{'fraction_non_alphanumeric': 0.07424154223130762, 'fraction_numerical': 0.017666887784802417, 'mean_word_length': 3.1632192977512257, 'pattern_counts': {'":': 0, '<': 18, '<?xml version=': 0, '>': 20, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 63, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Tensor products, characters, and blocks of finite-dimensional representations of quantum affine algebras at roots of unity

We establish several results concerning tensor products, q-characters, and the block decomposition of the category of finite-dimensional representations of quantum affine algebras in the root of unity setting. In the generic case, a Weyl module is isomorphic to a tensor product of fundamental representations and this isomorphism was essential for establishing the block decomposition theorem. This is no longer true in the root of unity setting. We overcome the lack of such a tool by utilizing results on specialization of modules. Furthermore, we establish a sufficient condition for a Weyl module to be a tensor product of fundamental representations and prove that this condition is also necessary when the underlying simple Lie algebra is sl(2). We also study the braid group invariance of q-characters of fundamental representations.


Introduction
Let g be a finite-dimensional simple Lie algebra over the complex numbers andg = g ⊗ C[t, t −1 ] its associated loop algebra. Following Drinfeld and Jimbo, one can consider the quantum groups U q (g) and U q (g) which are Hopf algebras over the field C(q) of rational functions in the indeterminate q. The latter is most often called a quantum affine algebra. Using C[q, q −1 ]-forms of U q (g) and U q (g), one defines specializations of the quantum groups at q = ξ ∈ C\{0}. If ξ is generic, i.e., not a root of unity, the "general behavior" of the resulting algebras does not depend on ξ or on the chosen form. However, if ξ is a root of unity, the resulting algebras depend drastically on the chosen form, as well as on ξ.
In this paper, we focus on the algebras obtained from Lusztig's form which is the one generated by the divided powers of the quantum Chevalley generators of U q (g) or U q (g). The resulting algebras will be denoted by U ξ (g) and U ξ (g), respectively. We study the category C ξ of (type 1) finite-dimensional representations of U ξ (g) with special attention to the root of unity setting (the order of the root of unity is assumed to be odd and relatively prime to the lacing number of g). We start by studying the concept of Weyl modules which had not been considered in the literature before in the root of unity context. Next, we obtain several results concerning specialization of modules and prove a version of the main result of [8] which gives a sufficient condition for a tensor product of simple objects of C ξ to be a quotient of the appropriate Weyl module. In the generic case, this result combined with results of Beck and Nakajima [5] were used in [12] to prove that the Weyl modules are isomorphic to certain tensor products of fundamental modules. We give an example showing that this is not always the case in the root of unity setting. We proceed by describing the block decomposition of C ξ . Finally, we show that the main result of [12] regarding the braid group invariance of the ℓ-characters of fundamental representations holds in the root of unity context as well.
The block decomposition of the category C ξ was first studied in [22] where it was proved that the blocks are parameterized by the so-called elliptic characters which are elements of the quotient of the ℓ-weight lattice of U ξ (g) by the ℓ-root lattice. However, in that paper ξ was assumed to be a nonzero complex number satisfying |ξ| = 1. The reason behind this assumption was that the original approach of [22] used analytic properties of the action of the R-matrix of U ξ (g) which could be guaranteed only for such ξ. A new approach, in connection with the ℓ-characters (more often known as q-characters), was developed in [12]. This approach was used to describe the blocks of C ξ for all generic ξ. We will mostly follow this later approach to describe the blocks of C ξ in the case that ξ is a root of unity. However, our results concerning tensor products show that one of the main techniques used in the generic context is not available in the root of unity one. Namely, an essential tool for proving that two objects having the same elliptic character must be in the same block was a corollary of [8, Theorem 4.4] stating that a tensor product of fundamental representations can always be reordered in such a way that it is a quotient of the corresponding Weyl module. In fact, as mentioned above, it was shown in [12] that such reordered tensor products are isomorphic to the corresponding Weyl module thus proving that every Weyl module is isomorphic to a tensor product of fundamental representations. The latter statement, as well as the aforemention corollary, is false (see Example 4.25) in the root of unity setting. Because of this, our version of [8, Theorem 4.4] in the root of unity context (Theorem 4.33) requires a somewhat restrictive hypothesis. In any case, even if such hypothesis could be removed (which is the case for g of type A 1 ), the technique of tensor products could not be used as in the generic context.
To overcome this issue, we consider specialization of modules. Namely, we prove that the irreducible quotient of the specialization of an irreducible constituent of a Weyl module is an irreducible constituent of the specialization of that Weyl module. Using this and combinatorial arguments, we are then able to "lift" simple objects having the same elliptic character from the root of unity to the formal q context in such a way that the modules thus obtained have the same elliptic character again. We then use the results of [12] showing that these lifted modules are linked by a certain sequence of Weyl modules. By specializing this sequence back to the root of unity context we are then able to show that the starting two modules must have been in the same block. We remark that this proof also works for ξ = 1 in which case C ξ is the category of finite-dimensionalg-modules. The blocks for ξ = 1 were described in [11] using a different approach (for showing that two objects with the same elliptic character must be in the same block). Hence, our proof is an alternate one to the proof of the block decomposition theorem in the classical context as well. In fact, the proof works for any generic value of ξ and, therefore, it can be thought of as a uniform proof. We note that the elliptic characters were called spectral characters in [11] since their elliptic behavior degenerated when ξ = 1.
The theory of ℓ-characters is one of the most interesting topics of the finite-dimensional representation theory of quantum affine algebras. It was initiated in [26] under the name q-characters. There are several papers dedicated to the study of ℓ-characters of the finite-dimensional irreducible U ξ (g)-modules and other important subclasses of representations such as standard or Weyl modules (see [24,25,27,28,42,43,44] and references therein). The ℓ-characters can be encoded in a ring homomorphism from the Grothendieck ring R ξ of C ξ to the integral group ring Z[P ξ ] of the ℓ-weight lattice. In [42], under the assumption that g is simply laced, Nakajima defined certain polynomials similar to Kazhdan-Lusztig polynomials by studying cohomology of quiver varieties. This lead him to define a function χ ξ,t : called the t-analogue of the ℓ-character ring homomorphism. It turns out that at t = 1 this function specializes to the ℓ-character ring homomorphism. Moreover, the definition of χ ξ,t was axiomatized in a purely combinatorial manner leading to an algorithm for computing the ℓ-characters of the irreducible or of the standard modules. This algorithm was used in [43] to give explicit formulas for the t-analogues of the ℓ-characters of the standard modules when g is of type A or D and ξ is not a root of unity. The formulas were presented in terms of tableaux and a connection with the theory of crystals was discovered. The algorithm was also used with the help of a supercomputer to compute the t-analogues of the ℓ-characters of the fundamental modules when g is of type E in [44] (also for generic ξ). In [27,28], Hernandez proved a conjecture of Nakajima saying that the existence of the function χ ξ,t could be established using only its axiomatic description (without the use of geometry). This allowed him to extend the concept of t-analogues of ℓ-characters for general g. However, due to the lack of a definition of the quiver varieties in general, a and FAPESP. D.J. also thanks Anna Lachowska for motivating discussions at the initial stage of this paper. We thank Hiraku Nakajima for several references, pointing out a mistake in §4.7 in the first draft of the paper, and very kind and helpful communication. 1. The Quantum Loop Algebras at Roots of Unity 1.1. Notation and Basics on Simple Lie Algebras. Throughout, C, Z, Z ≥0 denote the sets of complex numbers, integers, and non-negative integers. For any integer m, Z >m denotes the set of all integers greater than m. Given a ring A, the underlying multiplicative group of units is denoted by A × . The dual of a vector space V is V * . The symbol " ∼ =" denotes isomorphisms. Tensor products of vector spaces are always taken over the underlying field unless otherwise indicated.
Let g be a simple Lie algebra of rank n over the complex numbers with a fixed triangular decomposition g = n + ⊕ h ⊕ n − .
Let I = {1, . . . , n} be an indexing set of the vertices of the Dynkin diagram of g and R + the set of positive roots. The simple roots are denoted by α i , the fundamental weights by ω i , while Q, P, Q + , P + denote the root and weight lattices with corresponding positive cones, respectively. Let also h i be the coroot associated to α i and denote by x ± i a basis element of the root space corresponding to ±α i , i ∈ I. Equip h * with the partial order λ ≤ µ iff µ−λ ∈ Q + . Denote by W the Weyl group of g, by s i the simple reflections, by ℓ(w) the length of w ∈ W, and let w 0 be the longest element of W. Let C = (c ij ) i,j∈I be the Cartan matrix of g, i.e., c ij = α j (h i ), and let D = diag(d i : i ∈ I) where the numbers d i are coprime positive integers such that DC is symmetric. We suppose the nodes of the Dynkin diagram of g are labeled as in Table 1 below and let I • be the indexing set of the black nodes. We shall need the following well-known lemma which can be deduced from the results of [29].
If a is a Lie algebra over C, define its loop algebra to beã = a ⊗ C C[t, t −1 ] with bracket given by [x ⊗ t r , y ⊗ t s ] = [x, y] ⊗ t r+s . Clearly a ⊗ 1 is a subalgebra ofã isomorphic to a and, by abuse of notation, we will continue denoting its elements by x instead of x ⊗ 1. We haveg =ñ − ⊕h ⊕ñ + andh is an abelian subalgebra. The elements x ± i ⊗ t r and h i ⊗ t r will be denoted by x ± i,r and h i,r , respectively.
Let U (a) denote the universal enveloping algebra of a Lie algebra a. Then U (a) is a subalgebra of U (ã). Moreover, if a is a direct sum of two of its subalgebras, say a = b ⊕ c, then multiplication establishes an isomorphism of vector spaces x → −x, and ǫ : a → C, x → 0, can be uniquely extended so that U (a) becomes a Hopf algebra with comultiplication △, antipode S, and counit ǫ. Given a Hopf algebra H, we shall denote by H 0 the augmentation ideal of H, i.e., the kernel of its counit.
For each i ∈ I and r ∈ Z, define elements Λ i,r ∈ U (h) by the following equality of formal power series in the variable u:

Quantum Loop
Algebras. Let C(q) be the ring of rational functions in an indeterminate q and define ] q i . The quantum loop algebra U q (g) of g is the algebra over C(q) with generators x ± i,r (i ∈ I, r ∈ Z), k ±1 i (i ∈ I), h i,r (i ∈ I, r ∈ Z\{0}) and the following defining relations: for all sequences of integers r 1 , . . . , r m , where m = 1 − c ij , S m is the symmetric group on m letters, and ψ ± i,r are determined by equating powers of u in the following equality of formal power series: Denote by U q (ñ ± ) , U q (h) the subalgebras of U q (g) generated by {x ± i,r }, {k ±1 i , h i,s }, respectively. Let U q (g) be the subalgebra generated by x ± i,0 , k ±1 i and define U q (n ± ), U q (h) in the obvious way. Note that ψ ± i,0 = k ±1 i and that U q (g) is the quantum group as defined in [38]. Henceforth, we may denote x ± i,0 by x ± i . Multiplication establishes isomorphisms of C(q)-vectors spaces: We will also consider the subalgebra U q (g i ) generated by k ±1 i , h i,r , x ± i,s for all r, s ∈ Z, r = 0. The algebra U q (g i ) is isomorphic to U q i (sl 2 ). The subalgebra U q (g i ) is defined similarly.
We shall make use of the following proposition whose proof is straightforward.
Let a ∈ C(q) × . Then, there exists a unique C(q)-algebra automorphism ̺ a of U q (g) such that ̺ a is the identity on U q (g) and ̺ a (x ± i,r ) = a r x ± i,r .
Note that where the division is that of formal power series in u with coefficients in U q (g). Although we are denoting the elements x ± i,r , h i,r , Λ i,r of U q (g) by the same symbols as their classical counterparts ing, this will not create confusion as it will be clear from the context. Notice also that U q (h) is generated by k ±1 i and Λ i,r , i ∈ I, r ∈ Z, as a C(q)-algebra.
Let U A (g) be the A-subalgebra of U q (g) generated by the elements (x ± i,r ) (m) and k i for all i ∈ I, r ∈ Z, m ∈ Z ≥0 .
Proof. Let θ be the maximal root of g and write θ = i∈I θ i α i . If g is such that θ i = 1 for some i ∈ I, the proof is essentially given in [4] (the proof is written down completely only in the case g is simply laced but, as pointed out in the proof of [7, Lemma 2.1], the same proof goes through for g of type B or C). In the three remaining cases (E 8 , F 4 , and G 2 ) we have min{θ i : i ∈ I} = 2. Following the reasoning of the proof of [4, Proposition 2.6], part of the argument can be modified to cover these cases by using [37, Lemma 2.7].
. It was proved in [19, Proposition 6.1] that multiplication establishes an isomorphism of A-modules The subalgebras U A (g), U A (n ± ), and U A (h) are defined similarly and the corresponding statements to the ones above are obtained analogously. We note that [ k i ;c m ] ∈ U A (g) and record the following relations (cf. [36]): Given ξ ∈ C × , denote by ǫ ξ the evaluation map A → C sending q to ξ and by C ξ the A-module obtained by pulling-back ǫ ξ . Set for a = g, n ± , h,g,ñ ± ,h. The algebra U ξ (a) is called the restricted specialization of U q (a) at q = ξ. We shall denote an element of the form x ⊗ 1 ∈ U ξ (a) with x ∈ U A (a) simply by x. If ξ is not a root of unity, the algebra U ξ (g) is isomorphic to the algebra given by generators and relations analogous to those of U q (g) with ξ in place of q and its representation theory is parallel to that of U q (g). We shall be particularly interested in the cases ξ = ζ where ζ is a nontrivial root of unity and ξ = 1. In the former case, the algebras U ζ (a) are also known as Lusztig's quantum groups. As for the later: 15,40]). U (g) (resp. U (g)) is isomorphic to the quotient of U 1 (g) (resp. U 1 (g)) by the ideal generated by k i − 1.
Corollary 1.5. The category of U 1 (g)-modules (resp U 1 (g)-modules) on which k i acts as the identity operator for all i ∈ I is equivalent to the category of allg-modules (resp. g-modules).
Henceforth, assume l is an odd integer ≥ 3 and assume l is relatively prime to 3 if g is of type G 2 . Let ζ ∈ C be a primitive l-th root of unity and set ζ i = ζ d i for i ∈ I. The hypotheses on l imply that ζ 2 i is a root of 1 of order l as well. In particular, l is minimal such that [l] ζ i = 0. If g is not of type G 2 , let C ′ be the set consisting of q and all nonzero complex numbers which are not roots of unity of even order. If g is of type G 2 we also exclude from C ′ the roots of unity whose orders are multiples of 3. Henceforth, for simplicity of notation, the expression "root of unity" will pertain only to nontrivial roots of unity and not to 1. Unless stated otherwise, ξ will always denote an arbitrary element of C ′ and we set l = 1 when ξ is not a root of unity for notational convenience. An element of C ′ which is not a root of unity will be referred to as a generic element (so 1 and q are generic). If ξ ∈ C ′ satisfies ξ k = 1 for some k ∈ Z >0 , then ξ is said to have finite order. Otherwise, ξ has infinite order. Thus, 1 is the only generic element of finite order.
Remark. Although several of the results that follow can be proved for even roots of unity as well, some of the proofs (and even the precise statements) require many more technicalities. For simplicity and uniformity, we shall keep the above made assumptions on ξ.
It is well known that U ξ (g) is a Hopf algebra. However, if ξ = 1, an expression for the comultiplication ∆ of U ξ (g) in terms of the generators x ± i,r , h i,r , k ±1 i is not known (the definition is given in terms of the Chevalley generators which we do not need to consider here). The following partial information obtained in [3,4,17,21] suffices for our purposes (see also Lemma 7.5 of [19]). First we recall that Let X ± be the C(ξ)-span of the elements x ± i,r for all i ∈ I, r ∈ Z. Similarly, given i ∈ I, let X ± (i) be the C(ξ)-span of the elements x ± j,r for all j ∈ I\{i}, r ∈ Z.
Proposition 1.6. The restriction of ∆ to U ξ (g i ) satisfies: for every r ∈ Z >0 .
If V is a representation of a Hopf algebra H, its dual space V * is equipped with the structure of a representation by using the antipode. We shall not need the precise expression for the antipode of U q (g).
Given i ∈ I and s ∈ Z, define Lemma 1.7. For every i ∈ I and s ∈ Z, we have Here (X − i,s (u)) (k−l) is understood to be zero if k < l and the subindex k on the right hand side means the coefficient of u k in the given power series. Also, Proof. For s ∈ {0, −1} this was proved in [19, §5]. The general case follows by applying the algebra automorphism determined by the assignment ( The proofs of the next two lemmas can be found in [38]. Lemma 1.9. The following identities hold in U ζ (g): (c) k 2l i = 1 and k l i is in the center of U ζ (g).
In particular, U ζ (g) is generated by x ± i,r and (x ± i,r ) (l) with i ∈ I, r ∈ Z, and U ζ (h) is generated by the elements k i , k i l , Λ i,r with i ∈ I, r ∈ Z.
} as a C-algebra (with identity).
Let U fin ξ (g) be the subalgebra of U ξ (g) generated by U ξ (h) and x ± i,r , i ∈ I, r ∈ Z. Define U fin ξ (g) similarly. If ξ is not a root of unity, U fin ξ (g) = U ξ (g) and U fin ξ (g) = U ξ (g), but for ξ = ζ they are proper subalgebras.
Remark. The algebras U fin ζ (g) and U fin ζ (g) are enlargements (in the Cartan part only) of the so-called small quantum groups and small quantum affine algebra (see [25, §2.4] for instance).
1.4. Braid group action and some identities. In this section we state some identities that will be needed later. Given a C(q)-associative algebra A, x, y ∈ A, and p ∈ C(q), define When p = 1 this coincides with the usual commutator of associative algebras and the subindex 1 will be omitted.
The braid group B associated to g is the multiplicative group generated by elements T i , i ∈ I, with defining relations: if c ij c ji = 1, The following proposition is well known.
Proposition 1.11. Let w = s i 1 · · · s i l be a reduced expression of w ∈ W . The assignment does not depend on the choice of a reduced expression for w.
Theorem 1.12. For every i ∈ I, there exists a unique algebra automorphism T i of U q (g) such that , and, for j = i, Moreover, the above defines a representation of B on U q (g), T i commutes with Ω, and T −1 Given i, j ∈ I such that c ji = −1, define the following elements of U q (g): Remark. Despite the ambiguity of the notation X ± α i +α j in the case c ij = c ji , this will not create confusion further on. Observe also that if c ji = −1 and c ij < −1, then d i = 1 and, hence, q i = q.
The next lemma is proved by a straightforward but tedious computation (cf.[39, §5]). Lemma 1.13. Let i, j ∈ I be such that c ji = −1. , Consider also the C(q)-algebra isomorphism Φ : Definition 1.14. Let i, j ∈ I be such that c ji = −1 and let R + ij be the set of positive roots of the subalgebra of g generated by x ± i and x ± j . For α ∈ R + ij which is not a simple root, let x ± α be the element on the left-hand side of the equalities in Lemma 1.13 which corresponds to X ± α . For α = α i , i ∈ I, set The first set of formulas in the next Lemma is obtained from [39, §5.3 and §5.4] by applying the anti-automorphism Ω while the second is obtained by applying Φ. Lemma 1.15. Let i, j ∈ I be such that c ji = −1. Then, The following is a particular case of [39, Proposition 5.7].
Theorem 1.16. Let g have rank 2 and fix an ordering on R + . Then, the corresponding ordered products of the elements (x ± α ) (k) , α ∈ R + , k ∈ Z >0 , form an A-basis of U A (n ± ).
Corollary 1.17. Let g have rank 2 and fix an ordering on R + . Then, the corresponding ordered products of the elements (x − α ) (k) , α ∈ R + , k ∈ Z >0 , form an A-basis of U A (n − ).
The proof the next lemma is straightforward.
Lemma 1.18. Let i, j ∈ I be distinct, r, s ∈ Z and U rs ij be the subalgebra of U q (g) generated by x − i,r and x − j,s . Then, the assignments It follows from the previous lemmas that (γ r,s α ) (k) , (γ r,s α ) (k) ∈ U A (ñ − ) for all α ∈ R + ij and all r, s ∈ Z. Moreover, by Lemma 1.13 and Theorem 1.16, Similar formulas for γ r,s α are obtained by replacing q with q −1 in (1.14).
Lemma 1.19. Let i, j ∈ I be such that c ji = −1 and r, s ∈ Z. Then, Proof. Let us first record the following identity which holds for any i ∈ I: This and the first identity of the lemma are immediate from the defining relation As for the second identity we proceed as follows. By the first identity and (1.14), we have 2α i +α j . In the second equality above we used (1.15), while the first identity was used once more in the third equality. The last equality holds since γ r,s The last identity is proved similarly and we omit the details.
Remark. The elements γ r,s α were first considered in [20] for simply laced g and in [7] for g of classical type. We warn the reader that there is a couple of typos in [7] regarding γ r,s 2α i +α j (denoted by γ ′ r,s (q) there): one in its definition and the other in the identity corresponding to the third identity of the above lemma.

Braid Group and ℓ-Lattices
2.1. The ℓ-Lattices. Given a field F consider the multiplicative group P F of n-tuples of rational functions µ = (µ 1 (u), · · · , µ n (u)) with values in F such that µ i (0) = 1 for all i ∈ I. We shall often think of µ i (u) as a formal power series in u with coefficients in F. Given a ∈ F × and i ∈ I, let ω i,a be defined by Clearly, if F is algebraically closed, P F is the free abelian group generated by these elements which are called fundamental ℓ-weights. It is also convenient to introduce elements ω λ,a , λ ∈ P, a ∈ F × , defined by If F is algebraically closed, introduce the group homomorphism (weight map) wt : P F → P by setting wt(ω i,a ) = ω i , where ω i is the i-th fundamental weight of g. Otherwise, let K be an algebraically closed extension of F and regard P F as a subgroup of P K . Then, define the weight map on P K as above and on P F by restriction (this clearly does not depend on the choice of K). Define the ℓ-weight lattice of U q (g) to be P q := P C(q) . The submonoid P + q of P q consisting of n-tuples of polynomials is called the set of dominant ℓ-weights of U q (g).
We will also use the notation λ + = λ. Two elements λ, µ of P + q are said to be relatively prime if λ i (u) is relatively prime to µ j (u) in C(q) [u] for all i, j ∈ I. Every ν ∈ P q can be uniquely written in the form where the division is that of formal power series in u.
The next proposition is easily checked.
Let P be the subgroup of P q generated by ω i,a for all i ∈ I and all a ∈ C × (equivalently, P = P C ). Set also P + = P ∩ P + q . From now on we will identify P q with its image in (U q (h)) * under Ψ. Similarly, given ξ ∈ C × , P can be identified with a subset of U ξ (h) * by using the same expression for Ψ ν (Λ i,±r ) and Observe also that every element λ ∈ P q such that λ i (u) splits in C(q)[u] for all i ∈ I can be uniquely decomposed as For notational convenience, henceforth we set P ξ = P when ξ = q. It will be convenient to introduce the following notation. Given i ∈ I, a ∈ C(ξ) × , r ∈ Z ≥0 , define Even though the dependence on ξ is missing in the notation above, this will not create a confusion. Suppose ξ has infinite order and observe that if f (u) ∈ C(ξ)[u] splits in C(ξ)[u] and f (0) = 1, then there exist unique m ∈ Z ≥0 , a 1 , . . . , a m ∈ C(ξ) × , and r 1 , . . . , r m ∈ Z ≥1 such that In particular, given λ ∈ P + ξ such that The factorization (2.6) is called the ξ i -factorization of f and the factors f i,a k ,r k are called the ξ i -factors of f . Two polynomials are said to be in ξ i -general position if the set of ξ i -factors of their product is the union of the corresponding sets of ξ i -factors (counted with multiplicities).
for all k, j and all 0 ≤ p < r k .
We shall also define the notion of weak ξ i -resonant order by asking that for all k, j and all 0 ≤ p < min{r k , s j }.
One also defines the concept of general position for a finite collection of polynomials by requiring that they are pairwise in general position.
Remark. If m 1 = m 2 = 1, then either (f, g) or (g, f ) is in ξ i -resonant order. In general, both (f, g) and (g, f ) may not be in weak ξ i -resonant order. Also, given a finite collection of polynomials f 1 , · · · , f m whose ξ i -factorizations have only one ξ i -factor, then they can be arranged to be in ξ i -resonant order. The concept of resonant order was introduced in [8], but with different terminology. Namely, the polynomial g was said to be in general position with respect to f if (2.8) is satisfied. The terminology "resonant" was used in [22]. Notice that f and g are in general position iff both (f, g) and (g, f ) are in weak ξ i -resonant order.
We also denote by φ l the induced group homomorphism P q → P q . Observe that, if a, b ∈ C(q) are such that b l = a, then and that every λ ∈ P + admits a unique decomposition of the form Set P + l = {λ ∈ P + : λ = λ ′ }. Denote by P + A be the subset of P q consisting of n-tuples of polynomials with coefficients in A and P A the corresponding subgroup. Let P ++ A be the submonoid of P + A consisting of n-tuples of polynomials with invertible leading coefficients and P s A = {λ ∈ P + q : the roots of λ i (u) lie in A × for all i ∈ I} ⊆ P ++ A . We may also use the terminology λ splits in A × to mean λ ∈ P s A . Notice that, if λ ∈ P ++ A , then λ i (u) splits in A for all i ∈ I iff λ splits in A × .
Given f ∈ A(u), letf be the element of C(u) obtained from f by evaluating q at a given ξ ∈ C × . Similarly, given λ ∈ P A , let λ be the element of P obtained from λ by evaluating q at ξ.
For the remainder of the section we assume ξ = ζ. In this case we have It is now easy to see that, for every i ∈ I and f (u) ∈ C[u] satisfying f (0) = 1, there exist unique m 0 , m 1 ∈ Z ≥0 , a 1 , . . . , a m 0 , b 1 , . . . , b m 1 ∈ C × , and r 1 , . . . , r m 0 , s 1 , . . . , s m 1 ∈ Z >0 , such that In particular, given λ ∈ P + , there exist unique m, The factorization (2.13) is called the ζ i -factorization of f . The factors f i,a k ,r k are called the quantum ζ i -factors of f while the factors (1 − b k u l ) s k are called the Frobenius ζ i -factors. We will refer to both kinds of factors simply by the ζ i -factors of f . Two polynomials are said to be in ζ i -general position if the ζ i -factors of their product are the union of their ζ i -factors (counted with multiplicities). Let j for all k, j and all 0 ≤ p < r k .
The concepts of general position and (weak) ζ i -resonant order for a finite ordered collection of polynomials is defined in the obvious way.

2.2.
Braid group action on the ℓ-weight lattices. The following proposition was proved in [6,8].
Proposition 2.2. The following formulas define an action of B on P q : In particular, wt(T w µ) = wwt(µ) for all w ∈ W. Moreover, T i µ ∈ P A for µ ∈ P A .
By composing the above action with the evaluation map ǫ ξ , ξ ∈ C × , one gets an action of B on P. Hence, if µ ∈ P, there is an ambiguity in the notation T i µ. It will always be clear from the context if we mean the "q-action" or the "ξ-action" for some ξ ∈ C × . Lemma 2.3. Suppose i ∈ I and w ∈ W are such that ℓ(s i w) = ℓ(w) + 1. Then, (T w λ) i (u) ∈ C(ξ) [u] for every λ ∈ P + ξ .
Proof. It follows from [8, Proposition 3.2]. Alternatively, see the the proof of Corollary 4.8 below.
Recall that w 0 defines a Dynkin diagram automorphism such that w 0 · i = j iff w 0 ω i = −ω j for i, j ∈ I. Given λ ∈ P + q , denote by λ * any of the three elements defined by the corresponding action of B given by the following expression A straightforward but tedious computation working with a reduced expression for w 0 shows that where h ∨ is the dual Coxeter number of g and r ∨ = max{c ij c ji : i, j ∈ I, i = j} is the lacing number of g.
Fix a reduced expression for w 0 , say The concept of ξ-resonant order for m-tuples of elements of P + ξ is defined in the obvious way.
Remark. If ξ is not a root of unity, condition (a) above follows from condition (b) (see the proof of [8, Theorem 4.4]).
Remark. Because of the closing remark of the previous subsection, there is no "sufficiently" general analogue of Lemma 2.4(b) in the root of unity context. The lack of a root of unity analogue of part (a) of Lemma 2.4 and, consequently also of part (b), will imply the corresponding lack of a root of unity analogue of Corollary 4.23 below.
We now use the braid group action to define the ℓ-analogue of simple roots. These objects were defined originally in [26,25] (without using the braid group) and in [12]. Given a ∈ C(ξ) × , define the simple ℓ-root α i,a by The subgroup of P ξ generated by the simple ℓ-roots α i,a , i ∈ I, a ∈ C(ξ) × , is called the ℓ-root lattice of U ξ (g) and will be denoted by Q ξ . Let also Q + ξ be the submonoid generated by the simple ℓ-roots. Quite clearly wt(α i,a ) = α i . Define a partial order on P ξ by One easily checks the following proposition (cf. [12, §3.3]).
Proposition 2.5. The action of the braid group on P ξ preserves Q ξ .
2.3. The group of elliptic characters. This subsection will be used only in §4. 6. Set The elements of the group Γ ξ are called elliptic characters. The motivation for this name will be explained in the remark after Proposition 2.6 below. We denote by γ ξ the canonical projection P ξ → Γ ξ .
Remark. We warn the reader that there were a couple of typos in [12, §A.2] in the definition of the sets corresponding to our T k . They have been corrected above. Proposition 2.6. Assume g is not of type D n with n even. Then, the group Γ ξ is isomorphic to the (additive) abelian group with generators {χ a : a ∈ C(ξ) × } and relations j∈T k for all a ∈ C(ξ) × and k = 1, 2, 3. If g is of type D n with n even, then Γ ξ is isomorphic to the (additive) abelian group with generators {χ ± a : a ∈ C(ξ) × } and relations Proof. The proposition was proved in [12] for ξ = q and the proof is analogous for the other cases. The isomorphism is given by sending the image of ω i,a in Γ ξ to the generator χ a for i ∈ I • (in the case I • has two elements, the image of ω n−1,a is sent to χ − a while the image of ω n,a is sent to χ + a ).
Remark. Observe that the relations above imply χ a = χ aξ 2r ∨ h ∨ for every a ∈ C(ξ) × (and similarly for χ ± a ). This is the motivation for the terminology elliptic characters. Namely, if ξ ∈ C × is not a root of unity, the parametrizing set for the distinct generators of the form χ a is the elliptic curve C × /ξ 2r ∨ h ∨ Z . For ξ = 1, the elliptic behavior degenerates and the elements of Γ 1 are usually called spectral characters. Notice that Γ 1 is isomorphic to the group of functions with finite support from C × to P/Q (cf. [11]). If ξ = ζ, the elliptic curve parametrizing the distinct generators χ a depends on the greatest common divisor of 2r ∨ h ∨ and the order l of ζ. For instance, let g = sl 2 so that l is relatively prime to 2r ∨ h ∨ = 4. One can check that we have χ a = χ aζ m for any m ∈ Z and, hence, the curve is C × /ζ Z . Moreover, we also have 2χ a = 0 for every a ∈ C × . In particular, Γ ζ is a quotient of Γ 1 in this case. On the other hand, if g = sl 3 and l = 3 = r ∨ h ∨ , the curve remains the same as in the generic case.
We record the following lemma proved "in between the lines" in [12,22] for ξ of infinite order. For ξ ∈ {ζ, 1} the proof is analogous. Given i ∈ I, let P ξ,i be the subgroup of P ξ generated by ω i,a with a ∈ C(ξ) × and P + ξ,i = P + ξ ∩ P ξ,i . Similarly one defines P ξ,J and P + ξ,J for any J ⊆ I.

The Irreducible and Highest-Weight Representations
3.1. Weight Modules. We now review the finite-dimensional representation theory of U ξ (g). We include the sketch of some of the proofs which we believe to be relevant for the development of the paper. The omitted details on the material of this subsection can be found in [2,15,38,40].
Let V be a U ξ (g)-module. Given µ ∈ P define If ξ has infinite order, the second condition in the definition of V µ is redundant. The space V µ is said to be the weight space of V of weight µ and the nonzero vectors of µ are called weight vectors of weight µ. The module V is said to be a weight module (of type 1) if It is well-known that every finite-dimensional U ξ (g)-module is isomorphic to the tensor product of a weight module of type 1 with a one-dimensional representation. We shall only consider representations of type 1 with finite-dimensional weight spaces and will refer to them simply as weight modules. The abelian tensor category of finite-dimensional U ξ (g)-weight modules will be denoted by C ξ .
Remark. Let V be a U ζ (g)-weight module and note that k l i acts as the identity operator on V for all i ∈ I, and that ζ µ(h i ) i ∈ {ζ j i : j = 0, 1, ..., l − 1} for every i ∈ I and µ ∈ P . Also, if v ∈ V µ and c ∈ Z, Observe that the commutation relations (1.7) imply that Quite clearly, every finite-dimensional U ξ (g)-module is integrable.
If V is a weight module, define the character of V to be the element char(V ) of the integral group ring Z[P ] given by Here, e µ , µ ∈ P , denote the basis elements of Z[P ] so that e µ e ν = e µ+ν . The action of W on P can be naturally extended to an action of W on Z[P ].
Proof. Since V is integrable, the formula Proceeding as in [40, Chapter 5] one proves that T i is a linear automorphism of V .
If v ∈ V is a weight vector such that U ξ (n + ) 0 v = 0, then v is called a highest-weight vector. If V is generated by a highest-weight vector of weight λ, then V is said to be a highest-weight module of highest weight λ. The concept of lowest-weight module is defined similarly by replacing U ξ (n + ) 0 with U ξ (n − ) 0 . If ξ is not a root of unity, the condition U ξ (n + ) Lemma 1.9). The next proposition is standard. Proposition 3.3. Let V be a highest-weight module. Then, V has a unique maximal proper submodule and, hence, a unique irreducible quotient. In particular, V is indecomposable.
Definition 3.4. Given λ ∈ P + , let W ξ (λ) be the universal U ξ (g)-module with respect to the following property: W ξ (λ) is generated by a highest-weight vector v of weight λ satisfying (x − i ) (m) v = 0 for all i ∈ I and m > λ(h i ). The module W ξ (λ) is called the Weyl module of highest weight λ. The unique irreducible quotient of W ξ (λ) is denoted by V ξ (λ).
Remark. If ξ is not a root of unity, the condition ( For every λ ∈ P + , W ξ (λ) is an integrable U ξ (g)-module. Furthermore, it is the universal finite-dimensional highest-weight U ξ (g)-module of highest weight λ.
Proof. Quite clearly the weight spaces of W ξ (λ) are finite-dimensional and integrability follows easily from the quantum Serre's relations and the condition (x − i ) (m) v = 0 for all i ∈ I and m > λ(h i ). Then, by Proposition 3.2, the set of weights of W ξ (λ) is invariant under the Weyl group action. In particular, W ζ (λ) µ = 0 only if µ ∈ P (λ) (cf. Lemma 1.1) which implies that W ζ (λ) is finite-dimensional.
On the other hand, let V be a finite-dimensional highest-weight U ξ (g)-module of highest weight λ.
It is quite easy to see that any finite-dimensional irreducible U ξ (g)-module (of type 1) is highestweight. Moreover, since every element of P is conjugate to a dominant weight, it follows from Proposition 3.2 that the highest weight must be in P + . This proves: Corollary 3.6. Every simple object from C ξ is isomorphic to V ξ (λ) for some λ ∈ P + .
The proof of the next theorem is standard.
Theorem 3.7. The irreducible constituents of V ∈ C ξ (counted with multiplicities) are completely determined by char(V ).
The following result is now immediate from Lemma 1.1 and the definition of dual representation.
Corollary 3.8. For every λ ∈ P + , V ξ (λ) is a lowest-weight module of lowest weight w 0 λ. In particular, We also record the following theorem.
Remark. Due to Corollary 1.5, the above results for U 1 (g) recover the basic results on the finitedimensional representation theory of g. We shall use the notation V (λ) for the g-module corresponding to V 1 (λ).
Theorem 3.10. Suppose ξ is not a root of unity and let V be an object from C ξ . Then, V is completely reducible. In particular, V ξ (λ) ∼ = W ξ (λ) for every λ ∈ P + .
If ξ = ζ, Theorem 3.10 is false and the category C ζ is not semisimple. For a brief overview of the character theory in this case, see [15, §11.2].

ℓ-Weight
Modules. This section is based on [15, 19, 25] and we refer to these papers for the omitted details.
Let V be a U ξ (g)-module. A nonzero vector v ∈ V is said to be an ℓ-weight vector if there exists λ ∈ P ξ and k ∈ Z >0 such that (η − Ψ λ (η)) k v = 0 for all η ∈ U ξ (h). In that case, λ is said to be the ℓ-weight of v. Let V λ denote the subspace spanned by all ℓ-weight vectors of ℓ-weight λ. V is said to be an ℓ-weight module if V = L µ ∈ P ξ V µ . Denote by C ξ the category of all finite-dimensional U ξ (g)-ℓ-weight modules. Quite clearly C ξ is an abelian category. Observe that if V is an ℓ-weight module, then V is also a U ξ (g)-weight module and The following proposition was proved in [26,25].
Remark. Note that not every finite-dimensional U q (g)-module lying in C q is an ℓ-weight module since C(q) is not algebraically closed. Instead, they are quasi-ℓ-weight modules in a sense similar to that developed in [32] for hyper loop algebras. We shall not need this concept here.
An ℓ-weight vector v of ℓ-weight λ is said to be a highest-ℓ-weight vector if ηv = Ψ λ (η)v for every η ∈ U ξ (h) and U ξ (ñ + ) 0 v = 0. A module V is said to be a highest-ℓ-weight module if it is generated by a highest-ℓ-weight vector. The notion of a lowest-ℓ-weight module is defined similarly. If V is a highest-ℓ-weight module of highest ℓ-weight λ, then (1.7) implies (3.5) dim(V wt(λ) ) = 1 and The next proposition is easily established using (3.5).
Proposition 3.13. If V is a highest-ℓ-weight module, then it has a unique proper submodule and, hence, a unique irreducible quotient. In particular, V is indecomposable.
Proof. The latter statement is clear from the former and Proposition 3.14. The proof for ξ = ζ and r ∨ ≤ 2 is similar. The case g of type G 2 seems not to have been considered in the literature for ξ = 1 (however, see [5] for the connection with the so-called extremal weight modules). Below we review the main parts of the proof in general and provide the extra technical details for type G 2 . In between the lines, we make minor corrections to the argument used in [7].
Set V = W ξ (λ) and λ = wt(λ) and let v be a highest-ℓ-weight vector of V . Proceeding as in the previous subsection, one easily proves that V is an integrable U ξ (g)-module and, hence, has finitely many weight spaces. It remains to show that all weight spaces are finite-dimensional. The proof is similar to that of [20, Proposition 4.4]. Namely, we show by induction on the height of η ∈ Q + that there exists N (η) ∈ Z ≥ 0 such that V λ−η is spanned by elements of the form (x − i 1 ,s 1 ) (k 1 ) · · · (x − im,sm ) (km) v with j k j α i j = η and |s j | ≤ N (η) for all j = 1, . . . , m. Clearly such vectors without the restriction on s j span V λ−η . Moreover, by Lemma 1.9, we can assume k j ∈ {1, l}.
The proof of this in the case η = kα i for some i ∈ I and some k ∈ Z ≥0 is similar to that given in [20, Proposition 4.4] or rather to that given in [31,Theorem 3.11] (in the context of hyper loop algebras) and we omit the details. In particular, this shows the induction on the height of η starts. We remark that the proof of this particular case makes use of Lemma 1.7. In [20, Proposition 4.4] only the special cases of the lemma mentioned in our proof above was used alongside multiplication by elements h i,s to shift the parameter s. In the root of unity case this multiplication has to be replaced by the more general version of Lemma 1.7 as stated here because h i,s may be zero (see Lemma 1.10). Now suppose the height of η is larger that 1 and consider a vector of the form (x − i 1 ,s 1 ) (k 1 ) · · · (x − im,sm ) (km) v with k 1 > 0 and j>1 k j > 0. By the induction hypothesis we can assume that |s j | ≤ N (η − α i 1 ) for j > 1. We will show that all such vectors are in the span of vectors as above with |s j | ≤ max{N (η − α i ) : i ∈ I} + r ∨ := N (η). If i 1 = i 2 this easily follows from [20, Lemma 4.3].
For i 1 = i 2 we perform a sub-induction on N (η) − |s 1 |. Assume g is of type G 2 and c i 1 ,i 2 = −3 (the other cases are similar and simpler). Fix the order on R + and suppose that s 1 ≥ N (η). Then, by Theorem 1.16, (1.14), and Lemma 1.19, there exist f, g : Notice that, since l = 3 and k 1 , k 2 ≤ l, we have r < l above and, hence, One now easily uses the induction hypotheses to complete the proof in this case. The case s 1 ≤ −N (η) is dealt with similarly. Namely, one first uses Corollary 1.16 to write for some function f as before and then uses (1.14) and Lemma 1.19 as before (this time Lemma 1.19 will cause s 1 to raise).

3.3.
The ℓ-characters. Let Z[P ξ ] be the integral group ring over P ξ . If V is an ℓ-weight module, the ℓ-character of V is defined to be the following element of Z[P ξ ]: The valuation map ǫ ξ induces a ring homomorphism Z[P q ] → Z[P] also denoted by ǫ ξ .
Remark. The ℓ-characters were originally defined in [26,25] where they were called q-characters and ζ-characters, respectively. To keep a more uniform terminology, we prefer to call them ℓ-characters as in [12] which stresses the fact that they are generating functions for the dimensions of the ℓ-weight spaces.
The proof of the following theorem is analogous to that of Theorem 3.7.
Theorem 3.20. Let V ∈ C ξ . The irreducible constituents (counted with multiplicities) of V are completely determined by char ℓ (V ).
It turns out that the ℓ-characters of objects from C ξ are not invariant with respect to the braid group action on P ξ in general. In fact, the theory of ℓ-characters is much more intricate than that of characters and it is not yet completely understood unless ξ = 1. Even the following basic theorem proved in [24,25] (for ξ = 1) is not simple to be proved (see also [12] for a different approach when g is of classical type). For ξ = 1 this theorem follows easily from Theorem 4.10 below (see also [11]). Theorem 3.21. Let V be a finite-dimensional highest-ℓ-weight module of highest ℓ-weight λ ∈ P + ξ . If λ i (u) splits in C(ξ)[u] for every i ∈ I, then V is an ℓ-weight module and V µ = 0 only if µ ≤ λ. More precisely, if λ = m j=1 ω λ j ,a j for some λ j ∈ P + and a j ∈ C(ξ) × and if V µ = 0, then for some i k ∈ I, j k ∈ {1, . . . , m}, and 0 ≤ b k < r ∨ h ∨ . In particular, Remark. If λ i (u) does not split for some i ∈ I (in particular ξ = q), V is not an ℓ-weight module, but rather a quasi-ℓ-weight module, as mentioned previously. An analogue of the above theorem in this case can be established by following the ideas of [32]. Observe that it follows that W q (λ) ∈ C q iff the roots of λ i (u) (if any) are in C(q) for all i ∈ I.
Remark. While C ξ is a braided tensor category for any value of ξ, C ξ is not braided when ξ = 1.

Specialization of modules.
In this subsection, let ξ ∈ C ′ \{q}. Recall that an A-lattice of a C(q)-vector space V is the A-span of a basis of V . If V is a U q (g)-module (resp. U q (g)-module) and L is an A-lattice of V , we will say that L is a U A (g)-admissible lattice (resp. U A (g)-admissible lattice) if it is a U A (g)-submodule (resp. U A (g)-submodule) of V . Given a U A (g)-admissible lattice (resp. U A (g)-admissible lattice) of a module V , define ThenL is a U ξ (g)-module (resp. U ξ (g)-module) and dim C (L) = dim C(q) (V ). Given v ∈ L, we shall denote an element of the form v   Proof. Quite clearlyL is a highest-weight U ξ (g)-module and, hence, a quotient of W ξ (λ). The isomorphism now follows from Proposition 4.1 and Theorems 3.9 and 3.10.

Recall the definitions of P ++
A and P s A given in §2.1. Recall also that, for µ ∈ P A , µ denotes the element of P obtained from µ by applying the evaluation map ǫ ξ . Theorem 4.3. Let V be a U q (g)-highest-ℓ-weight module with highest-ℓ-weight λ ∈ P ++ A , v a highestℓ-weight vector of V , and L = U A (g)v. Then, L is a U A (g)-admissible lattice of V,L is a quotient of W ξ (λ), and char(L) = char(V ).
Proof. This is essentially a corollary of the proof of Theorem 3.16 (with ξ = q) as observed in [20]. However, the case g of type G 2 is reasonably more subtle and it seems that it has not been considered before. Namely, L is clearly invariant under the action of U A (g) and we shall see that it follows from the proof of Theorem 3.16 that L is finitely generated over A. Assuming this, since A is a principal ideal domain and L is obviously torsion free, it follows that L is a free A-module. Moreover, since C(q) is the fraction field of A, an A-basis of L is linearly independent over C(q) showing that the rank of L is less than or equal to the dimension of V . On the other hand, it follows from the proof of Theorem 3.16 again that L contains a C(q)-basis of V . The first statement follows immediately, while the second follows after observing thatL is a highest-ℓ-weight module of highest ℓ-weight λ. The third statement is immediate from the first together with Proposition 4.1(a).
If g is not of type G 2 , proceeding as in the proof of Theorem 3.16 it immediately follows that L is generated over A by finitely many elements of the form (x − i 1 ,r 1 ) (k 1 ) · · · (x − im,rm ) (km) v with (i j , s j , k j ) ∈ I × Z × Z ≥0 . If g is of type G 2 , let us take a closer look at (3.6) (and its analogue for s 1 < 0 which we shall not write down): Applying Lemma 1.19 we also get for some g ′ : Z 4 → A. Observe that if (k 1 , k 2 ) is not of the form (3k, 2k), then r < k 1 /3 in (4.2) and the induction hypotheses of the proof of Theorem 3.16 apply and imply that the element (x − i 1 ,s 1 ) (k 1 ) · · · (x − im,sm ) (km) v belongs to the A-span of similar elements with |s j | ≤ N (η) where η = j k j α i j . Thus, assume (k 1 , k 2 ) = (3k, 2k). By multiplying (3.6) by ([3] q ) k [k] q ! and using the induction hypotheses on the height of η and on s 1 as before, one easily sees that the elements ( . Hence, L ′ is a finitely generated A-submodule of V . On the other hand, it is clear that L ′ spans V over C(q) and, hence, we conclude that L ′ is an A-lattice of V as before. From now on, given a highest-ℓ-weight module V as in the above proposition, we will denote by V the U ξ (g)-moduleL as constructed there.
Proof. Let B = {v 1 , . . . , v m } be an A-basis of L and, hence, also a C(q)-basis of V . Given i ∈ I and r ∈ Z, let A i,r be the matrix of the action of Λ i,r on V with respect to B. Then, the entries a s,t of A i,r are in A and the characteristic polynomial p(u) of the action of . It follows that the entries of the matrix of the action of Λ i,r on L with respect to the basis B are ǫ ξ (a s,t ) and, hence, the characteristic polynomial is p(u). The proposition follows.
highest-ℓ-weight module with highest ℓ-weight λ ∈ P s A and L is as in Theorem 4.3.
Given V ∈ C ξ , set Proof. Since V ξ (λ) is a quotient of V q (λ), it suffices to show that the latter is irreducible. If this were not the case, V q (λ) would have to contain an irreducible submodule and, hence, ǫ ξ (wt ℓ (V q (λ))) would have to contain an element of P + ξ other than λ.
Remark. Notice that Theorem 3.21 implies that, if wt ℓ (V q (λ)) ∩ P + q = {λ} and either ξ is not a root of unity or ξ = ζ and l > 2r ∨ h ∨ , then the hypothesis of the above corollary is satisfied.
Let g = sl 2 , I = {i}, v be a highest-weight vector of V q (λ), λ ∈ P + , and L = U A (g)v. Then it is easy to see that In particular, the image of this set inL is a C-basis ofL. Now let g be general again, v and L be as in Theorem 4.3, and λ = wt(λ). Fix a reduced expression s i N · · · s i 1 for w ∈ W, and consider Proceeding inductively on j, it follows from Lemma 1.1(b) and (4.4) that m j ∈ Z ≥0 , Corollary 4.7. Let V be a finite-dimensional highest-ℓ-weight U ξ (g)-module of highest ℓ-weight λ ∈ P + ξ . Then, V Twλ = V wwt(λ) for all w ∈ W.
Proof. For ξ = q this is Proposition 4.6 again. Otherwise, since P + ξ ⊆ P s A , the result is immediate from Propositions 4.6 and 4.4.
Corollary 4.8. Let ξ ∈ C ′ and λ ∈ P + ξ be such that λ i (u) splits in C(q)[u] for all i ∈ I and suppose that i 0 ∈ I and w ∈ W are such that ℓ(s i 0 w) = ℓ(w) + 1. Then, T w λ ≤ λ and (T w λ) i 0 (u) ∈ C(ξ) [u] splits. Moreover, the roots of T w λ i 0 (u) form a subset of ∪ k a k ξ Z where a k runs through the set of roots of λ i (u), i ∈ I.
Proof. The proof goes inductively on ℓ(w). If ℓ(w) = 0 there is nothing to be done. Otherwise, let j ∈ I and w ′ ∈ W be such that w = s j w ′ and ℓ(w ′ ) = ℓ(w)−1. By the induction hypothesis, T w ′ λ ≤ λ. In particular, by the very definition of α i,a , T w ′ λ = µν −1 for unique relatively prime µ, ν ∈ P + ξ such that µ i (u), ν i (u) split in C(ξ) [u] for all i ∈ I. Also by the induction hypothesis, (T w ′ λ) j (u) ∈ C(ξ)[u], i.e., ν j (u) = 1 and the roots of µ j (u) lie in the set ∪ k a k ξ Z .
Let V = V ξ (λ) and v w be defined as in (4.7). It follows from Lemma 1.
and, hence, that U ξ (g j )v w ′ is finite-dimensional U ξ (g j )-highest-ℓ-weight module of highest-ℓ-weight (T w ′ λ) j (u). It now follows from Theorem 3.21 that T w λ ≤ T w ′ λ. Moreover, by Lemma 1.1(b) again, . Using Theorem 3.21 once more the last statement follows.
Remark. For ξ of infinite order, the above corollary was first stated in [12]. As observed in [9], the proof of [12] was incomplete. The approach we used above is a representation theoretic alternative to the one presented in [9, §2.10] which relies instead purely on the Weyl group action on P and the braid group action on P ξ . Conjecture 4.9. Let λ ∈ P + and V = W q (λ). Then, V ∼ = W ξ (λ).
Conjecture 4.9 is the natural affine analogue of Corollary 4.2. In the case ξ = 1, it was originally conjectured in [20] and proved in [10] for g of type A and in [23] for simply laced g. Moreover, it was pointed out by Nakajima that the results of [34,35,41] can be used to prove the conjecture in the case ξ = 1. We shall leave the discussion about the general case of the above conjecture to the future. We only mention that in the case ξ = 1 the proof can be reduced to the case λ = ω λ,a with λ ∈ P + and a ∈ C × . This follows from the second statement of part (b) of Theorem 4.10 below. If ξ = ζ, however, it is unclear to us at this moment if such a reduction is possible.

Evaluation modules.
Given a g-module V , let V (a) be theg-module obtained by pulling-back the evaluation map ev a :g → g, a ∈ C × . Such modules are called evaluation modules. If V = V (λ) we use the notation V (λ, a) for the corresponding evaluation module. The next theorem can be found in [20]. (a) If λ = ω λ,a for some λ ∈ P + and some a ∈ C × , then V (λ) ∼ = V (λ, a).
In the remainder of this subsection we assume g = sl n+1 and ξ ∈ C ′ \{q}. Consider the C(q)-algebra U ′ q (g) given by generators x ± i , k ±1 µ with i ∈ I, µ ∈ P , and the following defining relations: There is an obvious monomorphism of algebras is said to be of type 1 if the generators k ν , ν ∈ P , act diagonally with eigenvalues of the form q (ν,µ) for some µ ∈ P where (·, ·) is the bilinear form such that (α i , α j ) = c ij . It is not difficult to see that restriction establishes an equivalence of categories from that of type 1 finite-dimensional U ′ q (g)-modules to C q . From now on we shall identify these two categories using this equivalence. One can construct the algebras U ′ A (g) and U ′ ξ (g) similarly as U A (g) and U ξ (g). The next proposition was proved in [33, §2] and [16, Proposition 3.4].
Theorem 4.11. Suppose g is of type A n . There exists an algebra homomorphism ev : U q (g) → U ′ q (g) such that, if λ ∈ P + and V is the pull-back of V q (λ) by ev, then there exists m(λ) ∈ Z such that V is isomorphic to V q (λ) where Moreover, the image of U A (g) by ev is contained in U ′ A (g).
Given a ∈ C(q) × , let where ̺ a is defined in Proposition 1.2. Denote by V q (λ, a) the pull-back of V q (λ) by the evaluation map ev a . It is easy to see that V q (λ, a) ∼ = V q (λ) where It turns out that, for g not of type A, there is no analogue of Theorem 4.11. In fact, it is known (see [7] for instance) that there exists i ∈ I and m ∈ Z ≥0 such that the action of U q (g) on V q (mω i ) cannot be extended to one of U q (g).

Proof.
Let v and L be as in Theorem 4.3. It follows that L = U A (g)v and, therefore, the irreducible quotient of V q (λ, a) is isomorphic to V ξ (λ) as a U ξ (g)-module. The result now follows easily.
We now give explicit formulas for the action of the elements of U ξ (g) on evaluation modules in the case g = sl 2 . In this case one can normalize ev so that m(λ) = 0 for all λ ∈ P + and, hence, Lemma 4.14. The following holds in W ξ (λ, a): for every r, s ∈ Z, s > 0.
The following result is easily established.

4.4.
Highest-ℓ-weight tensor products. The goal of this subsection is to prove a root of unity analogue of the following theorem which is the main result of [8].
Corollary 4.23. Let ξ have infinite order. Given a 1 , . . . , a k ∈ C(ξ), there exists σ ∈ S k such that As an application of this corollary alongside results of [5], the following theorem was proved in [12].
Theorem 4.24. Suppose ξ has infinite order and that λ = k j=1 ω i j ,a j ∈ P + ξ for some i j ∈ I, a j ∈ C(ξ) × . Then, there exists σ ∈ S k such that W ξ (λ) ∼ = V ξ (ω i σ(1) ,a σ(1) ) ⊗ · · · ⊗ V ξ (ω i σ(k) ,a σ(k) ). In other words, if λ i (u) is either constant or splits for all i ∈ I, then the associated Weyl module can be realized as a tensor product of fundamental representations.
We now give a counterexample showing that both Corollary 4.23 and Theorem 4.24 do not hold in the case ξ = ζ.
Example 4.25. Let g = sl 2 , I = {i}, ξ = ζ, l = 3, and suppose a 1 , a 2 , a 3 ∈ C × are such that a σ(3) ) for any σ ∈ S 3 . For proving this, it suffices to show that are not highest-ℓ-weight for any a ∈ C × . Let v 1 , v 2 , v 3 be highest-ℓ-weight vectors for each factor of these tensor products, v = v 1 ⊗ v 2 ⊗ v 3 , and W = U ζ (g)v. It suffices to check that the dimension of W ω i is less than 3. In fact, one easily checks using Lemma 4.14 and Proposition 1.6(c) that for the later tensor product W ω i is 1-dimensional while for the former W ω i is 2-dimensional.
With a little more work one can compute composition series for the above tensor products. Namely, aζ ) and there exists a non-split short exact sequence On the other hand, if there exists a non-split short exact sequence Definition 4.26. Let g = sl 2 , I = {i}, and λ = k j=1 ω i,a k for some k ∈ Z >0 and a j ∈ C × for j = 1, . . . , k. We say that λ is ζ-regular if there exists σ ∈ S k such that W ζ (λ) ∼ = V ζ (ω i,a σ(1) ) ⊗ · · · ⊗ V ζ (ω i,a σ(k) ). In general, given g, let w 0 = s i N · · · s i 1 be a reduced expression for w 0 . We say that λ ∈ P + is ζ-regular (with respect to this reduced expression of w 0 ) if (T i j−1 · · · T i 1 λ) i j is ζ i j -regular for all j = 1, . . . , N .
For the next two lemmas we will use the following notation. Let λ, µ ∈ P + ξ , λ = wt(λ), µ = wt(µ), and v λ , v µ be highest-ℓ-weight vectors of highest-ℓ-weight modules V, W of highest ℓ-weight λ and µ, respectively. Let also v w λ be a generator of V wλ for w ∈ W.
Proof. The proof is a refinement of that of [8, Lemma 4.2]. Since (x − i,r ) (k) v w 0 λ = 0 for all i ∈ I, r, k ∈ Z, k > 0, it follows from Proposition 1.6 that ( Next we prove that v λ ⊗W ⊆ U ξ (g)(v λ ⊗v µ ) and, under the hypothesis of the "moreover part" of the for all m ∈ Z ≥0 , k j ∈ Z >0 , i j ∈ I, r j ∈ Z (for the "moreover part" it suffices to prove this with k j = 1 for all j = 1, . . . , m and with U fin ζ (g) in place of U ξ (g)). We proceed by induction on the height of j k j α i j . Induction clearly starts for m = 0. We shall write the proof for the first statement only since the second one is proved similarly.
We consider only the case r 1 ≥ 0 since the case r 1 < 0 is similar. The proof is by induction on k 1 with a further subinduction on r 1 . Let v ′ ∈ W . By Proposition 1.6 we have where ̟ is a sum of vectors which belong to with the height of m ′ j=1 k ′ j α i ′ j strictly smaller than that of m j=1 k j α i j , while ̟ ′ is a sum of vectors belonging to Hence, by the induction hypothesis on the height we have ̟ ∈ U ξ (g)(v λ ⊗ v µ ), while the same is true for ̟ ′ by the subinduction hypothesis on k 1 and r 1 (observe that if k 1 = 1 then k = 0 and if r 1 = 0 then ̟ ′ = 0 which shows that both subinductions start). Since ( This completes the proof of (4.13).
Proof. If ξ = 1 this is part of Theorem 4.10 and if ξ has infinite order this follows from [14,Lemma 4.10] which will be reviewed below. Thus, assume ξ = ζ. We first prove the "moreover" part. We proceed by induction on m observing that there is nothing to prove when m = 1. Let v j , j = 1, . . . , m, be highest-ℓ-weight vectors for V ζ (λ j ) and v ′ = v 2 ⊗· · ·⊗v m . By the induction hypothesis, V ′ = U fin ζ (g)v ′ . The proof runs parallel to that of [14,Lemma 4.10]. Let I = {i}. By Theorem 4.21, it suffices to consider the case λ j = ω i,a j ,n j for all j = 1, . . . , m and some a j ∈ C × and 0 < n j < l. Given k ∈ {0, . . . , n j }, set v k j = (x − i ) k v j so that {v k j : k = 0, . . . , n 1 } is a basis of V (λ j ). By Lemma 4.29, it suffices to show that for all k = 0, . . . , n 1 − 1. It is not difficult to see, using Proposition 1.6, that there exist a r,s ∈ C such that In fact, a r,s can be explicitly computed using Lemma 4.14. It turns out that the determinant of the matrix (a r,s ), 0 ≤ r, s ≤ m, is different from zero iff a 1 /a j = ζ −(n 1 +n j −2k) and a j /a ℓ = ζ −(n j +n ℓ ) for all 1 < j < ℓ ≤ m (see the proof of [14,Lemma 4.10]). Since (λ 1 , . . . , λ m ) is in ξ-resonant order, (4.15) follows.
To prove the first statement, write λ j = λ ′ j φ l (λ ′′ j ) as in (2.11) and notice that Theorem 4.16(b) and Corollary 4.20(a) imply that Since we have already proved the "moreover" part of the theorem, it follows that V is a highest-ℓ-weight module of highest-ℓ-weight λ. Let v λ , v µ be highest-ℓ-weight vectors of V and W = V ζ (φ l (µ)), respectively. By the "moreover" part of Theorem 4.16(b), x − i,r acts trivially on W and, hence, We are done by Lemma 4.29.
We now prove a stronger version of Proposition 4.30 for two-fold tensor products of evaluation representations. For this purpose only, if l = 1, i.e., if ξ is generic, set P + 1 = P + (recall that P + 1 = {0} in our usual notation). Thus, let g = sl 2 , I = {i}, a, b ∈ C(ξ) × , and λ, µ ∈ P + l be such that is in ξ-resonant order as well, but this may not be the case otherwise. Proof. We use the notation of Lemma 4.14. Namely, let v λ 0 , v µ 0 be highest-weight vectors of V ξ (λ) and V ξ (µ), respectively, and set v ν This is done similarly to the proof of [14,Lemma 4.10]. Since in the present situation that proof simplifies significantly, let us be precise. Induction obviously starts when k = 0. Hence, assume 0 It suffice to show that det(a j,m ) = 0. Using Proposition 1.6 and Lemma 4.14 one computes: Therefore, which vanishes iff either a/b = ξ −(λ(h i )+µ(h i )−2k) or [k + 1] ξ = 0. The former does not happen because k < min{λ(h i ), µ(h i )}. As for the latter, if ξ is generic [k + 1] ξ = 0 regardless of the value of k ≥ 0, while for ξ = ζ this follows since k < min{λ(h i ), µ(h i )} < l. The induction step is proved.
If λ(h i ) ≤ µ(h i ) (which is assumed in the proof of [14,Lemma 4.10]), the proof is completed by using Lemma 4.29. Otherwise, let V = V ξ (λ, a) ⊗ V ξ (µ, b). By Lemma 4.29 again, it suffices to show . To prove this, in turn, by Proposition 3.2, it suffices to show that . We now use the first part of the proof to show that Observe that the vectors v j := v λ k−j ⊗ v µ j , j = 0, . . . , k, form a basis of V λ+µ−kα i . The proof of (4.17) is again done by induction on k. Since for k = 0 there is nothing to prove, assume 0 ≤ k < µ(h i ) is such that (4.17) holds. This implies v j ∈ U ξ (g)(v λ 0 ⊗ v µ 0 ) for all 0 ≤ j ≤ k. Using Proposition 1.6 and Lemma 4.14 once more we get Remark. As of the moment we do not know if the above proposition remains true or not in greater generality. Thus, we pose the following question. Can "ξ-resonant order" be replaced by "weak ξresonant order" in Theorem 4.22? By looking at the proof of Theorem 4.22, one sees that it suffices to answer this for g = sl 2 . In other words, it suffices to show a version of Proposition 4.31 for tensor products with arbitrary number of factors. The generalization of the first part of the proof of Proposition 4.31 is identical to the proof of [14,Lemma 4.10] (this is implicitly used in the proof of Theorem 4.22). Hence, it remains to generalize the argument of the second part of the proof.
The next lemma is easily established.
Proof. The proof runs parallel to that of Theorem 4.22 with a few modifications. We again proceed by induction on m and observe that there is nothing to prove when m = 1. Once more, let v j , j = 1, . . . , m be highest-ℓ-weight vectors for V ζ (λ j ) and v ′ = v 2 ⊗ · · · ⊗ v m . By the induction hypothesis, V ′ = U fin ζ (g)v ′ . Fix a reduced expression for w 0 , say s i N · · · s i 1 . By Lemma 4.29, it suffices to show that -module of highest-ℓ-weight (λ k ) i j by Lemma 4.32. By hypothesis, the (m − 1)-tuple ((λ 2 ) i j , . . . , (λ m ) i j ) is in ζ i j -resonant order. In particular, it follows from Lemma 4.32 and Proposition 4.30 that On the other hand, U ζ (g i j )v is a quotient of W ζ ((T i j−1 · · · T i 1 λ 1 ) i j ) by Lemma 4.28. Since λ 1 is ζ-regular, W ζ ((T i j−1 · · · T i 1 λ 1 ) i j ) is isomorphic to a tensor product of the form V ζ (ω i j ,a 1 ) ⊗ · · · ⊗ V ζ (ω i j ,a k ) for some a 1 , . . . , a k ∈ C × and where k = wt(T i j−1 · · · T i 1 λ 1 )(h i j ). The assumption that (T i j−1 · · · T i 1 λ 1 ) i j , (λ 2 ) i j , . . . , (λ m ) i j is in ζ i j -resonant order implies that (ω i j ,a 1 , . . . , ω i j ,a k , (λ 2 ) i j , . . . , (λ m ) i j ) is also in ζ i j -resonant order. It follows from Proposition 4.30 that which implies (4.18).
We now give a counterexample showing that the "loop" analogue of the first statement of Proposition 4.1(a) does not hold in general.
Let v and w be highest-ℓ-weight vectors of V q (ω i,a ) and V q (ω i,b ), respectively, and set . It is known that {v 1 , v 2 } is an A-basis for the zero weight space L 0 := L ∩ V 0 . Using Lemma 4.14 and Proposition 1.6, one easily computes that We will show that there exists no A-basis of L 0 formed by ℓ-weight vectors provided (b − aq 2 ) / ∈ A × and a = b (a = b is the only choice for the pair (a, b) so that any nonzero vector of V 0 is an ℓ-weight vector). By contradiction, suppose there exist α, β ∈ C(q) such that {αv ⊗ w 1 , βv 1 ⊗ w} is an A-basis of L 0 . Then, the matrix A whose columns are the coordinates of v 1 , v 2 with respect to this basis must have entries in A and its determinant must be in A × . Using (4.19) one easily sees that It follows that α −1 , β −1 ∈ A and, hence, det(A) ∈ A × iff α, β, (b − aq 2 ) ∈ A × . Now, there are plenty of choices for a, b ∈ A × such that b − aq 2 / ∈ A × .

4.5.
On Jordan-Hölder constituents. We now prove several results regarding the irreducible constituents of U ξ (g)-modules obtained by specializing U q (g)-modules. We begin with the following general fact. Recall the definitions of I • ⊆ I from §1.1 (Table 1) and of P + ξ,I• given at the end of §2.3. Theorem 4.36. Every simple object of C ξ is isomorphic to the quotient of a submodule of a tensor product of the modules W ξ (ω i ) for i ∈ I • .
Proof. For simplicity, we write the proof of the theorem for the case that I • is a singleton, i.e., that g is not of type D 2m . Thus, let i denote the unique element of I • . For ξ = 1 the result is well-known and the proof for ξ not a root of unity is analogous. Moreover, every simple object of C ξ is an irreducible summand of a tensor power of V ξ (ω i ) = W ξ (ω i ). We now consider the case ξ = ζ.
Let λ ∈ P + and m ∈ Z ≥0 be such that V q (λ) is a summand of V q (ω i ) ⊗m . Fix highest-weight vectors v j of the j-th factor of this tensor product, let L j = U A (g)v j , j = 1, . . . , m, and L = L 1 ⊗ · · · ⊗ L m . Quite clearly L ∼ = ⊗ j L j ∼ = W ζ (ω i ) ⊗m . Hence, it suffices to show that there exists v ∈ L such that x + k v = 0 for all k ∈ I, U q (g)v ∼ = V q (λ), and the imagev of v in L is nonzero. The A-module L λ has an A-basis formed by elements of the form Here i = (i 1,1 , · · · , i r 1 ,1 ; · · · ; i 1,m , · · · , i rm,m ) and similarly for k. It easily follows that there is an A-linear combination of such vectors, say v ′ , satisfying the two first properties required of v. Write and observe that there exist c ′ i, k ∈ C[q] and f ∈ A such that the nonzero c ′ i, k are relatively prime in Since the c ′ i, k are relatively prime, it follows that ǫ ζ (c ′ i, k ) = 0 for at least one value of the pair ( i, k). The theorem follows.
In order to prove an affine analogue of the above theorem we will need the following proposition which will also be used to prove the main result of §4.6. Proposition 4.37. Let λ ∈ P s A , V a nontrivial quotient of W q (λ), and ξ ∈ C ′ \{q}. If V q (µ) is an irreducible constituent of V with multiplicity m, then V ξ (µ) is an irreducible constituent of V with multiplicity at least m.
Proof. By an obvious induction on the length of V , it suffices to show the proposition in the case V q (µ) is a submodule of V . The argument is essentially the same as that of [31,Proposition 4.16]. Namely, let v be a highest-ℓ-weight vector of V and L = U A (g)v. Then, from the proof of Theorem 4.3, L has an A-basis consisting of vectors which are A-linear combinations of elements of the form Let v 1 , · · · , v r be an A-basis for L µ where µ = wt(µ). Any highest-ℓweight vector for V q (µ) is a solution r j=1 c j v j , for some c j ∈ C(q), of the linear system for all i ∈ I, s ∈ Z, k ∈ Z >0 . By (induction) assumption, there exists a nontrivial solution for this system. Since L is admissible and the ℓ-weights of V are in P A (by Theorem 3.21 given that λ ∈ P s A ), it follows that there exists a solution with the c j lying in A. Hence, there is also a solution with c j ∈ C[q] and such that the nonzero c j are relatively prime. This completes the proof similarly to that of Theorem 4.36.
Theorem 4.38. For every λ ∈ P + ξ , there exists µ ∈ P + ξ,I• such that V ξ (λ) is an irreducible constituent of W ξ (µ). Moreover, if λ ∈ P s A , then µ can be chosen to be in P s A as well.
Proof. For ξ = q, it follows from the results of [18,22] that V q (λ) is a constituent of a tensor product of the form V q (ω i 1 ,a 1 ) ⊗ · · · V q (ω im,am ) with i j ∈ I • and a j ∈ C × (a j ∈ A × if λ ∈ P s A ). The theorem now follows in this case by Corollaries 4.23 and 3.22.
For ξ ∈ C ′ , consider the module V q (λ) and apply the theorem for the case ξ = q. Thus, let µ ∈ P + ξ,I• ∩ P s A be such that V q (λ) is a constituent of W q (µ). By Proposition 4.37, V ξ (λ) is a constituent of W q (µ). 4.6. Blocks. Recall the notation introduced in §2.3. We shall need the following proposition which follows from the results of [12,22]. Proposition 4.39. Let µ ∈ P + q,I• and λ = µτ q,k,a for some k ∈ {1, 2, 3} and some a ∈ C(q) × . If V is an irreducible constituent of W q (λ), then V is an irreducible constituent of W q (µ).
An object V ∈ C ξ is said to have elliptic character γ ∈ Γ ξ if V µ = 0 implies γ ξ (µ) = γ. Denote by C γ ξ the abelian subcategory of C ξ consisting of representations with elliptic character γ. The main result of this section is the following theorem. This theorem was first proved in [22] in the case that ξ ∈ C × satisfies |ξ| = 1 using analytic properties of the action of the R-matrix of U ξ (g). An R-matrix free approach was given in [12, §8] for the case that ξ is not a root of unity. For ξ = 1, the theorem was proved in [11]. We now give a proof that works for any ξ ∈ C ′ \{q}, thus completing the proof of Theorem 4.40.
For a brief review of the theory of blocks of an abelian category see [22, §1]. The proof of Theorem 4.40 is immediate from the following two propositions.
Proposition 4.41. If V ∈ C ξ is indecomposable, then V ∈ C γ ξ for some γ ∈ Γ ξ . In other words, The proof of Proposition 4.41 in the case ξ = ζ is analogous to that given in [11,12] and we omit the details (see [12, §8.4]).
We now prove Proposition 4.42 for ξ = q. It suffices to show that, given λ, µ ∈ P + such that γ ξ (λ) = γ ξ (µ), there exists a sequence of indecomposable modules W 1 , . . . , W m satisfying: (a) V ξ (λ) is a constituent of W 1 and V ξ (µ) is a constituent of W m , (b) W j and W j+1 have at least one irreducible constituent in common.
Set µ ′ = ω ′ m , W ′ j = W q (ω ′ j ) and W j = W ′ j . By Proposition 4.39, W ′ j and W ′ j+1 have common irreducible constituents and, hence, so do W j and W j+1 by Proposition 4.37. Since W 1 is a quotient of W ξ (λ) and W m is a quotient of W ξ (µ) we are done.
In the case ξ = 1, the proof of Proposition 4.42 we gave above is an alternative one to that given in [11].
Proof. The first two statements have already been proved. The remaining statements for ξ of infinite order were proved in [12, Proposition 6.5]. In the root of unity case the proof can be carried out exactly as in the infinite order case after observing the following. Due to Lemma 1.10 we can work with elementsh i,r as it was done in [12, Proposition 6.5]. Moreover, observe as well that [rc ij ]q j [r]q i ∈ A for all i, j ∈ I and all nonzero integer r. Hence, we have the following identity in U A (g) Finally, observe that ζ r j + ζ −r j = 0 for all r ∈ Z and set Hence, we get the following identity in U ζ (g) (4.22) [h i,r , (x − j,s ) (k) ] = (ζ r j + ζ −r j ) −1 {rc ij } ζ [h j,r , (x − j,s ) (k) ] which replaces equation (6.19) of [12]. From here, all the steps of the proof can be performed exactly as in the case of ξ generic.
The following theorem can now be proved exactly as in [12, Theorem 6.1].
Theorem 4.47 provides an algorithm for computing a lower bound for char ℓ (V ξ (ω i,a )). For ξ of infinite order, it was used together with the knowledge of char(V ξ (ω i,a )) (see [7]) in [13] to compute char ℓ (V ξ (ω i,a )). Notice that it follows from Proposition 4.4 that ǫ ξ (char ℓ (V q (ω i,a ))) is an upper bound for char ℓ (V ξ (ω i,a )). We finish the paper by giving an example explaining how to use these facts and Corollary 4.5 to identify values of ξ for which (4.24) V ξ (ω i,a ) ∼ = V q (ω i,a ).
To do this, let V = V q (ω 2,a ), v be a highest-ℓ-weight vector of V , L = U A (g)v, and v w j be as in (4.7). In particular, v w j is an A-basis element of L and, hence, its image v w j ∈ V is nonzero. Let also W j = U q (g j )v w j and W j = U ζ (g j )v w j . Since v w j is a highest ℓ-weight vector for U q (g j ), W j is a quotient of the Weyl module for this subalgebra with highest ℓ-weight (T w j (ω 2,a )) j (and similarly for W j ). It follows from the above computations that V µ j ⊆ W j and that V 0 = n j=1 (W j ) 0 . We claim that W 1 must be a Weyl module. In fact, it is easy to see that V µ 1 ∩ W j = {0} iff j = 1. If W 1 were not a Weyl module, then it would be irreducible and so would be W 1 . This would imply that (µ 1 ) 1 (which is the the constant polynomial) would be an ℓ-weight of the irreducible U ζ (sl 2 )-module with highest ℓ-weight (1 − aζu)(1 − aζ −1 u). This contradicts Proposition 4.44.
Since v w 1 is an A-basis element of L, it follows that W 1 is also a Weyl module (alternatively, if W 1 were not a Weyl module, V would not be highest-ℓ-weight since V µ 1 ∩ W j = {0} if j > 1 and we would have a contradiction again). Therefore, there exists v ′ ∈ (W 1 ) 0 which generates a trivial submodule of W 1 . It is easy to see that v ′ also generates a trivial submodule of V . This shows that µ 1 is not an ℓ-weight of the irreducible quotient of V which is isomorphic to V ζ (ω 2,a ).

Remark.
The assumption that l is odd is not really essential above and one can easily deduce similar results for roots of unity of even order as well.
We believe that the above line of reasoning could be used alongside the results of [13] to obtain expressions for the ℓ-characters of the fundamental representations of U ζ (g) in terms of the braid group action for g of classical type. In order to keep the length of present text within reasonable limits, we postpone further discussion in this direction to a forthcoming publication.