Affinization of category O for quantum groups

Let g be a simple Lie algebra. We consider the category O-hat of those modules over the affine quantum group Uq(g-hat) whose Uq(g)-weights have finite multiplicity and lie in a finite union of cones generated by negative roots. We show that many properties of the category of the finite-dimensional representations naturally extend to the category O-hat. In particular, we develop the theory of q-characters and define the minimal affinizations of parabolic Verma modules. In types ABCFG we classify these minimal affinizations and conjecture a Weyl denominator type formula for their characters.


Introduction
Let g be a simple Lie algebra and q ∈ C × transcendental.In this paper we consider the category Ô of modules over the affine quantum group U q ( g) such that after the restriction to U q (g) the dimensions of the weight spaces are finite, and the set of non-trivial weights belongs to a finite union of cones generated by negative roots.This category was originally defined in [Her04].It is a tensor category which includes the finite-dimensional modules.The simple objects in Ô are highest weight U q ( g)-modules with highest ℓ-weights given by arbitrary sets of rational functions (f i ) i∈I with the property f i (0)f i (∞) = 1, i ∈ I, I being the set of nodes of the Dynkin diagram of g (see Theorem 3.6 below).Our motivation for the study of Ô is twofold.
First, many results from the category of finite-dimensional U q ( g)-modules can be easily extended to the much richer category Ô.For example, we have in Ô the classification of irreducibles by highest ℓ-weights, and the notions of fundamental modules, Kirillov-Reshetikhin modules and minimal affinizations.In type sl 2 , the irreducible modules are tensor products of evaluation modules.We define a theory of q-characters which gives an injective ring homomorphism from the Grothendieck ring of Ô to a certain formal ring possessing many properties which allow us to study it combinatorially.
Second, we are trying to find a new way to study the minimal affinizations of the finitedimensional modules.Minimal affinizations, which are analogs of the evaluation modules that exist only in type A, received a lot of attention, see [CP94b, Cha95, CP95, CP96a, CP96b, Her07, Mou10, MY11, MY12] but are still poorly understood in general.In the non-affine setting, important information about the finite-dimensional modules comes from the study of Verma modules, which have a much simpler structure.Inspired by this idea we initiate the study of minimal affinizations of Verma modules, which naturally leads us to the category Ô.
We establish the foundations of the theory of the category Ô, for the most part modifying the well-known methods initially developed by many authors for the finite-dimensional modules.As one notable exception, we give a proof of the classification of the minimal affinizations somewhat 1 different from the classical papers [Cha95,CP95]; see Theorem 5.7.We use the theory of qcharacters and treat types ABCFG simultaneously.In these types a minimal affinization is not only a minimal element with respect to the partial order defined in [Cha95], but the least element.In this paper we do not the consider the types, D and E, whose diagrams have a trivalent node.
Our main finding is that the minimal affinizations of the generic parabolic Verma modules (and many other modules) considered as U q (g)-modules indeed have a simple character similar to the Weyl denominator.For example, if λ = i∈I λ i ω i is a g-weight written in terms of the fundamental weights and none of λ i is an integer, we conjecture that the character of the minimal affinization of the Verma module with highest weight λ is given by where ∆ + is the set of positive roots of g and for a positive root α = i∈I α i ω i we define m α = max This formula and many similar formulae, see Conjecture 6.5, were found and partially checked with the help of a computer based on the use of the algorithm of [FM01].We give proofs only in some special cases, e.g. in types A n , B 2 , based on known results for finite-dimensional modules, but the simplicity of the answer suggests that a general proof may be not very difficult.
We would like to acknowledge the paper [HJ11] where the authors studied the stable limits of the Kirillov-Reshetikhin modules, which are minimal affinization of finite-dimensional modules with highest weights λ = nω i , as n → ∞.Those limits are representations of an algebra which is slightly different from the standard quantum affine group.Instead of going to that limit we study the analytic continuation with respect to n, see §3.4.That, in particular, allows us to stay with the standard quantum affine group.
The paper is structured as follows.After summarizing background material in §2, in §3 we define the category Ô, classify its simple objects (Theorem 3.6), and develop the theory of q-characters for Ô.We also briefly discuss analytic continuation ( §3.4) and the restricted duals of objects in Ô ( §3.5).In §4 we consider the case g = sl 2 and give a description of the simple objects in Ô in terms of tensor products of evaluation modules.Affinizations, and in particular minimal and least affinizations, are introduced in §5.Theorem 5.7 classifies least affinizations in types ABCFG.Finally, §6 contains a series of conjectural formulae for the U q (g)-characters of least affinizations of parabolic Verma modules, and of certain other representations.
Acknowledgements.From October 2010 until December 2011, the research of CASY was funded by the EPSRC, grant number EP/H000054/1.The research of EM is supported by the NSF, grant number DMS-0900984.Computer programs to calculate q-characters were written in FORM [Ver].

Background
2.1.Cartan data.Let g be a complex simple Lie algebra of rank N and h a Cartan subalgebra of g.We identify h and h * by means of the invariant inner product •, • on g normalized such that the square length of the maximal root equals 2. With I = {1, . . ., N }, let {α i } i∈I be a set of simple roots, and {α ∨ i } i∈I , {ω i } i∈I , {ω ∨ i } i∈I the corresponding sets of, respectively, simple coroots, fundamental weights and fundamental coweights.Let C = (C ij ) i,j∈I be the Cartan matrix.We have the maximal number of edges connecting two vertices of the Dynkin diagram of g.Thus r ∨ = 1 if g is of types A, D or E, r ∨ = 2 for types B, C and F and r ∨ = 3 for G 2 .Let r i = 1 2 r ∨ α i , α i .The numbers (r i ) i∈I are relatively prime integers.We set the latter is the symmetrized Cartan matrix, Let Q (resp.Q + ) and P (resp.P + ) denote the Z-span (resp.Z ≥0 -span) of the simple roots and fundamental weights respectively.Let ≤ be the partial order on h * (and in particular on P and Q) Let ∆ ⊂ Q be the set of roots of g and ∆ + = ∆ ∩ Q + the set of positive roots.Let g denote the untwisted affine algebra corresponding to g.Let C = (C ij ) i,j∈{0}∪I be the extended Cartan matrix, α 0 be the extra simple root of g, r 0 = 1 2 r ∨ α 0 , α 0 , D = diag(r 0 , r 1 , . . ., r N ) and B = D C.
Fix a transcendental q ∈ C × .For each i ∈ I let Define the q-numbers, q-factorial and q-binomial: 2.2.Quantum Affine Algebras.The quantum affine algebra U q ( g) in the Drinfeld-Jimbo realization [Dri87,Jim85] is the unital associative algebra over C with generators (x ± i ) i∈ I , (k ±1 i ) i∈ I subject to the relations The algebra U q ( g) can be endowed with the coproduct, antipode and counit given by There exists another presentation of U q ( g), due to Drinfeld [Dri88].In this presentation U q ( g) is generated by ) and central elements c ±1/2 , subject to the following relations: ) for all sequences of integers n 1 , . . ., n s , and i = j, where Σ s is the symmetric group on s letters, and φ ± i,n 's are determined by equating coefficients of powers of u in the formula (2.3) Note that φ + i,−n = φ − i,n = 0 for all n ∈ Z >0 , and φ ± i,0 = k ±1 i .We have x ± i,0 = x ± i for all i ∈ I.The subalgebra of U q ( g) generated by (k i ) i∈I , (x ± i ) i∈I is a Hopf subalgebra of U q ( g) and is isomorphic as a Hopf algebra to U q (g), the quantized enveloping algebra of g.In this way, U q ( g)modules restrict to U q (g)-modules.The Cartan involution of U q (g) is defined by We shall need the following quantum-affine analog φ of the Cartan involution.By definition, [Cha95], φ is the algebra automorphism whose action on generators is: Let Û ± ⊂ U q ( g) be the subalgebras generated by (x ± i,r ) i∈I,r∈Z , and U ± ⊂ U q (g) the subalgebras generated by (x ± i ) i∈I .Let Û 0 ⊂ U q ( g) be the subalgebra generated by c ±1/2 , (k i ) i∈I and (h i,r ) i∈I,r∈Z =0 , and U 0 ⊂ U q (g) the subalgebra generated by (k i ) i∈I .We have the following triangular decompositions of U q (g) and of U q ( g) [CP94a]: It is known [Dam98] that on representations of U q ( g) on which c acts as the identity, (2.9) 3. The category Ô 3.1.Definition of Ô.Let : h * → (C × ) N be the surjective homomorphism of abelian groups such that i∈I λ i ω i := (q λ 1 1 , q λ 2 2 , . . ., q λ N N ).
By a slight overloading, we use the word weight to refer to an N -tuple Since q is not a root of unity, the restrictions of to P and in particular to Q + are injective; let P and Q + denote their respective images.Then h * inherits from h * the usual partial order: We call V ̺ the weight space of weight ̺, and nonzero elements v ∈ V ̺ weight vectors of weight ̺.
We say Definition 3.1.We say a U q (g)-module V is in category O if: (i) V is a weight module all of whose weight spaces are finite-dimensional.
(ii) There exist a finite number of weights ̺ 1 , . . ., ̺ k ∈ h * such that every weight of V is in Let us define an ℓ-weight module to be any U q ( g)-module on which the actions of the generators (h i,r ) i∈I,r∈Z =0 commute pairwise.Proposition 3.2.Every simple U q ( g)-module V whose restriction as a U q (g)-module is in O is an ℓ-weight module.Moreover it can be obtained by twisting, by an automorphism of U q ( g), a module in which c 1/2 acts as the identity.
Proof.Since the invertible central element c 1/2 acts as a multiple of the identity on any simple module, there exists a τ ∈ C such that c 1/2 .v= τ v for all v ∈ V .Then each weight space V ̺ carries a representation of the 3-dimensional Lie algebra generated by h i,r , h j,s and (c − c −1 ).By Definition 3.1 part (i), V ̺ is finite-dimensional.The Weyl algebra C[x, p]/ xp − px − 1 does not admit finite-dimensional representations.Therefore τ 2 − τ −2 = 0. Hence c − c −1 acts as zero on V .This proves the first part.If τ 2 = −1 then the map defines an automorphism of U q ( g); twisting by it we indeed arrive at a module on which c 1/2 acts as the identity.On the other hand if τ = −1, we may twist by the automorphism of U q ( g) defined by c 1/2 → −c 1/2 , k i → k i , h i,r → h i,r , x ± j,s → (∓1) s x ± j,s , with the same result.
Definition 3.3.We say a U q ( g)-module is in category Ô if its restriction as a U q (g)-module is in category O and c 1/2 acts as the identity on V .Definitions 3.1 and 3.3 were stated in [Her04].The category O is a subcategory of the abelian monoidal category of all U q (g)-modules.It is clear that O is closed under taking quotients, submodules and finite direct sums, and tensor products.Therefore O is an abelian monoidal category.
Likewise, Ô is an abelian monoidal subcategory of the category of all U q ( g)-modules.Every V ∈ Ob Ô is an ℓ-weight module.
Remark 3.4.Because we wish O to be closed under tensor products, we do not require that every object V of O be finitely generated as a U q (g)-module.Similarly, inside our category Ô there is a subcategory consisting of those objects that are finitely generated as U q ( g)-modules.This subcategory contains all simple objects of Ô (these are classified in Theorem 3.6 below) and is strictly smaller than Ô.It is an interesting question whether this subcategory is closed under taking tensor products.
3.2.Classification of simple objects.Given V ∈ Ob Ô, the decomposition (3.2) into weight spaces can be refined as follows.An ℓ-weight is any N -tuple of sequences of complex numbers γ ≡ (γ ± i,±r ) i∈I,r∈Z ≥0 , such that γ + i,0 γ − i,0 = 1 for every i ∈ I. Given an ℓ-weight γ we define its weight to be wt(γ Then for every weight ̺ of V we have, c.f. (2.3), where the sum is over all ℓ-weights of weight ̺.We call V γ the ℓ-weight space of ℓ-weight γ.We say γ is an ℓ-weight of V if dim(V γ ) > 0. If v ∈ V γ is nonzero and moreover φ ± i,±r .v= γ ± i,±r v for all i ∈ I, r ∈ Z ≥0 , then v is called an ℓ-weight vector of ℓ-weight γ.Every ℓ-weight space contains an ℓ-weight vector.If v ∈ V is nonzero and x + i,r .v= 0 for all i ∈ I, r ∈ Z, then we say the vector v is singular.
We say V ∈ Ob Ô is a highest ℓ-weight representation of highest ℓ-weight γ if V = U q ( g).v for some singular ℓ-weight vector v ∈ V γ .By (2.7) dim(V γ ) = 1, so v is unique up to scale; we call it the highest ℓ-weight vector of V .Definition 3.5.We say an ℓ-weight f = (f ± i,±r ) i∈I,r∈Z ≥0 is rational if there is an N -tuple of complex-valued rational functions (f i (u)) i∈I of a formal variable u such that, for each i ∈ I, f i (u) is regular at 0 and ∞, f i (0)f i (∞) = 1, and in the sense that the left-and right-hand sides are the Laurent expansions of f i (u) about 0 and ∞, respectively.Let R be the set of rational ℓ-weights.R forms an abelian group, the group operation (f , g) → f g being given by component-wise multiplication of the corresponding tuples of rational functions.
In what follows, we do not always distinguish between a rational ℓ-weight f and the corresponding tuple (f i (u)) i∈I of rational functions.Note that in terms of the latter, we have For every weight ̺, let V (̺) be the irreducible U q (g)-module with highest weight ̺.Recall that V (̺) is unique up to isomorphism and is finite-dimensional if and only if ̺ ∈ P + ; see [CP94a], chapter 10.
For every rational ℓ-weight f , let us write L(f ) for the irreducible U q ( g)-module with highest ℓ-weight f .By definition, L(f ) is unique up to isomorphism.Moreover L(f ) and L(f ′ ) are not isomorphic unless f = f ′ .Every highest ℓ-weight U q ( g)-module with highest ℓ-weight f ∈ R has L(f ) as a quotient.
Recall [CP94b] that L(f ) is finite-dimensional if and only for each i ∈ I the rational function f i (u) is of the form f i (u) = q deg P i i P i (uq −2 i )/P i (u) for some polynomial P i (u) with constant coefficient 1, called a Drinfeld polynomial.Observe that this is a stronger condition than wt(f ) ∈ P + .
We can now state the following theorem, which classifies the simple objects in Ô.
Theorem 3.6.The map f → L(f ) defines a bijection between R and the isomorphism classes of simple objects in Ô.
Proof.Suppose V ∈ Ob Ô is irreducible.Then it follows from part (ii) of Definition 3.1 that V contains a singular ℓ-weight vector, say v. Since V is irreducible, V = U q ( g).v, so V is a highest ℓweight representation.Thus it is enough to show that a highest ℓ-weight irreducible representation V is in Ô if and only if its highest ℓ-weight f is rational.We shall first show that for each Here it is understood that The remaining equations of Finally, the constraint that Hence f + i (z) and f − i (z) are indeed of the required form.Conversely, suppose f + i (z) and f − i (z) are as above for some N > 0. Then a similar calculation shows that x + i,s N j=1 a j x − i,M +j .v= 0 for all M ∈ Z. Since V has no singular vectors which are not scalar multiples of v, it follows that N j=1 a j x − i,M +j .v= 0, i.e. that for all M ∈ Z, the vectors {x − i,M +j .v: j = 1, 2, . . ., N } are linearly related.By applying this result finitely many times, any given vector x − i,r .vcan be expressed as a linear combination of, say, the vectors To complete the proof, we note that if This follows by an induction on height(α) exactly as in [CP94b], §5, proof of case (b).
The "only if" part of the theorem was proved in [Her04], Lemma 14.
3.3.q-Characters.Recalling the definition of the group R of rational ℓ-weights, Definition 3.5, let us define a subgroup Q ⊂ R, the group of l-roots, as follows.For each j ∈ I and a ∈ C × , define for each i ∈ I.Note that wt(A j,a ) = α j .We call each A j,a a simple l-root.The reader should be warned that in [FR98,FM01] what we call A j,a was instead labelled A j,aq j .Let Q be the subgroup of R generated by A i,a , i ∈ I, a ∈ C × .Note that Q is a free group, i.e. the A j,a are algebraically independent.Let Q ± be the monoid generated by A ±1 i,a , i ∈ I, a ∈ C × .We call the latter the positive/negative l-roots.
There is a partial order ≤ on R in which f ≤ g if and only if gf −1 ∈ Q + .It is compatible with the partial order (3.1) on h * in the sense that f ≤ g implies wt f ≤ wt g.Definition 3.7.The q-character of V ∈ Ob Ô is the formal sum of its ℓ-weights, counted with multiplicities: One also has the usual U q (g)-character map Proposition 3.8.Suppose f and g are ℓ-weights of V ∈ Ob Ô, and i ∈ I. Then Proof.Let (v k ) 1≤k≤dim V f be a basis of V f in which the action of the φ ± i,r is upper-triangular, in the sense that for all i ∈ I and 1 (The leading order is u 1 : recall that φ ± i,0 act diagonally.)Let (w k ) 1≤k≤dim Vg be a basis of V g in which the action of the φ ± i,r is lower-triangular, in the sense that for all i ∈ I and 1 where x + i (z) := r∈Z z −r x + i,r .On resolving this equation in the basis of V g above and taking the w ℓ component, we have Then there is a smallest K such that (x + i (z).vK ) g = 0 and then a smallest L such that λ K,L (z) = 0.So (3.5) gives, in particular, This must hold for all j ∈ I.For each j ∈ I, (3.6) is an equation of the form 0 n ∈ C for all n ∈ Z ≥0 .Equivalently, for each i ∈ I, it is a countably infinite set of first order recurrence relations on the series coefficients of λ K,L (v).There are non-zero solutions if and only if there is an a ∈ C × such that b as an equality of power series in u.Similar arguments hold for φ − i (u).
This proposition has a number of important corollaries.First, Corollary 3.9.Suppose f and g are ℓ-weights of V ∈ Ob Ô, and v ∈ V f and w ∈ V g are nonzero.
If i ∈ I and a ∈ C × are such that w ∈ span r∈Z x ± i,r .vand As is the case for finite-dimensional U q ( g)-modules, the q-characters of the simple objects in Ô have the following "cone" property.
In particular, all ℓ-weights of L(f ) are rational.
Let Groth( Ô) be the Grothendieck ring of Ô.For all Theorem 3.11.χ q defines an injective ring homomorphism in Groth( Ô), i.e. whenever there is a short exact sequence 0 → V → W → U → 0 of U q ( g)-modules.
To show that χ q (V ⊗ W ) = χ q (V )χ q (W ) we argue as in [FR98].For each ℓ-weight f of V , let (v f ,k ) 1≤k≤dim V f be a basis of V f in which the action of the φ ± i,r is upper-triangular, c.f. (3.4); and likewise for each ℓ-weight g of W let (w g,k ) 1≤k≤dim Wg be an upper-triangular basis of W g .Then it follows from (2.9) that (v f ,k ⊗ w g,ℓ ) is a basis of (V ⊗ W ) f g in which the action of the φ ± i,r is upper-triangular.Thus, ℓ-weights are multiplicative across tensor products, and their multiplicities are additive, as required.
The classes [L(f )] ∈ Groth( Ô), f ∈ R, of the irreducible representations are linearly independent, because their images under χ q are linearly independent.Injectivity of χ q follows from this.
We also need the following proposition.
, by Theorem 3.6, enough to show that v ⊗ w is a singular ℓ-weight vector in L(f ) ⊗ L(g) and has ℓ-weight f g.That v ⊗ w has ℓ-weight f g follows from (2.9).That v ⊗ w is singular follows exactly as in the case of finite-dimensional modules, c.f. [CP94a].Finally, L(g

Analytic continuation.
In this subsection we observe that if the rational highest ℓ-weight f depends rationally on an additional parameter x ∈ C then the normalized U q (g)-character of L(f ), is the same for almost all x.In fact, for each positive integer n, χ(L(f )) modulo weights µ such that height(λ − µ) > n, is the same for all but finitely many x.
We use the following standard lemma from linear algebra.
Lemma 3.14.Let V, W be complex vector spaces, with dim W < ∞, and let A i (u) : V → W , i ∈ N, be a countable set of linear operators rationally depending on a complex parameter x.Let Then there exists a finite set S ⊂ C such that for all x 1 , x 2 ∈ C \ S and x 3 ∈ S we have Proposition 3.15.Let f i (u, x), i ∈ I, be rational functions of u and x such that for each x ∈ C, f i (u, x) defines a rational ℓ-weight f (x).Then for all α ∈ Q + there exists a finite set S ⊂ C such that for all x 1 , x 2 ∈ C \ S and x 3 ∈ S we have Proof.By induction on height(α), making use of Lemma 3.14.
3.5.Dual modules.Given V ∈ Ob O we shall write V * for the restricted left dual of V .That is, V * is the space of linear maps λ : V → C with finite support on a weighted basis of V , equipped with the left U q (g)-action given by (x.λ)(v) = λ(S(x).v).It is clear that V * is a weight module whose weight spaces are all finite-dimensional.
Let R(g) denote the irreducible lowest ℓ-weight U q ( g)-module with lowest ℓ-weight g.
Proposition 3.16.For all f ∈ R, L(f Proof.L(f ) * is irreducible and so isomorphic to some R(g); we shall now show that g = f −1 .Indeed, by definition the following diagram commutes for all x ∈ U q ( g): Now suppose we take λ to be the lowest weight vector in L(f ) * and v to be the highest weight vector in L(f ).Note λ(v) = 0.It follows from (2.9) that identically, which can hold only if the rational functions f i (u) and g i (u) obey g i (u)f i (u) = 1, as claimed.
Given a rational ℓ-weight f , let us define f † by Note that f † is again a rational ℓ-weight, and that (f † ) † = f .From (2.5-2.6)one sees that R(f −1 ) φ ∼ = L(f † ), where φ denotes the pull-back via the Cartan involution.Hence we have the following.
4. Description of irreducibles in category Ô when g = sl 2 Throughout this section, g = sl 2 .Recall [Jim85, CP91] that for any a ∈ C × there is a homomorphism of algebras ev a : U q ( sl 2 ) → U q (sl 2 ) such that ev a (c 1/2 ) = 1 and These maps are called evaluation homomorphisms, and the pull-backs of U q (sl 2 )-modules by the ev a are called evaluation modules.One of the first key results in the theory of finite-dimensional representations of quantum affine algebras is that every irreducible U q ( sl 2 )-module is isomorphic to a tensor product of evaluation modules [CP91].In this section we give the analogous description of the irreducibles in Ô.
4.1.Strings.Let V (µ) a ∈ Ob Ô denote the pull-back via ev a of the irreducible U q ( sl 2 )-module It is irreducible, with highest ℓ-weight given by the rational function We refer to any rational function of u of this form as a string.
Definition 4.1.We say that two strings S µ (a) and are either disjoint, or one is contained in the other.
We say that the string S µ (a) starts at aq −µ−1 and ends at aq µ−1 .We call a string finite if it starts to the left of its end, where we say a is to the left of b if a ∈ bq −2Z ≥0 .Thus S µ (a) is finite if and only if µ ∈ Z ≥0 .If S µ (a) is finite we associate it with the finite set aq µ−1−2Z ≥0 ∩ aq −µ−1+2Z ≥0 = {aq −µ−1 , aq −µ+1 , . . ., aq µ−3 , aq µ−1 }, and we say b is inside S µ (a) if it belongs to this set.In this language, two strings are in general position if and only if (1) if neither string is finite then neither string starts to the left of the end of the other; (2) if one string is finite and the other is not, then the non-finite string neither starts nor ends inside the finite one; (3) if both strings are finite then their sets are either disjoint or one is contained in the other.
The final part is the usual condition for finite-dimensional representations, c.f. [CP91,CP94b].The reader should be warned that in [CP91] the q-string corresponding to (4.2) is defined not to include the element aq −µ−1 , in contrast to our convention.Proposition 4.2.If L(f ) ∈ Ob Ô then the corresponding rational function f (u) can be written in the form in such a way that each pair (S µ i (a i ), S µ j (a j )), 1 ≤ i < j ≤ r, is in general position.

Irreducible tensor products.
The following proposition, stated in [CP91] for finite-dimensional modules, remains valid in Ô.
Proposition 4.4.There is a basis (v i ) 0≤i≤dim(V (µ)a)−1 of V (µ) a on which the action of the generators x ± 1,k is given by: Proof.The check is straightforward, using (4.1) and the usual basis of the irreducible U q (sl 2 )-module V (µω 1 ).
Proposition 4.5.In the dual basis (v * i ) 0≤i≤dim(V (µ)a)−1 of (V (µ) a ) * the action of the generators x ± 1,k is given by: Proof.By direct calculation, making use of the relation ev a •S = S • ev aq 2 satisfied by the antipode of U q ( sl 2 ) [CP91].(One checks this equality on the Chevalley generators (2.1), using the relations Theorem 4.6.Let a 1 , . . ., a r ∈ C × and µ 1 , . . ., µ r ∈ C × .The tensor product is irreducible if and only if each pair (S µ i (a i ), S µ j (a j )), 1 ≤ i < j ≤ r, is in general position.
The module V (µ) a ⊗ V (ν) b has a proper submodule containing Ω 0 if and only if (V (ν) b ) * ⊗ (V (µ) a ) * has a factor module containing Ω * 0 ; that is, if and only if (V (ν) b ) * ⊗ (V (µ) a ) * has a submodule not containing Ω * 0 .Given Proposition 4.5, a similar calculation to the one referred to above shows that this is the case if and only if By inspection one verifies that this condition holds precisely when (S µ (a), S ν (b)) are in general position.
Turning to the general case, for the "only if" part we argue as follows.Suppose some pair (S µ i (a i ), S µ j (a j )), 1 ≤ i < j ≤ r, is not in general position.If the tensor product (4.4) is irreducible then it is irreducible for all orderings of the tensor factors, c.f. Proposition 3.13.So it is enough to show it is reducible for some ordering of the tensor factors.Pick any ordering in which V (µ i ) a i and V (µ j ) a j are adjacent; then the tensor product is indeed reducible, because it has a factor V (µ i ) a i ⊗ V (µ j ) a j which is reducible, as above.
Now we prove the "if" part.The argument is essentially as in [CP91], and is by an induction on the number r of tensor factors.We have the case r = 2 above.For the inductive step, we may suppose that V (µ 1 ) a 1 ⊗ • • • ⊗ V (µ r−1 ) a r−1 is generated as a U q ( sl 2 )-module by a tensor product of highest weight vectors of the tensor factors, ar is generated as a U q ( sl 2 )-module by the vectors (Ω ′ ⊗ v i ) 0≤i≤dim(V (µr)a r .We now argue by induction on i that Ω ′ ⊗ v i ∈ U q ( sl 2 ).Ω, where Ω := Ω ′ ⊗ v 0 .This is trivially true for i = 0.For the inductive step, suppose Ω ′ ⊗ v j ∈ U q ( sl 2 ).Ω for all 0 ≤ j ≤ i, and consider the action of x − 1,k on Ω ′ ⊗ v i .Recall from [CP91] the following property of the comultiplication of the quantum loop algebra Therefore if we let d k,j be the eigenvalue of φ + 1,k on the highest weight space of V (µ 1 ) a 1 ⊗• • •⊗V (µ j ) a j , set b r := a r q µr−2i and b j := a j q µ j for each 1 ≤ j < r, and define then for all k ∈ Z ≥1 we have

.5)
Note that A = (A k,j ) 1≤k≤r,0≤j≤r−1 is a square matrix.To complete the proof it is enough to show that det A = 0, for then equation (4.5) allows Ω ′ ⊗ v i+1 to be expressed as a linear combination of x − 1,k .(Ω′ ⊗ v i ), 1 ≤ k ≤ r, which completes the inductive step on i, and consequently also the inductive step on r.
It was shown in [CP91] that If det A = 0 then there exist j, k such that either (1) 1 ≤ j < k < r and a j = q µ j +µ k a k , or else (2) 1 ≤ j < k = r and a j = q µ j +µr−2i a r .
The first of these is impossible since S µ j (a j ) and S µ k (a k ) are in general position.For the second, by making use of the freedom noted above to reorder the tensor factors, we may assume that dim V (µ r ) ar ≤ dim(V (µ j ) a j ) for all 1 ≤ j ≤ r.That is, if any of the tensor factors have finite dimension, then none have dimension lower than V (µ r ) ar .Now if µ r ∈ Z ≥0 then i < µ r and so (2) is also ruled out since S µ j (a j ) and S µ k (a k ) are in general position.

Least affinizations
In this section our main result is Theorem 5.7, which classifies the least affinizations in types ABCF and G.

Definition of least affinizations.
It is natural to consider affinizations of the simple objects of O, in the sense of the following definition, which is adapted directly from the case of finitedimensional representations discussed in [Cha95].
Two affinizations of V (µ) are equivalent if they are isomorphic as U q (g)-modules.
Proposition 5.2.For all f ∈ R, L(f ) and L(f † ) are isomorphic as U q (g)-modules.
Proof.For any V ∈ Ob O, (V * ) ϕ ∼ = V as U q (g)-modules, where (V * ) ϕ is the pull-back via the Cartan involution ϕ of V * .This is clear since Consequently, the result follows from Corollary 3.17.
Recall that an element X of a partially ordered set (P, ≺) is said to be minimal if there is no Y ∈ P such that Y ≺ X, and is said to be least if Y X for all Y ∈ P .A partially ordered set has at most one least element.If a least element does exist then it is the unique minimal element.
For each µ ∈ h * , there is a partial order on the equivalence classes of affinizations of V (µ), defined as follows.Let L(f ), L(f ′ ) ∈ Ob Ô be two affinizations of V (µ).We say that the class of L(f ) weakly precedes that of L(f ′ ) if and only if for all α ∈ Q + either This partial order is given in terms of the dimensions of U q (g)-weight spaces, but it could equivalently have been defined in terms of multiplicities of U q (g)-module composition factors.
Definition 5.3.A minimal (resp.least) affinization of V (µ) ∈ Ob O is an equivalence class of affinizations of V (µ) which is minimal (resp.least) with respect to this partial order.By a slight overloading we say also that any representative of such a class is a minimal (resp.least) affinization.
Proof.If f is of this form, then all non-highest weights of L(f ) are dominated by Note that the least affinization of the irreducible U q (sl 2 )-module V (µω 1 ), µ ∈ C, is an evaluation module, V (µ) a (in the notation of §4).
In view of Proposition 3.15, minimal (resp.least) affinizations with generic highest weights are limits (or rather analytic continuations) of minimal (resp.least) affinizations of finite-dimensional modules.Namely, let λ = i∈I λ i ω i .Let I = J ⊔K. Fix λ j , j ∈ J, to be equal to given nonnegative integers.Let be the normalized character of a least affinization L(f ) of V (λ).
Corollary 5.6.There exists a limit lim where λ k run over N, and this limit is equal to χ(λ) with generic Proof.This follows from Proposition 3.15.
Note that in particular there exists a limit of the normalized characters of Kirillov-Reshetikhin modules and it is equal to the normalized character of the Kirillov-Reshetikhin module with generic nontrivial component.Compare [HJ11].
In fact, in a similar way, there exists an analytic continuation of χ q (L(f )) with 5.2.Classification of minimal affinizations in types ABCFG.For the remainder of this section we suppose g is of type ABCF or G.We pick a straight labelling of the Dynkin diagram in which In these cases the following theorem, which is the main result of §5, shows that every simple object V ∈ Ob O has a least affinization.
The rest of §5 is devoted to proving this theorem.After some preliminary lemmas in §5.3, in §5.4 we treat the case in which λ has support at the two end nodes of the Dynkin diagram; in §5.5 we treat the case in which λ has support at the two end nodes and one other node.Finally, the theorem is proved in §5.6.
Let τ a : U q ( g) → U q ( g), a ∈ C × , be the automorphism defined by its action on generators according to where t a : R → R is defined by (t a f i )(u) := f i (au).It follows from Theorem 5.7 that if L(f ) and L(f ′ ) are least affinizations of V (λ) then there exists an a ∈ C × such that either f = t a (f ′ ) or f = t a (f ′ † ), c.f. (3.7).
Our strategy for identifying least affinizations relies on computing "the top" part of the qcharacter, in the following sense.For each 0 ≤ M ≤ N = rank(g), define We write χ q (L(f ))| M for the q-character of L(f ) truncated to include only those terms whose weights lie in wt(f )W M .In certain cases, we shall compute χ q (L(f ))| M for each 0 ≤ M ≤ N .That is, informally speaking, we shall consider all weights that can be reached from the highest weight by lowering at most once in each simple direction.As it turns out, this is sufficient to distinguish least affinizations from others.

Preliminary lemmas.
Let ÛJ be the subalgebra generated by (x + i,r ) i∈J,r∈Z , (x − i,r ) i∈J,r∈Z , (h i,r ) i∈J,r∈Z\{0} and (k ±1 i ) i∈J subject to the relations (2.2).Given a rational ℓ-weight f = (f i (u)) i∈I , we write f J for the ℓ-weight (f i (u)) i∈J of ÛJ , and L(f J ) for the irreducible ÛJ -module with highest ℓ-weight f J .
Similarly, let U J be the subalgebra generated by (x + i,0 ) i∈J , (x − i,0 ) i∈J and (k ±1 i ) i∈J .Given a weight ρ ∈ h * of U q (g) we write ρ J be for weight (ρ i ) i∈J of U J , and V (ρ J ) for the irreducible U J -module with highest weight ρ J .
Lemma 5.8.Let I 1 , I 2 , . . ., I k be subdiagrams of I such that the corresponding diagram subalgebras g I 1 , g I 2 , . . ., g I k of g are simple and pairwise commuting.Let L(f ) ∈ Ob Ô with highest ℓ-weight vector v. Then Proof.Let k = 1.Suppose w ∈ ÛI 1 .v is a singular vector with respect to ÛI 1 .Then, on weight grounds, w is a singular vector with respect to U q ( g).Therefore, since L(f ) is irreducible, w is proportional to v. Hence ÛI 1 .v is irreducible.
For general k the lemma follows by the mutual commutativity of the g I k .
Lemma 5.9 (The restriction lemma).Let J ⊆ I.An affinization L(f ) ∈ Ob Ô of V (µ) is least only if the simple ÛJ -module through a highest ℓ-weight vector v of L(f ) is a least affinization of V (µ J ).
Proof.Suppose there is a J ⊆ I such that ÛJ .v is not least.Then there is another affinization of V (µ J ), say L(s J ), whose equivalence class does not weakly succeed that of L(f J ).It follows that the class of L(g) does not weakly succeed that of L(f ), where the rational ℓ-weight g = (g i (u)) i∈I,r∈Z ≥0 of U q ( g) is given by Corollary 5.10.A U q ( g)-module L(f ) ∈ Ob Ô is a least affinization only if f i (u) is a string -c.f.(4.2) -for each i ∈ I.
Proof.We use the restriction lemma with J = {i}.Let v be a highest ℓ-weight vector of L(f ).
It follows from the results of §4 that Û{i} .vwt(f )α i −1 has dimension 1 if f i (u) is a string and dimension ≥ 2 otherwise.
In the following lemma, we use ≃ to denote equality up to a multiplicative constant.
In particular Proof.Given Corollary 3.9, it is enough to show that statements (i), (ii) and (iii) hold in the irreducible Û{i} -module W whose highest weight is v.So in the rest of this proof, we work in W .Let ϕ i,s := φ + i,s − φ − i,s .In case (i) we have ϕ i,s .v= αa s (1 − c a ) for some α ∈ C × and thus, for all r, s ∈ Z, (ϕ i,r+s − a r ϕ i,s ).v = 0, and hence x + i,s (x − i,r .v− a r w) = 0, where w a = x − i,0 .v∈ W .
In case (ii) we shall show that for all r, s ∈ Z, ).v = 0, which in turn holds because Finally in case (iii) we shall show that for all r, s ∈ Z, x + i,s (x − i,r v − (a r w + ra r w ′ )) = 0 where This is true, given that for some γ ∈ C × .Similar direct calculations show that, in each case (i), (ii) and (iii), the given vectors in W have the ℓ-weights claimed, and are not in r∈Z ker(x + i,r ) and hence are not zero.Given any a, c ∈ C × , define: a 1 := a, and a i+1 := a i q −B i,i+1 for each 1 ≤ i ≤ N − 1, (5.2) and then, for 0 ≤ K, S ≤ N , let Proposition 5.12.Suppose f ∈ R is of the form for some µ, ν ∈ C × , and f j (u) = 1 for all 1 < j < N .For each 0 ≤ M < N , (5.7) Let us consider the generic case in which all of these rational functions are in lowest terms as written -i.e.there are no cancellations -and in which none have poles of second order.Now certainly, Now by considering 0 = [x + K+1,s , x − K,r ].v K−1,N −K we see that w ∈ s∈Z ker x + K+1,s .So w = 0. Similarly y = 0. Therefore u spans L(f ) f K,N−K , and dim(L(f ) f K,N−K ) = 1, as required.
For f 0,N and f N,0 the logic is simpler because there is no need to identify vectors.On ℓ-weight grounds and by Lemma 5.11 part (ii) one finds dim(L(f ) f 0 It remains to consider the exceptional cases in which cancellations or coincident poles occur in the functions (f K−1,N −K ) K (u) as written in (5.6-5.7).In view of Lemma 5.11, the dimensions of the 'outermost' ℓ-weight spaces L(f ) f 0,N and L(f ) f N,0 drop to zero under exactly the conditions specified in (5.4).On the other hand one finds that there are no conditions under which the dimensions of the ℓ-weight spaces L(f for some a, c ∈ C obeying at least one of the following equations: (5.9) Proof.If L(f ) obeys one of the two conditions (5.9) then L(f † ) obeys the other, and by Proposition 5.2, both define the same equivalence class.The result is then immediate from Proposition 5.12 Remark 5.14.If V (µω 1 + κω N ) is finite-dimensional, i.e. µ, κ ∈ Z ≥0 , at most one of the equations (5.9) can hold.However, for infinite-dimensional modules they are not mutually exclusive.In type A 2 , for example, the least affinization of V = V (µω 1 − µω 2 ) is the class of L(f ), where This is an evaluation module, so L(f ) ∼ = V as U q (g)-modules.By the usual Weyl character formula, all weight spaces of V are one-dimensional.
and, for 0 ≤ K, L < j and 0 ≤ R, S < N + 1 − j, Proposition 5.15.Suppose the rational ℓ-weight f is of the form affinizations of V (µω 1 + κω N ) strictly succeed this class.So it is indeed least.Note in particular that it strictly precedes the class defined by By Proposition 5.12, (I-IV) are the only affinizations that are least for the subdiagrams {1, . . ., j} and {j, . . ., N }.The term (5.11) vanishes in cases (I) and (II) but not in cases (III) and (IV).
5.6.Proof of Theorem 5.7.First we restate Theorem 5.7 in the following form.We pick and fix, for this subsection, a λ ∈ h Now, in view of Corollaries 5.13 and 5.17, this statement is equivalent to the following proposition.
Proposition 5.18.An affinization L(f Proof.The "only if" part follows from Lemma 5.9 and Corollaries 5.10, 5.13 and 5.17.For the "if" part, suppose f is such that conditions (1-3) hold.Then f is of the form given in (5.13).
When K = 1 for every weight µ = λ of L(f ), then µ ≤ λ − α i 1 and the result is clear.So suppose that λ has support at K ≥ 2 nodes.Let L(s), s ∈ R, be any affinization of V (λ) and w a highest ℓ-weight vector of L(s).Define We first make the following observation: If A (2) = A (3) = ∅ then L(s) and L(f ) are isomorphic as U q (g)-modules.
(5.14) Indeed, by Lemma 5.9 and Corollaries 5.13 and 5.17, A (2) = A (3) = ∅ holds only if f and s are of the form given in (5.13).Then, as noted in §5.2, there exists an a ∈ C × such that either f = t a (f ′ ) or f ∼ = t a (f ′ † ), c.f. (3.7), and therefore (5.14) follows from Proposition 5.2.Now we consider the case that A (2) = ∅ or A (3) = ∅.We shall show that the class of L(f ) strictly precedes that of L(s) in the partial order.By Proposition 5.12, for all k ∈ A It remains to compare weight spaces with weights that are not dominated by weights of the form λ − α i k − α i k +1 − . . .− α i k+1 , k ∈ A (2) , or λ − α i k − α i k +1 − . . .− α i k+2 , k ∈ A (3) .Let µ be any such weight.Then there exist simple, pairwise commuting, diagram subalgebras g 1 , . . ., g T of g with corresponding subdiagrams I 1 , . . ., I T , and elements α (t) ∈ Q + It for each 1 ≤ t ≤ T , such that µ = λ − T t=1 α (t) and such that, for each 1 ≤ t ≤ T , {i k , i k + 1, . . .i k+1 } ⊆ I t for all k ∈ A (2) and {i k , i k + 1, . . ., i k+2 } ⊆ I t for all k ∈ A (3) .It follows from the observation (5.14) above that for each 1 ≤ t ≤ T , L(s It ) is isomorphic to L(f It ) as a U q (g)-module.Therefore, by Lemma 5.8, we have: Hence the class of L(f ) strictly precedes that of L(s) in the partial order, as required.

Character conjectures
In this section we give a series of three conjectures, of increasing generality, for the classical (i.e.U q (g)-) character of certain irreducible representations in Ô.Our main interest is in the least affinizations of Verma modules, and these provide our starting point.Computer experiments, using the algorithm of [FM01], suggest that their characters have a simple form, similar to the Weyl denominator.Conjecture 6.1 (Least affinization of the generic Verma module).Suppose g is of type ABCF or G. Let L(f ) ∈ Ob Ô be a least affinization of V (λ), where λ = i∈I λ i ω i with λ i / ∈ Z for any i ∈ I. Then .
This conjecture is known to hold in at least two special cases: Proposition 6.2.Conjecture 6.1 is true in types A n , n ∈ Z ≥1 , and B 2 .
Proof.In type A, least affinizations are evaluation modules.So the least affinization L(f ) of an irreducible Verma module V is isomorphic to V as a U q (g)-module.The formula for χ(L(f )) in Conjecture 6.1 is therefore correct, because it agrees with the usual character formula for Verma modules, i.e. the Weyl denominator.(Note that in type A, max i∈I ω ∨ i , α = 1 for all α ∈ ∆ + .)In type B 2 , for all k, ℓ ∈ Z ≥0 the least affinization L(f ) of V (kω 1 + ℓω 2 ) has as its U q (g)-module decomposition [Cha95] where every pole and every zero of f (a) i (u) lies in aq Z .Conjecture 6.5.Suppose that for each a ∈ C × /q Z and each i ∈ I there is an n a (i) ∈ Z ≥−1 , an X ∈ C and an r ∈ Z such that f (1) for all i, j ∈ S (a) with i = j, we have U (a) ∩ {i, j} = ∅; (2) for every i ∈ S (a) , j ∈ U (a) such that {i, j} ∩ (S (a) ∪ U (a) ) = {i, j}, there is a straight labelling (i = j 1 , j 2 , j 3 , . . ., j K = j) of {i, j} such that Note that condition (3) is redundant except when the node j is trivalent.To see that Conjecture 6.5 does entail Conjecture 6.3, let us give the following.
Proof of Conjecture 6.3 assuming Conjecture 6.5.Let i 1 < i 2 < • • • < i K be the nodes of J. Without loss of generality, suppose f obeys condition (I) in Theorem 5.7.(If not, reverse the ordering of the Dynkin diagram.)Now we apply Conjecture 6.5.For each 1 ≤ k ≤ K there is exactly one a ∈ C × /q Z such that i k ∈ S (a) ; and for this a, U (a) = {i k+1 } when k < K, while U (a) = ∅ when k = K.Thus Conjecture 6.5 implies , where for convenience we define ω ∨ i K+1 := 0. The result follows provided we can show that And indeed, this equality is a consequence of the following statement, which can be seen by caseby-case inspection: Let g be of type ABCF or G and rank N , and pick a straight labelling of the nodes of the Dynkin diagram; then for any positive root α, the N -tuple ( ω ∨ i , α ) 1≤i≤N is unimodal,

w a − b r w b =
This follows from ((a − b)ϕ i,r+s + (a r b − b r a)ϕ i,s + (b r − a r )ϕ i,s+1

5. 4 .
The case of two nodes.We write δ a,b :=