume that (i) holds. Applying 5.1, we see that 2 (n) − (n) (x+ (x1 ) = (−1)n q −n k n Pn + y, 0) ˆ 2 ) by the induction where y is a sum of terms already known to belong to Uqres (sl n n−1 hypothesis (i) and (ii) . n Now assume that (ii) holds. Applying 5.1 again, we get (n+r) − (n) (x+ (x1 ) = (−1)n q −n(n+r) k n Dn+ (ξ (r) ) + z, 0) ˆ 2 ) by the where z is a sum of terms which are already known to belong to Uqres (sl n n ˆ 2 ) for all r ∈ N. The induction hypothesis (i) and (ii) . Hence, Dn+ (ξ (r) ) ∈ Uqres (sl − result for Dn follows, since Dn− = −T ◦ Φ ◦ Dn+ , ˆ 2 ).  and T and Φ preserve Uqres (sl Lemma 5.1 is a consequence of the next lemma. ˆ 2 ): Lemma 5.3. Let n, r ∈ N. The following identity holds in Uq (sl Dn (ξ (r) )x− 1 = −q −n−r [n + 1]q kDn+1 (ξ (r−1) )+ n+1 X (r) q m−1 [m]q x− ). m Dn−m+1 (ξ 27 We complete the proof of 5.1, assuming 5.3. We proceed by induction on s, the case s = 0 being trivial. By the induction hypothesis and 5.3, we have min(r,s) [s + (r) − (s+1) 1]q (x+ (x1 ) 0) = X t=0 X − (s−t) t (−1)t q −t(r+s−t) Dm (ξ )k D(ξ (r−t) )x− 1 X − (s−t) t (−1)t q −t(r+s−t) Dm (ξ )k m+n=t m,n≥0 min(r,s) = X t=0 × m+n=t m,n≥0 n −q n−r+t [n + 1]q kDn+1 (ξ (r−t−1) ) ) n+1 X (r−t) + q l−1 [l]q x− ) . l Dn−l+1 (ξ l=1 Looking at the expression which left-multiplies Dn (ξ (r−t) ), we see that we are reduced to proving that − − [s + 1]q Dt−n (ξ (s+1−t) )q −t(r+s+1−t) = Dt−n (ξ (s+1−t) )q −(t−1)(r+s+1−t)−n−r+t [n]q + t−n X Dv− (ξ (s−t) )[t − v − n + 1]q q −t(r+s−t)−t−v−n x− t−v−n+1 , v=0 or, on simplifying, that [s − n + − 1]q Dt−n (ξ (s+1−t) ) = t−n X q −v [t − v − n + 1]q Dv− (ξ (s−t) )x− t−v−n+1 . v=0 After suitably re-labelling the indices, one sees that this is equivalent to the identity in 4.1(b). To complete the proof of 5.1, we are thus reduced to giving the Proof of 5.3. We proceed by induction on r. When r = 0, the identity becomes P n x− 1 = n+1 X q m−1 [m]q x− m Pn−m+1 . m=1 This follows from 2.4 (take r = 1 in 3.4 and apply Φ to both sides, noting that 28 Assuming the result for r, we have [r + 1]q Dn (ξ (r+1) )x− 1 = n X + LPm Dn−m (ξξ (r) )x− 1 m=0 = X + (r) − P m x+ )x1 l Dn−m−l (ξ X + (r) − [Pm , x+ )x1 l ]Dn−m−l (ξ X + + 2 (r) − (q 2 x+ )x1 l+2 Pm−2 − (q + 1)xl+1 Pm−1 )Dn−m−l (ξ m+l≤n m,l≥0 = + (r) − x+ )x1 l Dn−l (ξ l=0 m+l≤n m,l≥0 = n X m+l≥n m,l≥0 + n X l=0 (23) + n −q −n−r+l [n − l + 1]q kDn−l+1 (ξ (r−1) ) x+ l n−l+1 X ) (r) q s−1 [s]q x− ) , s Dn−l−s+1 (ξ s=1 on using 3.3 and the induction hypothesis. The first term on the right-hand side of (23) is equal to q 2 n−2 X (r) − )x1 x+ l+2 Dn−l−2 (ξ − (q + 1) l=0 = q2 n−2 X 2 n−1 X (r) − )x1 x+ l+1 Dn−l−1 (ξ l=0 x+ l+2 l=0 − (q 2 + 1) ( −q −n+l+2−r [n − l − 1]q kDn−l−1 (ξ (r−1) ) + n−1 X n−l−1 X (r) q m−1 [m]q x− ) m Dn−l−m−1 (ξ m=1 x+ l+1 l=0 ( −q −n+l+1−r [n − l]q kDn−l (ξ (r−1) ) + n−l X (r) q m−1 [m]q x− ) , m Dn−l−m (ξ m=1 on using the induction hypothesis again. We now distinguish two types of term on the right-hand side of (23): (r−1) (a) The term involving x+ ), which is left-multiplied by m kDn−m+1 (ξ −q −n−r+m−2 [n−m+1]q −q −n−r+m [n−m+1]q +q −n+m−r−2 (q 2 +1)[n−m+1]q = 0, except that if m = 0 only the first term appears, and if m = 1 only the first and last terms appear. Hence, the contribution of this type of term is (r−1) (r−1) q −n−r+1 [n]q kx+ ) − q −n−r−2 [n + 1]q kx+ ). 1 Dn (ξ 0 Dn+1 (ξ − (r) ), which is left-multiplied by (b) The term involving x+ m xl Dn−m−l+1 (ξ q l−1 [l]q + q l+1 [l]q − q l−1 (q 2 + 1)[l]q = 0, except if m = 0 or 1, so the net contribution of this type of term is l−1 + − (r) l+1 + − ) (r) ) 29 Hence, −n−r+1 (r−1) (r−1) [r + 1]q Dn (ξ (r+1) )x− [n]q kx+ ) − q −n−r−2 [n + 1]q kx+ ) 1 =q 1 Dn (ξ 0 Dn+1 (ξ + n+1 X − (r) q l−1 [l]q x+ )− 0 xl Dn−l+1 (ξ l=1 n X − (r) q l+1 [l]q x+ ). 1 xl Dn−l (ξ l=1 On the other hand, we are trying to show that [r + 1]q Dn (ξ (r+1) )x− 1 = −q −n−r−1 = −q −n−r−1 [n + 1]q [r + 1]q kDn+1 (ξ (r) ) + [r + 1]q n+1 X (r+1) q m−1 [m]q x− ) m Dn−m+1 (ξ m=1 (r) [n + 1]q [r + 1]q kDn+1 (ξ ) (n−m+1 n+1 X X (r) + q m−1 [m]q x− x+ ) m l Dn−l−m+1 (ξ m=1 l=0 X + + 2 (r) (q 2 x+ ) s+2 Pl−2 − (q + 1)xs+1 Pl−1 )Dn−m−l−s+1 (ξ    ,   l+s≤n−m+1 l,s≥0 by repeating the argument leading to equation (23), which is equal to X + (r) q m−1 [m]q x− ) −q −n−r−1 [n + 1]q [r + 1]q kDn+1 (ξ (r) ) + m xl Dn−l−m+1 (ξ m+l≤n+1 l≥0,m≥1 + X + + 2 + 2 (r) q m−1 [m]q x− ). m (q xs+2 Pl−2 − (q + 1)xs+1 Pl−1 )Dn−l−m−s+1 (ξ s+l+m≤n+1 l≥0,m≥1 + As above, one sees that the term x− m xl , with l ≥ 0, m ≥ 1 and l + m ≤ n + 1, survives in the sum of the last two terms only if l = 0 or 1, giving [r + 1]q Dn (ξ (r+1) )x− 1 = −q −n−r−1 [n + 1]q [r + 1]q kDn+1 (ξ (r) )+ n+1 X + (r) ) q m−1 [m]q x− m x0 Dn−m+1 (ξ m=1 − n X + (r) q m+1 [m]q x− ). m x1 Dn−m (ξ m=1 So we must prove that (r−1) (r−1) q −n−r+1 [n]q kx+ ) − q −n−r−2 [n + 1]q kx+ ) 1 Dn (ξ 0 Dn+1 (ξ + q −n−r−1 [r + 1]q [n + 1]q kDn+1 (ξ (r) ) + − n+1 X m=1 n X q m−1 [m]q q m+1 [m]q + ψm Dn−m+1 (ξ (r) ) q − q −1 + ψm+1 −1 Dn−m (ξ (r) ) = 0. 30 The sum of the last two terms is equal to n + + X ψm+1 X ψm+1 + (r) Dn−m (ξ ) = Pl Dn−l−m (ξ (r) ) −1 −1 q − q q − q l+m≤n m=0 l,m≥0 = n X s X + ψm+1 + Ps−m Dn−s (ξ (r) ) −1 q−q s=0 m=0 n X −s−1 q =− + [s + 1]q kPs+1 Dn−s (ξ (r) ), s=0 by 3.1. So we are reduced to proving that (r−1) (r−1) q[n]q kx+ ) − q −2 [n + 1]q kx+ ) 1 Dn (ξ 0 Dn+1 (ξ + q −1 [r + 1]q [n + 1]q kDn+1 (ξ (r) ) n X + = q n+r q −s−1 [s + 1]q kPs+1 Dn−s (ξ (r) ). (24) s=0 Now, (r−1) )= x+ 0 Dn+1 (ξ n+1 X + (r−1) ) x+ 0 Pm Dn−m+1 (ξ m=0 = m n+1 XX + l (r−1) Pm−l x+ ), l q [l + 1]q Dn−m+1 (ξ m=0 l=0 by 3.4, which is equal to n+1 X m=0 Pm n−m+1 X + (r−1) q t [t + 1]q x+ ). t Dn−m−t+1 (ξ t=0 Similarly, (r−1) x+ )= 1 Dn (ξ n X m X + (r−1) ) q t [t + 1]q Pm−t x+ t+1 Dn−m (ξ m=0 t=0 = n+1 X Pm m=0 n−m X + (r−1) q t [t + 1]q x+ ). t+1 Dn−t−m (ξ t=0 Inserting these results in (24), and equating coefficients of kPm on both sides, we see that it suffices to prove that q[n]q n−m X + + (r−1) q t [t + 1]q x+ ) + q −1 [r + 1]q [n + 1]q Dn−m+1 (ξ (r) ) t+1 Dn−t−m (ξ t=0 −q −2 [n + 1]q n−m+1 X + (r−1) ) q t [t + 1]q x+ t Dn−t−m+1 (ξ t=0 n+r−m + (r) 31 The sum of the first and third terms is equal to [n]q n−m+1 X q t + (r−1) [t]q x+ ) t Dn−t−m+1 (ξ −q −2 [n + 1]q t=0 n−m+1 X + (r−1) (1 + q t+1 [t])x+ ) t Dn−t−m+1 (ξ t=0 + = −q −2 [n + 1]q [r]q Dn−m+1 (ξ (r) ) + n−m+1 X + (r−1) ) (q t [t]q [n]q − q t−1 [t]q [n + 1]q )x+ t Dn−t−m+1 (ξ t=0 + = −q −2 [n + 1]q [r]q Dn−m+1 (ξ (r) ) − q −n−1 n−m+1 X + (r−1) ). q t [t]q x+ t Dn−t−m+1 (ξ t=0 Thus, we are reduced to proving that + (q −1 [r + 1]q [n + 1]q − q n+r−m [m]q − q −2 [n + 1]q [r]q )Dn−m+1 (ξ (r) ) =q −n−1 n−m+1 X + (r−1) q t [t]q x+ ), t Dn−t−m+1 (ξ t=0 i.e. that n−m+1 X + + (r−1) ) = q n+r−m [n − m + 1]q Dn−m+1 (ξ (r) ). q t [t]q x+ t Dn−t−m+1 (ξ t=0 This is the statement of 4.1(a) (with n replaced by n − m + 1).  6. A Triangular Decomposition The following result is central to the construction of highest weight representations of Uqres (ĝ) in Sections 7 and 8. Proposition 6.1. We have Uqres (ĝ) = Uqres (ĝ)− .Uqres (ĝ)0 .Uqres (ĝ)+ . Proof. Let U ∆ denote the right-hand side of the equation in 6.1. Following the strategy in [13], Section 5, define the degree of the generators of Uqres (ĝ) as follows: (r) ) deg((x± i,n ) = r, deg(ki±1 ) = deg  ki ; r n  = 0, deg  hi,n [n]qi  = |n|. Further, call a monomial a finite product of the form product of (r) ′ (x− s i,n )
× product of ′ ki±1 s,  ′
hi,n ′
ki ; r (r) ′ s × product of (x+ s , s, and i,n ) [n]qi n and define its degree to be the sum of the degrees of the factors. Finally, any 32 said to have degree d. It follows from Proposition 6.1 in [1] that the degree is well defined. Note that, by (9), deg(Pi,n ) = |n|. Finally, for n ∈ N, let Un∆ be the subspace of U ∆ consisting of the elements of degree ≤ n. Proceeding by an evident induction on the degree, and using the analogue of 6.1 for Uqres (g) (see Section 1), it suffices to prove the following, for all r, s ∈ N, m, n ∈ Z, i, j ∈ I: i h hj,n + (r) ∆ ∈ Ur+|n|−1 ; (a) [n]q , (xi,m ) h j i h (r) ∆ (b) [n]j,n ∈ Ur+|n|−1 , (x− ; i,m ) q j (c) (r) (s) [(x+ , (x− i,m ) j,n ) ] ∆ ∈ Ur+s−1 , where [a, b] denotes ab − ba. We consider only the case n ≥ 0; the case n ≤ 0 is similar. By applying Tλ̌m , it i suffices to prove (a) when m = 0. Note also that (b) followshfrom (a) by applying i Ω. To prove (a) and (b), it therefore suffices to prove that for all i, j ∈ I, n, r > 0. Now,  hj,n [n]qj (r) ∈ U∆ , (x+ i,0 )  r 1 [naji ]qj X + s−1 + + r−s hj,n + r , (xi,0 ) = (x ) xi,n (xi,0 ) . [n]qj n [n]qj s=1 i,0 Our assertion follows, since it is clear that [naji ]qj [n]qj ∈ C[q, q −1 ], and by 4.6 we have r 1 X + s−1 + + r−s ∈ Uqres (ĝ)+ . (x ) xi,n (xi,0 ) [r]qi ! s=1 i,0 To prove (c), we can of course assume that i = j, so we might as well work in the case g = sl2 and drop the subscripts i, j as usual. We first show that (25) (r) − (s) (x+ Dn (ξ ) ∈ U ∆ 0) for all r, s ∈ N. We proceed by induction on n, the statement being obvious when n = 0. Assuming the result for n, we consider   n+r+1 + (r) + (n+1) − (n+s+1) (n+r+1) − (n+s+1) (26) (x0 ) (x0 ) (x1 ) (x+ = (x1 ) . 0) r q By 5.1 and 5.2, the right-hand side of (26) is in U ∆ , and by 5.1 the left-hand side equals (r) − (x+ Dn+1 (ξ (s)) + w, 0) (r) − where w belongs to U ∆ by the induction hypothesis and part (a). So (x+ Dn+1 (ξ (s) ) 0) ∆ ∈ U , and (25) is proved for n + 1. We now prove (c) by induction on s. As above, it suffices to prove that (27) (r) (s) ∆ [(x+ , (x− m) n) ] ∈ U . We need only consider the case m + n > 0. For, the case m + n < 0 is similar, and 33 that case is contained in (2). If s = 1, then by applying T m , we are reduced to (r) − proving that (x+ xn ∈ U ∆ for all r, n > 0. This follows by taking s = 1 in (25). 0) Assume then that s ≥ 2 and that (27) holds for smaller values of s. By 4.5, − (s) (ξ (s)) = q (n−1)s(s−1) (x− + z, D(n−1)s n) where z is a linear combination of products (s1 ) − (s2 ) (x− (xn2 ) ... , n1 ) where n1 , n2 , . . . and s1 , s2 , . . . are positive integers such that s1 , s2 , . . . < s and n1 s1 + n2 s2 + · · · = ns. Hence,   − (r) − (s) + (r) −(n−1)s(s−1) (s) + (r) [(x+ ) , (x [(x ) ] = q (ξ )] − [(x ) , D ) , z]. m n m m (n−1)s The first term on the right-hand side belongs to U ∆ by (25), and the second term belongs to U ∆ by the induction hypothesis. This completes the proof of 6.1.  Remark The arguments above actually prove the following statement: if r, s ∈ N, (r) − (s) m, n ∈ Z, m + n ≥ 0, then (x+ (xn ) is a linear combination of products m) − 0 + ± res ˆ ± 0 ˆ 2 )0 X X X , where X ∈ Uq (sl2 ) and X lies in the subalgebra of Uqres (sl h i generated by k ±1 , the k;u for all u ∈ Z, v ∈ N, and the PN for N ≥ 0. Analogous v statements in Uqres (ĝ) can be obtained by applying ϕi . 7. Representation Theory of Uǫres We first recall some facts about the representation theory of Uǫres (g) (see [5] and [15]). A representation V of Uǫres (g) is said to be of type I if, for all i ∈ I, ki acts semisimply on V with eigenvalues in ǫZi (ǫi = ǫdi ). Any finite-dimensional irreducible representation of Uǫres (g) can be obtained from a type I representation by tensoring with a one-dimensional representation on which the e± i act as zero and the ki act as ±1. If λ ∈ P , the weight space Vλ of a representation V of Uǫres (g) is defined by     ki ; 0 n0i 1 .v = ni v , Vλ = v ∈ V ki .v = ǫi v, ℓ where ni = λ(α̌i ), n = n0 + ℓn1 , 0 ≤ n0 < ℓ. We have (28) (r) (e± .Vλ ⊆ Vλ±rαi , i ) dim(Vw(λ) ) = dim(Vλ ), for all i ∈ I, r ∈ N, w ∈ W . A vector v in a type I representation V of Uǫres (g) is said to be a highest weight vector if there exists λ ∈ P such that + (r) 34 If, in addition, V = Uǫres (g).v, then V is said to be a highest weight representation with highest weight λ. For any λ ∈ P , there exists, up to isomorphism, a unique irreducible representation V (λ) of Uǫres (g) with highest weight λ. We have M V (λ) = V (λ)µ . µ≤λ Every finite-dimensional irreducible type I representation of Uǫres (g) is isomorphic to V (λ) for some (unique) λ ∈ P + . The following result of Lusztig [15] gives the structure of V (λ) for arbitrary λ ∈ P +. Theorem 7.1. Let λ ∈ P + . (a) If λ(α̌i ) < ℓ for all i ∈ I, then V (λ) is irreducible as a representation of Uǫfin (g). (b) If λ is divisible by ℓ, say λ = ℓλ1 , then V (λ) is isomorphic to the pull back of V (λ1 ) by F rǫ : Uǫres (g) → U (g). In particular, the weight space V (λ)µ 6= 0 only if λ − µ ∈ ℓQ+ and (hence) the e± i act as zero on V (λ) for all i ∈ I. 0 (c) In general, write λ = λ + ℓλ1 , where 0 ≤ λ0 (α̌i ) < ℓ for all i ∈ I. Then, as representations of Uǫres (g), V (λ) ∼ = V (λ0 ) ⊗ V (ℓλ1 ). We now give the analogous definitions for Uǫres (ĝ). Let Uǫres (ĝ)± be the subalge(r) for all i ∈ I, n ∈ Z, r ∈ N, and let Uǫres (g)0 be bras of Uǫres (ĝ) generated by (x± i,n ) h i ki ;0 ±1 the subalgebra generated by the ki , ℓ and Pi,n for all i ∈ I, n ∈ Z. Definition 7.2. A representation V of Uǫres (ĝ) is said to be of type I if V is of type I as a representation of Uǫres (g) and if c1/2 acts as 1 on V . If V is a type I representation of Uǫres (ĝ), a vector v ∈ V is said to be a highest weight vector if there exists a homomorphism of algebras Λ : Uǫres (g)0 → C such that (r) (x± .v = 0 i,n ) for all i ∈ I, n ∈ Z, r ∈ N, and x.v = Λ(x)v for all x ∈ Uǫres (g)0 . If, in addition, V = Uǫres (ĝ).v, then V is said to be a highest weight representation with highest weight Λ. Remarks 1. Since a Uǫres (ĝ)-highest weight vector v ∈ V is, in particular, a Uǫres (g)highest weight vector, there exists λ ∈ P + such that   ki ; 0 n0i = n1i , Λ(ki ) = ǫi , Λ ℓ where n = λ(α̌i ), ni = n0i + ℓn1i , 0 ≤ n0i < ℓ. The highest weight Λ is determined by the ni and the complex numbers {Λ(Pi,r )}i∈I,r∈Z (with Λ(Pi,0 ) = 1). 2. By the relations in 1.2 (and the fact that c1/2 acts as 1 on V ), we have (29) for all µ ∈ P . (r) .Vµ ⊆ Vµ+rαi , (x± i,n ) Uǫres (ĝ)0 .Vµ = Vµ 35 Proposition 7.3. For any algebra homomorphism Λ : Uǫres (ĝ)0 → C, there exists an irreducible representation V (Λ) of Uǫres (ĝ) with highest weight Λ. Moreover, V (Λ) is unique up to isomorphism. Proof. Let M (Λ) be the quotient of Uǫres (ĝ) by the left ideal generated by the x+ i,r for all r ∈ Z, together with the elements x −Λ(x)1 for all x ∈ Uǫres (ĝ)0 . It is obvious that M (Λ) is a highest weight representation of Uǫres (ĝ), with highest weight vector mΛ (say) which is the image of 1 in M (Λ). By 6.1, mΛ is, up to scalar multiples, the unique vector of maximal weight for Uǫres (g). It now follows by the usual arguments that M (Λ) has a unique irreducible quotient representation V (Λ), and that V (Λ) is, up to isomorphism, the unique irreducible representation of Uǫres (ĝ) with highest weight Λ.  The following multiplicative property of highest weights of representations of Uǫres (ĝ) will be of crucial importance later. Proposition 7.4. Let V ′ , V ′′ be type I representations of Uǫres (ĝ) and let v ′ ∈ V ′ , v ′′ ∈ V ′′ be highest weight vectors. Then, v ′ ⊗v ′′ is a highest weight vector in V ′ ⊗V ′′ and (30) Pi± (u).(v ′ ⊗v ′′ ) = Pi± (u).v ′ ⊗ Pi± (u).v ′′ . For the proof, we shall need the following lemma. Recall the algebra homomorˆ 2 ) → Uq (ĝ) (i ∈ I) and define ∆i to be the composite homomorphisms ϕi : Uq (sl phism ⊗ϕi ˆ 2 ) −∆ ˆ 2 ) ⊗ Uq (sl ˆ 2 ) −ϕ−i− −→ Uq (ĝ) ⊗ Uq (ĝ), Uq (sl → Uq (sl ˆ 2 ). Since ϕi (U res (sl ˆ 2 )) ⊂ U res (ĝ) where the first map is the comultiplication of Uq (sl q q and since Uqres (ĝ) is a C[q, q −1 ]-Hopf subalgebra of Uq (ĝ), it follows that ˆ 2 )) ⊂ U res (ĝ) ⊗ U res (ĝ). ∆i (Uqres (sl q q (s) | j ∈ I, m ∈ Z, s ∈ N}. Let X (+) be the C[q, q −1 ]-span of {(x+ j,m ) Lemma 7.5. For all i ∈ I, r ∈ N, (r) (r) ∆((x− ) − ∆i ((x− ) ∈ Uqres (ĝ)X (+) ⊗ Uqres (ĝ) + Uqres (ĝ) ⊗ Uqres (ĝ)X (+) . 1) i,1 ) Assuming this lemma for the moment, we give the Proof of 7.4. By 5.1, if r ≥ 0 and v = v ′ , v ′′ or v ′ ⊗v ′′ , 2 (r) − (r) kir Pi,r .v. (xi,1 ) .v = ǫ−r (x+ i i,0 ) Now, using 1.1 and the isomorphism in 1.2, one finds that − − − + + ∆i (x+ 0 ) = xi,0 ⊗ki + 1⊗xi,0 , ∆i (x1 ) = xi,1 ⊗1 + ki ⊗xi,1 . From this, one easily deduces (by induction on r) that (r) ∆i ((x+ ) 0) = r X s(r−s) ǫi + (r−s) s (s) (x+ ki , i,0 ) ⊗(xi,0 ) s=0 (r) ∆i ((x− )= 1) r X s(r−s) r−s − (s) (r−s) . ki (xi,1 ) ⊗(x− i,1 ) ǫi 36 From 7.5, and the obvious fact that Uqres (ĝ)X (+) ⊗ Uqres (ĝ) + Uqres (ĝ) ⊗ Uqres (ĝ)X (+) , specialised to q = ǫ, annihilates v ′ ⊗ v ′′ , it follows that 2 (r) − (r) (−1)r ǫ−r kir Pi,r .(v ′ ⊗v ′′ ) = ∆((x+ (xi,1 ) ).(v ′ ⊗ v ′′ ) i i,0 ) = r X s(r−s)+t(r−t) (−1)s+t ǫi (s) r−t − (t) ′ r−s s − (r−t) ′′ (x+ (xi,1 ) .v ⊗(x+ ki (xi,1 ) .v . i,0 ) ki i,0 ) s,t=0 Now, by (29), (s) − (t) ′ (x+ i,0 ) (xi,1 ) .v = 0 if s > t, (r−s) − (r−t) ′′ (x+ (xi,1 ) .v = 0 if r − s > r − t. i,0 ) So, by 5.1 again, (−1)r kir Pi,r .(v ′ ⊗v ′′ ) = = r X s=0 r X −2s(r−s)+r2 r−s + (s) − (s) ′ (r−s) − (r−s) ′′ ki (xi,0 ) (xi,1 ) .v ⊗kis (x+ .v (xi,1 ) 0) ǫi −2s(r−s)+r2 ǫi −(r−s)2 r ki Pi,r−s .v ′′ . 2 kir Pi,s .v ′ ⊗(−1)r−s ǫi (−1)s ǫ−s i s=0 Hence, Pi,r .(v ′ ⊗v ′′ ) = r X Pi,s .v ′ ⊗Pi,r−s .v ′′ . s=0 This proves relation (30) for Pi+ . The proof for Pi− is similar. The fact that v ′ ⊗v ′′ is a highest weight vector now follows from (29).  Proof of 7.5. In [8], we showed that (31) + − + ∆(x− i,1 ) − ∆i (x1 ) ∈ Uq (ĝ)Xi ⊗ Uq (ĝ) + Uq (ĝ) ⊗ Uq (ĝ)Xi , where Xi+ is the C(q)-span of {x+ j,m | m ∈ Z, j ∈ I, j 6= i}. We claim that this implies that (32) − r + + r ∆(x− i,1 ) − ∆i (x1 ) ∈ Uq (ĝ)Xi ⊗ Uq (ĝ) + Uq (ĝ) ⊗ Uq (ĝ)Xi for all r ≥ 1. The proof is by induction on r. For the inductive step, we have − r+1 r ∆(x− = ∆(x− i,1 ) i,1 ) ∆(xi,1 ) + + r ∈ ∆i (x− 1 ) + Uq (ĝ)Xi ⊗ Uq (ĝ) + Uq (ĝ) ⊗ Uq (ĝ)Xi

+ + − . ⊗ U (ĝ) + U (ĝ) ⊗ U (ĝ)X + U (ĝ)X ⊗ 1 + k ⊗ x × x− q q q q i i i i,1 i,1 Thus, to prove (32), it is enough to show that
− Uq (ĝ)Xi+ ⊗ Uq (ĝ) + Uq (ĝ) ⊗ Uq (ĝ)Xi+ (x− i,1 ⊗ 1 + ki ⊗ xi,1 ) + + 37 But this is clear, since by the relations in 1.2, + + + Xi+ x− i,1 ⊆ Uq (ĝ)Xi , Xi ki ⊆ Uq (ĝ)Xi . Now, since Uqres (ĝ) is a C[q, q −1 ]-Hopf subalgebra of Uq (ĝ), it is clear that (r) (r) ∆((x− ) − ∆i ((x− ) ∈ Uqres (ĝ) ⊗ Uqres (ĝ). 1) i,1 ) On the other hand, by 6.1, Uqres (ĝ) = Uqres (ĝ)− Uqres (ĝ)0 + Uqres (ĝ)X (+) . Thus, by (32), to prove the lemma, it is enough to show that (33)

Uqres (ĝ)− Uqres (ĝ)0 ⊗ Uqres (ĝ)− Uqres (ĝ)0 ∩ Uq (ĝ)X + ⊗ Uq (ĝ) + Uq (ĝ) ⊗ Uq (ĝ)X + = 0, where X + is the C(q)-span of {x+ j,m | j ∈ I, m ∈ Z}. But (33) is a straightforward consequence of the Poincaré–Birkhoff–Witt basis of Uq (ĝ) given in [1], Proposition 6.1.  The following lemma will be needed in Section 8. Lemma 7.6. Let i ∈ I and let Λ : Uǫres (ĝ)0 → C be an algebra homomorphism such ˆ 2 )).vΛ ⊂ V (Λ) is an irreducible that V (Λ) is finite-dimensional. Then, ϕi (Uǫres (sl i ˆ 2 ) isomorphic to V (Λi ), where Λi = Λ ◦ ϕi . representation of Uǫres (sl i ˆ 2 )-subrepresentation of ϕi (Uǫ (sl ˆ 2 )).vΛ . Proof. Let W be a proper irreducible Uǫres (sl i i λ( α̌ ) Let λ ∈ P + be such that Λ(ki ) = ǫi i . Let w ∈ W be any non-zero vector h ofi res res ˆ ±1 maximal weight for U (sl2 ) (the subalgebra of U (sl2 ) generated by k , k;0 ǫi and the (r) (x± 0) ǫi ℓ for all r ∈ N. Then, by (29), (r) (x+ .w = 0 i,n ) for all n ∈ Z, r ≥ 1. But since the weight space Wλ−η is zero unless η ∈ Q+ is a multiple of αi , we also have (r) .w = 0 for all n ∈ Z, r ≥ 1, j 6= i. (x+ j,n ) (r) for all n ∈ Z, r ≥ 1, j ∈ I is thus The space of vectors in W annihilated by (x+ j,n ) res non-zero, and since it is clearly preserved by Uǫ (ĝ)0 , it contains a Uǫres (ĝ)-highest weight vector. Since V (Λ) is irreducible as a representation of Uǫres (ĝ), this vector must be a multiple of vΛ . Hence, vΛ ∈ W , giving the desired contradiction.  We now give an explicit construction of some highest weight representations of ˆ 2 ). Uǫres (sl Proposition 7.7. For any non-zero a ∈ C(q), there is an algebra homomorphism ˆ 2 ) → U res (sl2 ) such that eva : Uqres (sl q ± ±1 ±1 ∓ −1 ± ± 38 Moreover, we have, for all n ∈ Z, (35) −n n n + −n n − n eva (x+ a k e , eva (x− a e k . n) = q n) = q See [4], Proposition 5.1, for the proof. ˆ 2 )) ⊆ U res (sl2 ), and hence that eva It is obvious that, if a ∈ C, eva (Uqres (sl q ˆ 2 ) → U res (sl2 ). From (34) we induces a homomorphism of algebras eva : Uǫres (sl ǫ deduce that 2 (r) eva ((x+ ) = ǫ−nr anr k nr (e+ )(r) , n) (36) 2 (r) eva ((x− ) = ǫ−nr anr (e− )(r) k nr . n) ˆ 2) If V is a representation of Uǫres (sl2 ), we denote by Va the representation of Uǫres (sl obtained by pulling back V by eva . If V is of type I, then so is Va . If V is a representation of U (sl2 ), we define V a similarly. Proposition 7.8. Let a ∈ C be non-zero, and let n ∈ N. If v is a highest weight vector in V (n), the (n+1)-dimensional irreducible type I representation of Uǫres (sl2 ), ˆ 2 ) with highest weight Λ then V (n)a is a highest weight representation of Uǫres (sl given by  h i  n k; 0 n = , Λ(k) = ǫ , Λ ℓ ǫ ℓ   n r r (37) . Λ(Pr ) = (−1) a |r| ǫ In particular, Pr .v = 0 if |r| > n. Proof. Let 0 6= v ∈ V (n)a be a Uǫres (sl2 )-highest weight vector. By (29), (s) (x+ .v = 0 for all s ∈ N, m ∈ Z. m) To prove the proposition, it therefore suffices to prove that   n r r v for all r ∈ Z. Pr .v = (−1) a |r| ǫ Assume that r > 0. Proceeding as in the proof of 7.4, 2 (r) − (r) (−1)r ǫ−r k r Pr .v = (x+ (x1 ) .v 0) = ǫ−r 2 +nr (e+ )(r) (e− )(r) .v (by (36))   2 k; 0 .v (by (2)) = ǫ−r +nr ar r hni 2 = ǫ−r +nr ar v. r ǫ Hence, r r hni 39 ˆ 2 ) and Finally, applying Ω to both sides of the identity in 5.1 (working in Uqres (sl then specialising), we obtain 2 (r) − (r) (−1)r ǫr k −r P−r .v = (x+ (x0 ) .v. −1 ) Repeating the argument in the preceding paragraph, one finds that P−r .v = (−1)r a−r hni r ǫ v.  Remark From equation (37), it is easy to show that ± Λ(P (u)) = n Y (1 − ǫn+1−2s a±1 u). s=1 8. Classification We begin the classification of the finite-dimensional irreducible representations of Uǫres (ĝ) with Proposition 8.1. Every finite-dimensional irreducible type I representation of Uǫres (ĝ) is highest weight. Proof. Let V be a finite-dimensional irreducible type I representation of Uǫres (ĝ). Since dim(V ) < ∞, there exists a common eigenvector 0 6= v ∈ V for the action of Uǫres (ĝ)0 , which acts by commuting operators on V . In particular, there exists λ ∈ P such that v ∈ Vλ . It follows that V = M Vµ , µ∈P since the right-hand side is non-zero and preserved by Uǫres (ĝ). Again since dim(V ) < ∞, there exists a maximal µ ∈ P , say µ0 , such that Vµ 6= 0. The action of Uǫres (ĝ)0 preserves Vµ0 , so there exists 0 6= v ′ ∈ Vµ0 that is a common eigenvector for Uǫres (ĝ)0 . Then, v ′ is a Uǫres (ĝ)-highest weight vector.  In view of this result, the following theorem completes the classification of the finite-dimensional irreducible type I representations of Uǫres (ĝ). Theorem 8.2. The irreducible representation V (Λ) of Uǫres (ĝ) is finite-dimensional if and only if there exists an I-tuple P = (Pi )i∈I ∈ ΠI such that Λ(ki ) = (38) deg(Pi ) ǫi , Λ(P + (u)) = Pi (u) , Λ  ki ; 0 ℓ  Λ(P − (u)) =  deg(Pi ) = ℓ Qi (u) ,  ǫi , 40 where Qi (u) = udeg(Pi ) Pi (u−1 ). Remarks 1. We shall often abuse notation by denoting the representation V (Λ) determined by P as in the theorem by V (P). 2. It is clear that V (P) ∼ = V (P′ ) as representations of Uǫres (ĝ) if and only if Pi′ is a (non-zero) multiple of Pi , for all i ∈ I. Proof of 8.2. Assume first that dim(V (Λ)) < ∞. Let vΛ be a Uǫres (ĝ)-highest weight vector in V (Λ). Since V (Λ) is, in particular, a finite-dimensional type I representation of Uǫres (g), there exist ni ∈ N such that   ki ; 0 n0i = n1i , Λ(ki ) = ǫi , Λ ℓ where ni = n0i + ℓn1i , 0 ≤ n0i < ℓ. Now, there is a C(q)-algebra homomorphism − to x− Uqi (sl2 ) → Uq (ĝ) that takes e+ to x+ i,1 , and k to cki (this is easily i,−1 , e res checked using 1.2). This map clearly takes Uqi (sl2 ) to Uqres (ĝ) and so induces a C-algebra homomorphism Uǫres (sl2 ) → Uǫres (ĝ). By considering V (Λ) as a represeni tation of Uǫres (sl2 ) via this homomorphism, we deduce that i (r) .vΛ = 0 (x− i,1 ) if r > ni (ni ) (ni ) .vΛ (for si (Λ) = Λ − ni αi .vΛ is a non-zero multiple of (x− and that (x− i,0 ) i,1 ) and by (28) the dimensions of the weight spaces for the action of Uǫres (sl2 ) are i preserved by the action of the Weyl group). This implies that  = 0 if r > ni + (r) − (r) (xi,0 ) (xi,1 ) .vΛ 6= 0 if r = ni . By 5.1, it follows that Pi,r .vΛ and hence that  = 0 if r > ni 6= 0 if r = ni , Pi+ (u).vΛ = Pi (u)vΛ for some polynomial Pi ∈ C[u] of degree ni . It remains to prove that (39) Λ(Pi− (u)) = Qi (u) . Qi (0) We show first that Λ(Pi− (u)) is uniquely determined by Pi (u). By repeating verbatim the proof of Theorem 3.4 in [4] (with q replaced by ǫi ), it follows that Pi (u) ± determines Λ(ψi,n ) for all n ∈ Z according to the formula ∓2 ±deg(Pi ) Pi (ǫi u) Λ(Ψ± i (u)) = ǫi Pi (u) . By 3.1, it suffices to prove that Pi determines Λ(Pi,−nℓ ) for all n > 0. We can assume that nℓ ≤ ni , otherwise Λ(Pi,−nℓ ) = 0 (this follows by an argument similar 41 (nℓ) .vΛ for m ∈ Z. These vectors are all non-zero, for Consider the vectors (x− i,m ) by equation (2) (and an application of Tλ̌i ), we get (nℓ) − (nℓ) (xi,m ) .vΛ (x+ i,−m )   1 ni ki ; 0 vΛ 6= 0. vΛ = = n nℓ  Since dim(V (Λ)) < ∞, there exists a linear relation ′ M X (nℓ) .vΛ = 0, am (x− i,m ) m=M for certain scalars am . We can assume that aM = −1. Then, we have ′ (40) (nℓ) (x− .vΛ = i,M ) M X (nℓ) am (x− .vΛ . i,m ) m=M +1 (nℓ) to both sides of equation (40) for m = −M, −M + 1, . . . , −M ′ + 1 Applying (x+ i,M ) and using 5.1 and the remark at the end of Section 6, we get a system of M ′ − M linear equations for the am with coefficients of the form Λ(Pi,r ) for r ≥ 0. It follows that the am are given by certain rational functions of the coefficients of Pi . (nℓ) to both sides of (40) now shows that Λ(Pi,−nℓ ) is given by Applying (x+ i,−M −1 ) a rational function of the coefficients of Pi . By 7.5, it suffices to prove (39) when g = sl2 . Dropping the subscript i and recalling that Λ(P − (u)) is uniquely determined by P (u), it suffices to verify it in any one irreducible representation V (Λ) with Λ(P + (u)) = P (u). Further, since both sides of (39) are multiplicative on tensor products by 7.4, we can assume that deg(P ) = 1. If P = 1 − au, where a ∈ C× , then by 7.8 we can take V (Λ) = V (1)a and we have Λ(P − (u)) = 1 − a−1 u. Since Q(u) = uP (u−1 ) = u − a, we get Λ(P − (u)) = Q(u) , Q(0) as required. Turning now to the converse, we must show that, if P = (Pi )i∈I ∈ ΠI , and if Λ is determined by P as in the statement of 8.2, the representation V (Λ) is finitedimensional. By 2.1, there exists a finite-dimensional representationP Vq (Λ) of Uq (ĝ) over C(q), with highest weight vector vΛ , such that vΛ has weight i∈I deg(Pi )λi for Uq (g) and Pi (u) vΛ . P + (u).vΛ = Pi (0) Further, since each Pi has coefficients in C, it follows that Uqres (ĝ)0 .vΛ = C[q, q −1 ]vΛ . By 6.1, res − 42 is preserved by the action of Uqres (ĝ): in fact, Wq (Λ) = Uqres (ĝ).vΛ . We shall prove that Wq (Λ) is a C[q, q −1 ]-lattice in Vq (Λ), i.e. that the natural map Wq (Λ) ⊗C[q,q −1 ] C(q) → Vq (Λ) is a C(q)-vector space isomorphism. For this, we note that Wq (Λ) is the direct sum of its weight spaces for Uqres (g): M Wq (Λ) = Wq (Λ)µ . µ∈P Choose a basis of Wq (Λ)µ as a C[q, q −1 ]-module for each µ ∈ P , and let B be the union of these bases. It is clear that B is linearly independent over C(q) (take a linear relation and clear denominators), and clear too that it spans Vq (Λ) over C(q) (because Uqres (ĝ) spans Uq (ĝ) over C(q)). This proves our assertion. Finally, define W (Λ) = Wq (Λ) ⊗C[q,q −1 ] C, via the homomorphism C[q, q −1 ] → C that takes q to ǫ, and denote by wΛ the image of vΛ ∈ Wq (Λ) in W (Λ). Obviously, W (Λ) = Uǫres (ĝ).wΛ and W (Λ) is finitedimensional (though not necessarily irreducible). Let M be a maximal proper res of W (Λ), and set V = W (Λ)/M . Clearly, wΛ has weight U Pǫ (ĝ)-subrepresentation res i∈I deg(Pi )λi for Uǫ (g), so the first two equations in the statement of 8.2 hold, and the third holds because it holds in Vq (Λ). The last equation is proved by the argument used earlier in the proof. Hence, V ∼ = V (Λ), and so V (Λ) is finitedimensional.  The following result is an immediate consequence of 7.4. Proposition 8.3. Let P, P′ ∈ ΠI , and assume that V (P) ⊗ V (P′ ) is irreducible as a representation of Uǫres (ĝ). Then, V (P) ⊗ V (P′ ) ∼ = V (P ⊗ P′ ). Corollary 8.4. Let V, V ′ be finite-dimensional irreducible type I representations of Uǫres (ĝ), and assume that V ⊗ V ′ is irreducible. Then, V ⊗ V ′ ∼ = V ′ ⊗ V as res representations of Uǫ (ĝ). Proof. This follows from 8.1, 8.2 and 8.3.  9. A Factorization Theorem In this section, we prove an analogue for Uǫres (ĝ) of the factorization theorem 7.1 for finite-dimensional irreducible representations of Uǫres (g). If P ∈ Π, let P = P 0P 1 be a factorisation such that (i) P 0 (u) is not divisible by 1 − auℓ for any non-zero a ∈ C; (ii) P 1 (u) = R(uℓ ) for some R ∈ Π. Such a factorisation obviously exists and is unique up to constant multiples. If 43 Theorem 9.1. For any P = (Pi )i∈I ∈ ΠI , we have V (P) ∼ = V (P0 ) ⊗ V (P1 ) as representations of Uǫres (ĝ). To prove this theorem, we shall need the following results describing the structure of the factors V (P0 ) and V (P1 ). Let Uǫfin (ĝ) be the subalgebra of Uǫres (ĝ) generated by Uǫres (ĝ)0 and the x± i,r for all i ∈ I, r ∈ Z. Theorem 9.2. Let P = (Pi )i∈I ∈ ΠI be such that each Pi is not divisible by 1−auℓ for any non-zero a ∈ C. Then, V (P) is irreducible as a representation of Uǫfin (ĝ). Theorem 9.3. Let P = (Pi )i∈I ∈ ΠI , where Pi (u) = Ri (uℓ ) for some R = (Ri )i∈I ∈ ΠI . Then, ∗ V (P) ∼ = F̂rǫ (V (R)), the pull-back of the irreducible representation V (R) of ĝ by the Frobenius homomorphism F̂rǫ : Uǫres (ĝ) → U (ĝ). Remark The representations V (R) were all described explicitly in 2.5, so 9.1 and 9.3 reduce the study of V (P) for all P ∈ ΠI to the case P = P0 to which 9.2 applies. For the latter case, we have an explicit desrciption of V (P) only when g = sl2 . This is given below (see 9.6). Proof of 9.2. We first prove the theorem when g = sl2 . For this, we need Proposition 9.4. Let r ∈ N, let m1 , . . . , mr be positive integers < ℓ, let a1 , . . . , ar ∈ C be non-zero, and let V = V (m1 )a1 ⊗ · · · ⊗ V (mr )ar . ˆ 2 ) if and only if, for all 1 ≤ Then, V is irreducible as a representation of Uǫres (sl s 6= t ≤ r, (41) as 6= ǫ±(ms +mt −2p) for all 0 ≤ p < min(ms , mt ). at ˆ 2 ). Moreover, in this case, V is irreducible as a representation of Uǫfin (sl Proof. We proceed by induction on r, beginning with the case r = 2 for which we change notation and consider V (m)a ⊗ V (n)b . Assume that m ≤ n (by 8.4, this is without loss of generality). The proof here is essentially the same as that in [4], Section 4.8. The crucial point is that, if m < ℓ, the structure of V (m) at an ℓth root of unity is the ‘same’ as its structure for generic q. Namely, V (m) has a basis ˆ 2 ) given by {v0 , v1 , . . . , vm } and action of Uǫres (sl k.vr = ǫm−2r vr , e+ .vr = [m − r + 1]ǫ vr−1 , e− .vr = [r + 1]ǫ vr+1 , with (e± )(ℓ) acting as zero (and vm+1 = v−1 = 0). As in [4], one checks that the only Uǫres (sl2 )-highest weight vectors in V (m) ⊗ V (n) are (scalar multiples of) the vectors p X (−1)r ǫr(n−r+1) [m − p + r]ǫ ![n − r]ǫ !vp−r ⊗ vr , wp = 44 ˆ 2 )0 and is for 0 ≤ p ≤ m, and that, if p > 0, wp is an eigenvector for Uǫres (sl annihilated by x+ r for all r ∈ Z if and only if b = ǫm+n−2p+2 . a (42) (ℓ) Note that (x+ r ) .wp = 0 for all r ∈ Z, since this vector has weight m + n − 2p + 2ℓ, which is > m + n. It follows that, if (42) holds for some 0 < p ≤ m, then wp ˆ 2 )-subrepresentation of V (m)a ⊗ V (n)b . The duality generates a proper Uǫres (sl argument used in [4] then shows that, if (43) b = ǫ−(m+n−2p+2) a for some 0 < p ≤ m, then V (m)a ⊗V (n)b contains a proper Uǫres (sl2 )-subrepresentation containing v0 ⊗ v0 . Conversely, if neither (42) nor (43) holds for any 0 < p ≤ m, the above argument ˆ 2 ) of shows that V (m)a ⊗ V (n)b is irreducible even under the subalgebra Uǫfin (sl res Uǫ (sl2 ), for neither it nor its dual representation contains a non-zero eigenvector of Uǫres (sl2 )0 that is annihilated by x+ r for all r ∈ Z. The proof for r ≥ 2 given in [4], Section 4.8, can now be repeated verbatim (with ˆ 2 ) if condition (41) holds. q replaced by ǫ) to show that V is irreducible under Uǫfin (sl The converse follows from 8.4 and the r = 2 case.  To see that the sl2 case of 9.2 follows from 9.4, define an ǫ-segment of length m, where 0 < m < ℓ, to be a set of the form Sm (a) = {aǫm−1 , aǫm−3 , . . . , aǫ−m+1 }, for some non-zero a ∈ C. Say that two ǫ-segments are in special position if their union is an ǫ-segment longer than both the given segments, and in general position otherwise. It is easy to see that two ǫ-segments Sms (as ) and Smt (at ) are in general position if and only if (41) holds. Moreover, as in [4], Proposition 4.7, one shows that, if P ∈ Π satisfies the conditions in 9.2, the set of roots of P , regarded as a set with multiplicities, can be written uniquely as a union of ǫ-segments in general position. It follows that P can be factorized uniquely as P (u) = ms r Y Y (1 − as ǫms +1−2p ), s=1 p=1 for some r, m1 , . . . , mr ∈ N and non-zero a1 , . . . , ar ∈ C. Hence, by 7.6 (and the remark which follows its proof), 8.3 and 9.4, V (P ) ∼ = V (m1 )a1 ⊗ · · · ⊗ V (mr )ar . ˆ 2 ). By 9.4 again, V (P ) is irreducible under Uǫfin (sl Returning now to the case of arbitrary g, we prove Theorem 9.2 in two steps: Step I: If P is as in the statement of 9.2, then V (P) = Uǫfin (ĝ).vP , where vP is a 45 For this, regard V (P) as a representation of Uǫres (g) and let M V (P) = V (P)λ−η η∈Q+ P be its weight decomposition, where λ = i∈I deg(Pi )λi ∈ P + is the Uǫres (g)-weight of vP . Note that V (P)λ = CvP . We show, by induction on η, that V (P)λ−η ⊆ Uǫfin (ĝ).vP . (44) We claim that this follows from the identities (45) n X −m(n+aij −1) (−1)m ǫi (m) (n−m) − =0 xj,s (x− (x− i,r ) i,r ) m=0 in Uǫres (ĝ), where n ≥ 1 − aij , r, s ∈ Z, and i 6= j. Indeed, (44) is obvious when η = 0. Assuming that (44) is proved for all η < η ′ ∈ Q+ , we prove it for η ′ . By 6.1, V (P)λ−η′ = X x− i,r .V (P)λ−η ′ +αi + X (ℓ) (x− i,r ) .V (P)λ−η ′ +ℓαi . i,r i,r Hence, it suffices to prove that (ℓ) fin (x− i,r ) .V (P)λ−η ′ +ℓαi ⊆ Uǫ (ĝ).vP for all i ∈ I, r ∈ Z. There is nothing to prove unless η ′ ≥ ℓαi , and if η ′ > ℓαi we have X V (P)λ−η′ +ℓαi = x− j,s .V (P)λ−η ′ +ℓαi +αj j,s by the induction hypothesis again. But then, by (45), (ℓ) (x− i,r ) .V (P)λ−η ′ +ℓαi = X − − (ℓ) − (x− i,r ) xj,s .V (P)λ−η ′ +ℓαi +αj ⊆ xi,r .V (P)λ−η ′ +αi +xj,s .V (P)λ−η ′ +αj , j,s and this is contained in Uǫfin (ĝ) by the induction hypothesis once again. We are thus reduced to proving that (ℓ) fin (x− i,r ) .vP ∈ Uǫ (ĝ).vP for all i, r. But this is clear, since by 7.6 and the sl2 case of 9.2 already proved, we have (ℓ) fin ˆ fin (x− i,r ) .vP ∈ ϕi (Uǫi (sl2 )).vP ⊆ Uǫ (ĝ).vP . To complete Step I, we must therefore prove the identities (45). We work in Uqres (ĝ) and then specialise. We proceed by induction on n. If n = 1 − aij , (45) is a special case of one of the defining relations in 1.2. Assuming that (45) holds for n, we multiply − −2n−aij 46 obtained by multiplying both sides of (45) on the right by x− i,r . The coefficient of − (n+1−m) − − (m) (xi,r ) xj,s (xi,r ) on the left-hand side of the resulting identity is o n −m(n+aij ) −2n−aij −(m−1)(n−1+aij ) −m(n−1+aij ) [n+1]qi . [m]qi = (−1)m qi [n − m + 1]qi + qi (−1)m qi After cancelling the factor [n + 1]qi , we thus have the identity (45) for n + 1. This completes Step I. Step II: Let P be as in the statement of 9.2, and let v ∈ V (P) be annihilated by x+ i,r for all i ∈ I, r ∈ Z. Then, v is a multiple of vP . Note that we have already proved this when g = sl2 (in the course of proving 9.2 itself when g = sl2 ). From now on let g be arbitrary and suppose for a contradiction that v ∈ V (P) is annihilated by x+ i,r for all i ∈ I, r ∈ Z and is not a multiple of vP . We can assume that v is a weight vector for Uǫres (g) and that, among vectors with these properties, v has maximal weight for Uǫres (g). By the analogue of (45) with x+ replacing x− (and with n = ℓ), we have − (ℓ) x+ j,s (xi,r ) .v = 0 for all i, j ∈ I, r, s ∈ Z. The maximal property of v implies that, for each i ∈ I, + (ℓ) (ℓ) r ∈ Z, (x+ i,r ) .v is a multiple of vP (possibly zero). We cannot have (xi,r ) .v = 0 for all i ∈ I, r ∈ Z, otherwise v would be a multiple of vP by the irreducibility (ℓ) of V (P). Hence, there is a unique index i ∈ I, say i0 , such that (x+ i,r ) .v is a non-zero multiple of vP . It follows that v has weight λ − ℓαi0 for Uǫres (g), where P ˆ 2 )).vP , which is isomorphic (sl λ = i∈I deg(Pi )λi . By 6.1, v belongs to ϕi0 (Uǫres i0 ˆ 2 ) to V (Pi ) by 7.6. By the sl2 -case of Step II, v is a as a representation of U res (sl ǫi0 0 multiple of vP . This contradiction completes Step II. We can now complete the proof of 9.2. If V (P) is reducible under Uǫfin (ĝ), it has an irreducible Uǫfin (ĝ)-subrepresentation W , say. Any non-zero vector v ∈ W of maximal weight for Uǫres (g) is obviously annihilated by x+ i,r for all i ∈ I, r ∈ Z. By Step II, v is a multiple of vP . This contradicts Step I.  We turn now to the proof of 9.3. We shall need the following lemma. Lemma 9.5. Let i ∈ I, r ∈ Z, n ∈ N. Then, (n) F̂rǫ ((x± )= i,r ) ( n/ℓ (x± i,r ) (n/ℓ)! if ℓ divides n 0 otherwise. ˆ Proof. It suffices to prove that the following diagram commutes for all i ∈ I: F̂r (46) ǫ Uǫres (ĝ) −−−− → U (ĝ)     Ti y yT i Uǫres (ĝ) −−−−→ U (ĝ) 47 Indeed, if Ti and T i are replaced by an automorphism τ ∈ T , the corresponding diagram obviously commutes. It follows that, if (46) commutes, then F̂rǫ ◦ Tλ̌i = T λ̌i ◦ F̂rǫ for all i ∈ I. This clearly implies the lemma, in view of the isomorphism in 1.2. To prove that the diagram (46) commutes, it suffices to prove that the homo(n) of Uǫres (ĝ), morphisms F̂rǫ ◦ Ti and T i ◦ F̂rǫ agree on the generators kj±1 and (e± j ) ˆ n ≥ 1. This is easy for k ±1 . For (e+ )(n) , we have (see Section 1) for all j ∈ I, j j −naij (47) (n) Ti ((e+ ) j ) = X + (−naij −r) + (n) + (r) (−1)r−naij ǫ−r (ej ) (ei ) i (ei ) r=0 if i 6= j (the proof when i = j is similar but easier). Applying F̂rǫ to the r th term in the sum gives zero unless n and r are both divisible by ℓ, say r = ℓr 1 , n = ℓn1 . (n) Hence, F̂rǫ (Ti (e+ ) = 0 unless n is divisible by ℓ, and j ) −n1 aij (ℓn1 ) ) F̂rǫ (Ti (e+ j ) = X 1 r1 =0 −n1 aij = X + −n aij −r (e+ )n (ej )r (e+ i ) (−n1 aij − r 1 )! n1 ! r1 ! 1 1 (−1)ℓr −ℓn aij ǫi−ℓr 1 (−1) r1 =0 1 1 1 + −n aij −r (e+ (e+ )n (ej )r i ) , (−n1 aij − r 1 )! n1 ! r1 ! 1 r1 −n1 aij 1 1 1 since ℓ is odd and ǫℓ = 1. By the classical analogue of (47), this last sum is equal to ! n1 ) (e+ j + (ℓn1 ) = ). Ti T ◦ F̂r ((e i ǫ j ) n1 ! (n) The proof for (e− is similar.  j ) We can now give the Proof of 9.3. We note first that it suffices to prove the theorem when g = sl2 . Indeed, this follows from 7.6 and its classical analogue, and the fact that ˆ 2 )), ˆ 2 ))) = ϕ (U (sl F̂rǫ (ϕi (Uǫres (sl i i which follows from 9.5. We take g = sl2 in the remainder of the proof. By 2.5, if R ∈ Π has factorisation R(u) = Y (1 − at u)rt , t where the at are distinct and the rt are ≥ 1, we have V (R) ∼ = O V (rt )at . 48 Since F̂rǫ is a homomorphism of Hopf algebras, the pull-back O ∗ ∗ (48) F̂rǫ (V (R)) ∼ F̂rǫ (V (rt )at ). = t We claim next that the diagram F̂r ǫ Uǫres (ĝ) −−−− → U (ĝ)   ev  evb y y bℓ (49) Uǫres (g) −−−−→ U (g) Frǫ commutes. To prove this, it suffices to check that evbℓ ◦ F̂rǫ and Frǫ ◦ evb agree on (r) and ki of Uǫres (ĝ), for all r ∈ N, i = 0, 1. This is clear except the generators (e± i ) ± (r) for (e0 ) . We have (r) evbℓ (F̂rǫ ((e± )) 0) = ( = ( r/ℓ evbℓ ((e± ) 0 ) (r/ℓ)! if ℓ divides r 0 otherwise ∓ r/ℓ ) b±r (e(r/ℓ)! if ℓ divides r 0 otherwise, whereas (r) Frǫ (evb ((e± )) = Frǫ (ǫ±r b±r (e∓ )(r) ) 0) ( ∓ r/ℓ ) ǫ±r b±r (e(r/ℓ)! if ℓ divides r, = 0 otherwise. Since ǫ±r = 1 if ℓ divides r, the commutativity of the diagram is proved. It follows from (48), the commutativity of the diagram (49), and 7.1(b) that ∗ F̂rǫ (V (R)) ∼ = O V (ℓrt )a1/ℓ , t t 1/ℓ where at is any ℓth root of at (the argument shows that the isomorphism class of V (ℓrt )a1/ℓ is independent of which ℓth root is chosen). By 7.8, the polynomial t associated to V (ℓrt )a1/ℓ is t ℓrt Y 1/ℓ (1 − ǫℓrt +1−2s at u) = s=1 ℓ Y 1/ℓ (1 − ǫ1−2s at u)rt = (1 − at uℓ )rt . s=1 ∗ By 7.4, F̂rǫ (V (R)) ∼ = V (P ), where P (u) = R(uℓ ).  Proof of 9.1. We show first that, for all i ∈ I, n ∈ Z, ± 0 1 (a) x± i,n acts on V (P )⊗V (P ) as xi,n ⊗ 1; ± (ℓ) 0 1 ± (ℓ) ± (ℓ) 49 For this, we recall the information about the comultiplication ∆ of Uq (ĝ) contained in [1], Proposition 5.3. For i ∈ I, r ∈ N, define − 12 r(r−1) λi,r = (−1)r qi 1 (qi − qi−1 )r [r]qi !, λ∗i,r = qi2 r(r−1) (qi − qi−1 )r [r]qi !, and Ri = ∞ X (r) λi,r Ti (e+ i ) ⊗ (r) , Ti (e− i ) Ri∗ = ∞ X (r) (r) . ⊗ (e− λ∗i,r (e+ i ) i ) r=0 r=0 (We have taken into account the fact that our comultiplication in Uq (ĝ) is opposite to that used in [1].) Note that, since each term in the above sums belongs to Uqres (ĝ), Ri and Ri∗ can be specialised. If w ∈ Ŵ , choose a reduced decomposition w = τ si1 . . . siN , and set (50) (Ri2 )Ri1 ), Rw = τ ⊗2 ((Ti1 Ti2 . . . TiN −1 )⊗2 (RiN ) . . . Ti⊗2 1 (51) ∗ Rw = (Ti−1 Ti−1 . . . Ti−1 )⊗2 (Ri∗1 ) . . . (Ti−1 )⊗2 (Ri∗N −1 )Ri∗N . N −1 2 N N Then, we have [1], for all i ∈ I, n ≥ 0, − − −1 n −1 ∆(x− i,n ) = Rnλ̌ (xi,n ⊗ 1 + c ki ⊗ xi,n )Rnλ̌i , (52) i ∆(x− i,−n ) (53) = (Rn∗ λ̌i )−1 (x− i,−n ∗ ⊗ 1 + c−n ki−1 ⊗ x− i,−n )Rnλ̌i , and the formulas for ∆(x+ i,±n ) can be obtained from (52) and (53) by using + Ω(x− i,∓n ) = xi,±n , ∆ ◦ Ω = Ω⊗2 ◦ ∆op , where ∆op is the opposite comultiplication of Uǫres (ĝ). We claim that Rnλ̌i and Rn∗ λ̌ act as the identity on V (P0 )⊗V (P1 ) for all i n ∈ Z. This clearly implies (a) and (b). Indeed, using 9.3 and setting ℓλ1 = P 1 1 + i∈I deg(Pi )λi , the weight space V (P )ℓλ1 −η is non-zero only if η ∈ Q is divisi(r) ble by ℓ, so (x− acts as zero on V (P1 ) unless r is divisible by ℓ. We deduce from i,n ) − 0 1 (52) and (53) that x− i,n acts on V (P )⊗V (P ) as xi,n ⊗ 1, and from the analogue −1 − (ℓ) (ℓ) (ℓ) acts as (x− ⊗ 1 + 1 ⊗ (x− of (11) for x− that (x− i,n ) i,n ) . The i,n ⊗ 1 + ki ⊗ xi,n ) + case of xi,n is similar. As for the claim, it is easy to see from (50) that Rnλ̌i is a finite product of expressions of the form (54) τ ⊗2 ∞ X (r) (r) λi,r (Ti1 Ti2 . . . TiM −1 )(e+ ⊗ (Ti1 Ti2 . . . TiM −1 )(e− . iM ) iM ) r=0 Now, τ si1 si2 . . . siM −1 (αiM ) is a real root of ĝ, so for any η ∈ ℓQ+ , ⊗2 − (r) 1 1 50 for some root α ∈ Q. By 9.3, the right-hand side is zero unless r is divisible by ℓ. Hence, all the terms in the sum (54) in which r is not divisible by ℓ are zero. But clearly λi,r = 0 if r is strictly positive and divisible by ℓ, so only the r = 0 term in (54) survives. The argument for Rn∗ λ̌ is similar. i With (a) and (b) proved, we can now show that, if vP0 ∈ V (P0 ) and vP1 ∈ V (P1 ) are Uǫres (ĝ)-highest weight vectors, we have V (P) = Uǫres (ĝ).(vP0 ⊗ vP1 ). Since V (P0 ) is irreducible under Uǫfin (ĝ) by 9.3, any w0 ∈ V (P0 ) is a linear combination of vectors of the form − − x− i1 ,m1 xi2 ,m2 . . . xit ,mt .vP0 with t ∈ N, i1 , i2 , . . . , it ∈ I, m1 , m2 , . . . , mt ∈ Z. By (a), − − − − − x− i1 ,m1 xi2 ,m2 . . . xit ,mt .(vP0 ⊗ vP1 ) = (xi1 ,m1 xi2 ,m2 . . . xit ,mt .vP0 ) ⊗ vP1 , hence w0 ⊗ vP1 ∈ Uǫres (ĝ).(vP0 ⊗ vP1 ). It follows that Uǫres (ĝ).(vP0 ⊗ vP1 ) ⊇ V (P0 ) ⊗ vP1 . 1 On the other hand, by 6.1 and the fact that x− i,n acts as zero on V (P ) for all i ∈ I, n ∈ Z, any vector w1 ∈ V (P1 ) is a linear combination of vectors of the form (ℓ) (ℓ) (ℓ) − . . . (x− (x− ju ,nu ) .vP1 , j1 ,n1 ) (xj2 ,n2 ) with u ∈ N, j1 , j2 , . . . , ju ∈ I, n1 , n2 , . . . , nu ∈ Z. We prove by induction on u that (ℓ) (ℓ) (ℓ) − . . . (x− Uǫres (ĝ).(vP0 ⊗ vP1 ) ⊇ V (P0 ) ⊗ (x− ju ,nu ) .vP1 . j1 ,n1 ) (xj2 ,n2 ) If u = 0, there is nothing to prove. Assume the result for u − 1, and let z 0 ∈ V (P0 ). By (b), (ℓ) − (ℓ) (ℓ) z 0 ⊗ (x− . . . (x− j1 ,n1 ) (xj2 ,n2 ) ju ,nu ) .vP1 − (ℓ) (ℓ) (ℓ) 0 . . . (x− = (x− ju ,nu ) .vP1 ) j1 ,n1 ) .(z ⊗ (xj2 ,n2 ) − (ℓ) 0 (ℓ) (ℓ) − ((x− . . . (x− j1 ,n1 ) .z ) ⊗ (xj2 ,n2 ) ju ,nu ) .vP1 . By the induction hypothesis, both terms on the right-hand side of this equation belong to Uǫres (ĝ).(vP0 ⊗ vP1 ), hence so does the left-hand side, completing the inductive step. We have now proved that V (P) is generated by vP0 ⊗ vP1 as a representation of Uǫres (ĝ). If V (P) is reducible, it contains a proper irreducible subrepresentation W , say, and hence, by 8.1, a Uǫres (ĝ)-highest weight vector v 6= 0. We can write v= X vp0 ⊗ vp1 , 51 where vp0 ∈ V (P0 ), vp1 ∈ V (P1 ) and the vp1 are linearly independent. By (a), for all n ∈ Z, X 0 1 0 = x+ .v = (x+ n n .vp ) ⊗ vp , p 0 0 fin so x+ n .vp = 0 for all p. Since V (P ) is irreducible as a representation of Uǫ (ĝ) by 9.3, this implies that each vp0 is a scalar multiple of vP0 . (For, the space of vectors res 0 in V (P0 ) annihilated by x+ i,n for all i ∈ I, n ∈ Z is preserved by Uǫ (ĝ) , hence contains a Uǫres (ĝ)0 -eigenvector.) So v = vP0 ⊗ z 1 for some z 1 ∈ V (P1 ). By (b), for all i ∈ I, n ∈ Z, + (ℓ) 1 (ℓ) 0 = (x+ i,n ) .v = vP0 ⊗ (xi,n ) .z , + 1 (ℓ) 1 so (x+ i,n ) .z = 0. Since xi,n acts as zero on V (P ) for all i ∈ I, n ∈ Z, this forces z 1 to be a multiple of vP1 . But then vP0 ⊗ vP1 ∈ W . In view of the first part of the proof, this contradicts the fact that W is a proper subrepresentation.  We conclude by giving an explicit realization of all the finite-dimensional irreˆ 2 ): ducible type I representations of Uǫres (sl ˆ 2) Theorem 9.6. Every finite-dimensional irreducible type I representation of Uǫres (sl is isomorphic to a tensor product (55) V (m1 )a1 ⊗ · · · ⊗ V (mr )ar ⊗ V (ℓn1 )b1 ⊗ · · · ⊗ V (ℓns )bs , where r, s, 0 ≤ m1 , . . . , mr < ℓ, n1 , . . . , ns ∈ N and a1 , . . . , ar , b1 , . . . , bs ∈ C are non-zero. Two such irreducible tensor products, with parameters r, s, m1 , . . . , mr , n1 , . . . , ns , a1 , . . . , ar , b1 , . . . , bs and r ′ , s′ , m′1 , . . . , m′r′ , n′1 , . . . , n′s′ , a′1 , . . . , a′r′ , b′1 , . . . , b′s′ , respectively, are isomorphic if and only if (i) r = r ′ and s = s′ and (ii) there are permutations ρ of {1, 2, . . . , r} and σ of {1, 2, . . . , s} such that m′t = mρ(t) , a′t = aρ(t) , n′t = nσ(t) , (b′t )ℓ = bℓσ(t) for all t. Conversely, a tensor product (55), where r, s, 0 ≤ m1 , . . . , mr < ℓ, n1 , . . . , ns ∈ N and a1 , . . . , ar , b1 , . . . , bs ∈ C are non-zero, is irreducible if and only if (i) for all 1 ≤ t 6= u ≤ r, at 6= ǫ±(mt +mu −2p) for all 0 ≤ p < min(mt , mu ), and au (ii) bℓ1 , . . . , bℓs are distinct. 52 References 1. Beck, J., Braid group action and quantum affine algebras, Commun. Math. Phys. 165 (1994), 555–568. 2. Beck, J. and Kac, V. G., Finite-dimensional representations of quantum affine algebras at roots of unity, J. Amer. 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Vyjayanthi Chari Department of Mathematics University of California Riverside CA 92521 USA email: [email protected] View publication stats Andrew Pressley Department of Mathematics King’s College Strand London WC2R 2LS UK email:[email protected]