# Universal Verma Modules and the Misra-Miwa Fock Space

Authors: Arun Ram; Peter Tingley
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

The Misra-Miwav-deformed Fock space is a representation of the quantized affine algebra Uv(𝔰𝔩^ℓ). It has a standard basis indexed by partitions, and the nonzero matrix
entries of the action of the Chevalley generators with respect to this basis are powers of v. Partitions also index the polynomial Weyl modules for Uq(𝔤𝔩N) as N tends to infinity. We explain how the powers of v which appear in the Misra-Miwa Fock space also appear naturally
in the context of Weyl modules. The main tool we use is the Shapovalov determinant for a
universal Verma module.


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## Body

## 1. Introduction

Fock space is an infinite dimensional vector space which is a representation of several important algebras, as described in, for example, [1, Chapter 14]. Here we consider the charge zero part of Fock space, which we denote by F, and its v-deformation Fv. The space F has a standard ℚ-basis {|μ〉∣λisapartition} and Fv:=F⊗ℚℚ(v). Following Hayashi [2], Misra and Miwa [3] define an action of the quantized universal enveloping algebra Uv(𝔰𝔩̂ℓ) on Fv. The only  nonzero matrix elements 〈μ|Fi̅|λ〉 of the Chevalley generators Fi̅ in terms of the standard basis occur whenμ is obtained by adding a single i̅-colored box to λ, and these are powers of v.We show that these powers ofv also appear naturally in the following context: partitions with at most N parts index polynomial Weyl modules Δ(λ) for the integral quantum group Uq𝒜(𝔤𝔩N). Let V be the standard N dimensional representation of Uq𝒜(𝔤𝔩N). If the matrix element 〈μ|Fi̅|λ〉 is  nonzero then, for sufficiently large N, (Δ𝒜(λ)⊗𝒜V)⊗𝒜ℚ(q) contains the highest weight vector of weightμ. There is a unique such highest weight vector vμ which satisfies a certain triangularity condition with respect to an integral basis of Δ𝒜(λ)⊗𝒜V. We show that the matrix element 〈μ|Fi̅|λ〉 is equal to vvalϕ2ℓ(vμ,vμ), where (·,·) is the Shapovalov form and valϕ2ℓ is the valuation at the cyclotomic polynomial ϕ2ℓ.Our proof is computational, making use of the Shapovalov determinant [4–6]. This is a formula for the determinant of the Shapovalov form on a weight space of a Verma module. The necessary computation is most easily done in terms of the universal Verma modules introduced in the classical case by Kashiwara [7] and studied in the quantum case by Kamita [8]. The statement for Weyl modules is then a straightforward consequence.Before beginning, let us discuss some related work. In [9], Kleshchev carefully analyzed the 𝔤𝔩N-1 highest weight vectors in a Weyl module for 𝔤𝔩N and used this information to give modular branching rules for symmetric group representations. Brundan and Kleshchev [10] have explained that highest weight vectors in the restriction of a Weyl module to 𝔤𝔩N-1 give information about highest weight vectors in a tensor product Δ(λ)⊗V of a Weyl module with the standard N-dimensional representation of 𝔤𝔩N. Our computations put a new twist on the analysis of the highest weight vectors in Δ(λ)⊗V, as we study them in their “universal” versions and by the use of the Shapovalov determinant. Our techniques can be viewed as an application of the theory of Jantzen [11] as extended to the quantum case by Wiesner [12].Brundan [13] generalized Kleshchev's [9] techniques and used this information to give modular branching rules for Hecke algebras. As discussed in [14, 15], these branching rules are reflected in the fundamental representation of 𝔰𝔩̂p and its crystal graph, recovering much of the structure of the Misra-Miwa Fock space. Using Hecke algebras at a root of unity, Ryom-Hansen [16] recovered the full Uv(𝔰𝔩̂ℓ) action on Fock space. To complete the picture, one should construct a graded category, where multiplication by v in the 𝔰𝔩̂ℓ representation corresponds to a grading shift. Recent work of Brundan-Kleshchev [17] and Ariki [18] explains that one solution to this problem is through the representation theory of Khovanov-Lauda-Rouquier algebras [19, 20]. It would be interesting to explicitly describe the relationship between their category and the present work. Another related construction due to Brundan-Stroppel considers the case when the Fock space is replaced by ∧mV⊗∧nV, where V is the natural 𝔤𝔩∞ module and m,n are fixed natural numbers.We would also like to mention very recent work of Peng Shan [21] which independently develops a similar story to the one presented here, but using representations of a quantum Schur algebra where we use representations of Uε(𝔤𝔩N). The approach taken there is somewhat different and in particular relies on localization techniques of Beĭlinson and Bernstein [22].This paper is arranged as follows. Sections2 and 3 are background on the quantum group Uq(𝔤𝔩N) and the Fock space Fv. Sections 4 and 5 explain universal Verma modules and the Shapovalov determinant. Section 6 contains the statement and proof of our main result relating Fock space and Weyl modules.


## 2. The Quantum GroupUq(𝔤𝔩N) and Its Integral Form Uq𝒜(𝔤𝔩N)

This is a very brief review, intended mainly to fix notation. With slight modifications, the construction in this section works in the generality of symmetrizable Kac-Moody algebras. See [23, Chapters 6 and 9] for details.


### 2.1. The Rational Quantum Group

Uq(𝔤𝔩N) is the associative algebra over the field of rational functions ℚ(q) generated by (2.1)X1,…,XN-1,Y1,…,YN-1,L1±1,…,LN±1,
with relations (2.2)LiLj=LjLi,LiLi-1=Li-1Li=1,XiYj-YjXi=δi,jLiLi+1-1-Li+1Li-1q-q-1,LiXjLi-1={qXj,ifi=j,q-1Xj,ifi=j+1,Xj,otherwise,LiYjLi-1={q-1Yj,ifi=j,qYj,ifi=j+1,Yj,otherwise,XiXj=XjXi,YiYj=YjYi,if|i-j|≥2,Xi2Xj-(q+q-1)XiXjXi+XjXi2=Yi2Yj-(q+q-1)YiYjYi+YjYi2=0,if|i-j|=1.
The algebra Uq(𝔤𝔩N) is a Hopf algebra with coproduct and antipode given by
(2.3)Δ(Li)=Li⊗Li,S(Li)=Li-1,Δ(Xi)=Xi⊗LiLi+1-1+1⊗Xi,S(Xi)=-XiLi-1Li+1,Δ(Yi)=Yi⊗1+Li-1Li+1⊗Yi,S(Yi)=-LiLi+1-1Yi,
respectively, (see [23, Section 9.1]).As aℚ(q)-vector space, Uq(𝔤𝔩N) has a triangular decomposition (2.4)Uq(glN)≅Uq(glN)<0⊗Uq(glN)0⊗Uq(glN)>0,
where the inverse isomorphism is given by multiplication (see [23, Proposition 9.1.3]). Here Uq(𝔤𝔩N)<0 is the subalgebra generated by the Yi for i=1,…,N-1, Uq(𝔤𝔩N)>0 is the subalgebra generated by the Xi for i=1,…,N-1, and Uq(𝔤𝔩N)0 is the subalgebra generated by the Li±1 for i=1,…,N.


### 2.2. The Integral Quantum Group

Let𝒜=ℤ[q,q-1]. For n,k∈ℤ>0 and c∈ℤ, let
(2.5)[n]:=qn-q-nq-q-1,x(k):=xk[k][k-1]⋯[2][1],[x;ck]:=∏s=1kxqc+1-s-x-1qs-1-cqs-q-s,
in ℚ(q,x). The restricted integral formUq𝒜(𝔤𝔩N) is the 𝒜-subalgebra of Uq(𝔤𝔩N) generated by Xi(k), Yi(k), Li±1 and [Li;ck] for 1≤i≤N,c∈ℤ,k∈ℤ>0. As discussed in [24, Section 6], this is an integral form in the sense that (2.6)UqA(glN)⊗AQ(q)=Uq(glN).
As with Uq(𝔤𝔩N), the algebra Uq𝒜(𝔤𝔩N) has a triangular decomposition (2.7)UqA(glN)≅UqA(glN)<0⊗UqA(glN)0⊗UqA(glN)>0,
where the isomorphism is given by multiplication (see [23, Proposition 9.3.3]). In this case, Uq𝒜(𝔤𝔩N)<0 is the subalgebra generated by the Yi(k), Uq𝒜(𝔤𝔩N)>0 is the subalgebra generated by the Xi(k), and Uq𝒜(𝔤𝔩N)0 is generated by Li±1 and [Li;ck] for 1≤i≤N, c∈ℤ, and k∈ℤ>0.


### 2.3. Rational Representations

The Lie algebra𝔤𝔩N=MN(ℂ) of N×N matrices has standard basis {Eij∣1≤i,j≤N}, where Eij is the matrix with 1 in position (i,j) and 0 everywhere else. Let 𝔥=span{E11,E22,…,ENN}. Let εi∈𝔥* be the weight of 𝔤𝔩N given by εi(Ejj)=δi,j. Define (2.8)hZ*:={λ=λ1ε1+λ2ε2+⋯+λNεN∈h*∣λ1,…,λN∈Z},(hZ*)+:={λ=λ1ε1+λ2ε2+⋯+λNεN∈hZ*∣λ1≥λ2≥⋯≥λN},P+:={λ=λ1ε1+λ2ε2+⋯+λNεN∈(hZ*)+∣λN≥0},R+:={εi-εj∣1≤i<j≤N},Q:=spanZ(R+),Q+:=spanZ≥0(R+),Q-:=spanZ≤0(R+)
to be the set of integral weights, the set of dominant integral weights, the set of dominant polynomial weights, the set of positive roots, the root lattice, the positive part of the root lattice, and the negative part of the root lattice, respectively.For an integral weightλ=λ1ε1+⋯+λNεN, the Verma moduleM(λ) for Uq(𝔤𝔩N) of the highest weight λ is (2.9)M(λ):=Uq(glN)⊗Uq(glN)≥0Q(q)λ,
where ℚ(q)λ=spanℚ(q){vλ} is the one dimensional vector space over ℚ(q) with Uq(𝔤𝔩N)≥0 action given by (2.10)Xi⋅vλ=0,Lj⋅vλ=qλjvλ,for1≤i≤N-1,1≤j≤N.Theorem 2.1 (see [23, Chapter 10.1]).
Ifλ∈(𝔥ℤ*)+ then  M(λ) has a unique finite dimensional quotient  Δ(λ) and the map  λ↦Δ(λ) is a bijection between  (𝔥ℤ*)+, and the set of irreducible finite dimensional  Uq(𝔤𝔩N)-modules.Asingular vector in a representation of Uq(𝔤𝔩N) is a vector v such that Xi·v=0 for all i.


### 2.4. Integral Representations

Theintegral Verma moduleM𝒜(λ) is the Uq𝒜(𝔤𝔩N)-submodule of M(λ) generated by vλ. The integral Weyl moduleΔ𝒜(λ) is the Uq𝒜(𝔤𝔩N)-submodule of Δ(λ) generated by vλ. Using (2.6) and (2.4), (2.11)MA(λ)⊗AQ(q)=M(λ),ΔA(λ)⊗AQ(q)=Δ(λ).
In general, Δ𝒜(λ) is not irreducible as a Uq𝒜(𝔤𝔩N) module.


## 2.1. The Rational Quantum Group

Uq(𝔤𝔩N) is the associative algebra over the field of rational functions ℚ(q) generated by (2.1)X1,…,XN-1,Y1,…,YN-1,L1±1,…,LN±1,
with relations (2.2)LiLj=LjLi,LiLi-1=Li-1Li=1,XiYj-YjXi=δi,jLiLi+1-1-Li+1Li-1q-q-1,LiXjLi-1={qXj,ifi=j,q-1Xj,ifi=j+1,Xj,otherwise,LiYjLi-1={q-1Yj,ifi=j,qYj,ifi=j+1,Yj,otherwise,XiXj=XjXi,YiYj=YjYi,if|i-j|≥2,Xi2Xj-(q+q-1)XiXjXi+XjXi2=Yi2Yj-(q+q-1)YiYjYi+YjYi2=0,if|i-j|=1.
The algebra Uq(𝔤𝔩N) is a Hopf algebra with coproduct and antipode given by
(2.3)Δ(Li)=Li⊗Li,S(Li)=Li-1,Δ(Xi)=Xi⊗LiLi+1-1+1⊗Xi,S(Xi)=-XiLi-1Li+1,Δ(Yi)=Yi⊗1+Li-1Li+1⊗Yi,S(Yi)=-LiLi+1-1Yi,
respectively, (see [23, Section 9.1]).As aℚ(q)-vector space, Uq(𝔤𝔩N) has a triangular decomposition (2.4)Uq(glN)≅Uq(glN)<0⊗Uq(glN)0⊗Uq(glN)>0,
where the inverse isomorphism is given by multiplication (see [23, Proposition 9.1.3]). Here Uq(𝔤𝔩N)<0 is the subalgebra generated by the Yi for i=1,…,N-1, Uq(𝔤𝔩N)>0 is the subalgebra generated by the Xi for i=1,…,N-1, and Uq(𝔤𝔩N)0 is the subalgebra generated by the Li±1 for i=1,…,N.


## 2.2. The Integral Quantum Group

Let𝒜=ℤ[q,q-1]. For n,k∈ℤ>0 and c∈ℤ, let
(2.5)[n]:=qn-q-nq-q-1,x(k):=xk[k][k-1]⋯[2][1],[x;ck]:=∏s=1kxqc+1-s-x-1qs-1-cqs-q-s,
in ℚ(q,x). The restricted integral formUq𝒜(𝔤𝔩N) is the 𝒜-subalgebra of Uq(𝔤𝔩N) generated by Xi(k), Yi(k), Li±1 and [Li;ck] for 1≤i≤N,c∈ℤ,k∈ℤ>0. As discussed in [24, Section 6], this is an integral form in the sense that (2.6)UqA(glN)⊗AQ(q)=Uq(glN).
As with Uq(𝔤𝔩N), the algebra Uq𝒜(𝔤𝔩N) has a triangular decomposition (2.7)UqA(glN)≅UqA(glN)<0⊗UqA(glN)0⊗UqA(glN)>0,
where the isomorphism is given by multiplication (see [23, Proposition 9.3.3]). In this case, Uq𝒜(𝔤𝔩N)<0 is the subalgebra generated by the Yi(k), Uq𝒜(𝔤𝔩N)>0 is the subalgebra generated by the Xi(k), and Uq𝒜(𝔤𝔩N)0 is generated by Li±1 and [Li;ck] for 1≤i≤N, c∈ℤ, and k∈ℤ>0.


## 2.3. Rational Representations

The Lie algebra𝔤𝔩N=MN(ℂ) of N×N matrices has standard basis {Eij∣1≤i,j≤N}, where Eij is the matrix with 1 in position (i,j) and 0 everywhere else. Let 𝔥=span{E11,E22,…,ENN}. Let εi∈𝔥* be the weight of 𝔤𝔩N given by εi(Ejj)=δi,j. Define (2.8)hZ*:={λ=λ1ε1+λ2ε2+⋯+λNεN∈h*∣λ1,…,λN∈Z},(hZ*)+:={λ=λ1ε1+λ2ε2+⋯+λNεN∈hZ*∣λ1≥λ2≥⋯≥λN},P+:={λ=λ1ε1+λ2ε2+⋯+λNεN∈(hZ*)+∣λN≥0},R+:={εi-εj∣1≤i<j≤N},Q:=spanZ(R+),Q+:=spanZ≥0(R+),Q-:=spanZ≤0(R+)
to be the set of integral weights, the set of dominant integral weights, the set of dominant polynomial weights, the set of positive roots, the root lattice, the positive part of the root lattice, and the negative part of the root lattice, respectively.For an integral weightλ=λ1ε1+⋯+λNεN, the Verma moduleM(λ) for Uq(𝔤𝔩N) of the highest weight λ is (2.9)M(λ):=Uq(glN)⊗Uq(glN)≥0Q(q)λ,
where ℚ(q)λ=spanℚ(q){vλ} is the one dimensional vector space over ℚ(q) with Uq(𝔤𝔩N)≥0 action given by (2.10)Xi⋅vλ=0,Lj⋅vλ=qλjvλ,for1≤i≤N-1,1≤j≤N.Theorem 2.1 (see [23, Chapter 10.1]).
Ifλ∈(𝔥ℤ*)+ then  M(λ) has a unique finite dimensional quotient  Δ(λ) and the map  λ↦Δ(λ) is a bijection between  (𝔥ℤ*)+, and the set of irreducible finite dimensional  Uq(𝔤𝔩N)-modules.Asingular vector in a representation of Uq(𝔤𝔩N) is a vector v such that Xi·v=0 for all i.


## 2.4. Integral Representations

Theintegral Verma moduleM𝒜(λ) is the Uq𝒜(𝔤𝔩N)-submodule of M(λ) generated by vλ. The integral Weyl moduleΔ𝒜(λ) is the Uq𝒜(𝔤𝔩N)-submodule of Δ(λ) generated by vλ. Using (2.6) and (2.4), (2.11)MA(λ)⊗AQ(q)=M(λ),ΔA(λ)⊗AQ(q)=Δ(λ).
In general, Δ𝒜(λ) is not irreducible as a Uq𝒜(𝔤𝔩N) module.


## 3. Partitions and Fock Space

We now describe thev-deformed Fock space representation of Uv(𝔰𝔩̂ℓ) constructed by Misra and Miwa [3] following work of Hayashi [2]. Our presentation largely follows [25, Chapter 10].


### 3.1. Partitions

A partitionλ is a finite length  nonincreasing sequence of positive integers. Associated to a partition is its Ferrers diagram. We draw these diagrams as in Figure 1 so that, if λ=(λ1,…,λN), then λi is the number of boxes in row i (rows run southeast to northwest ↖). Say that λ is contained inμ if the diagram for λ fits inside the diagram forμ and let μ/λ be the collection of boxes ofμ that are not in λ. For each box b∈λ, the contentc(b) is the horizontal position of b and the colorc¯(b) is the residue of c(b) modulo ℓ. In Figure 1, the numbers c(b) are listed below the diagram. The size|λ| of a partition λ is the total number of boxes in its Ferrers diagram.Figure 1
The partition(7,6,6,5,5,3,3,1) with each box containing its color for ℓ=3. The content c(b) of a box b is the horizontal position of b reading right to left. The contents of boxes are listed beneath the diagram so that c(b) is aligned with all boxes b of that content.The setP+ of dominant polynomial weights from Section 2.3 is naturally identified with partitions with at most N parts. If λ∈P+, then (3.1)Δ(λ)⊗Δ(ε1)≅⨁1≤k≤Nλ+εk∈P+Δ(λ+εk)
as Uq(𝔤𝔩N)-modules. The diagram of λ+εk is obtained from the diagram of λ by adding a box on row k, and Δ(λ+εk) appears in the sum on the right side of (3.1) if and only if λ+εk is a partition. See, for example, [26, Section 6.1, Formula 6.8] for the classical statement and [23, Proposition 10.1.16] for the quantum case.


### 3.2. The Quantum Affine Algebra

LetUv′(𝔰𝔩̂ℓ) be the quantized universal enveloping algebra corresponding to the ℓ-node Dynkin diagramMore precisely,Uv′(𝔰𝔩̂ℓ) is the algebra generated by Ei̅,Fi̅,Ki̅±1, for i̅∈ℤ/ℓℤ, with relations (3.2)Ki̅Kj̅=Kj̅Ki̅,Ki̅Ki̅-1=Ki̅-1Ki̅=1,Ei̅Fj̅-Fj̅Ei̅=δi̅,j̅Ki̅-Ki̅-1v-v-1,Ki̅Ej̅Ki̅-1={v2Ej̅,ifi̅=j̅,v-1Ej̅,ifi̅=j̅±1,Ej̅,otherwise,Ki̅Fj̅Ki̅-1={v-2Fj̅,ifi̅=j̅,vFj̅,ifi̅=j̅±1,Fj̅,otherwise,Ei̅Ej̅=Ej̅Ei̅,Fi̅Fj̅=Fj̅Fi̅,if|i̅-j̅|≥2,Ei̅2Ej̅-(v+v-1)Ei̅Ej̅Ei̅+Ej̅Ei̅2=Fi̅2Fj̅-(v+v-1)Fi̅Fj̅Fi̅+Fj̅Fi̅2=0,if|i̅-j̅|=1.
See [23, Definition Proposition 9.1.1]. The algebra Uv′(𝔰𝔩̂ℓ) is the quantum group corresponding to the  nontrivial central extension 𝔰𝔩̂ℓ′=𝔰𝔩ℓ[t,t-1]⊕ℂc of the algebra of polynomial loops in 𝔰𝔩ℓ.


### 3.3. Fock Space

Definev-deformed Fock space to be the ℚ(v) vector space Fv with basis {|μ〉∣λisapartition}. Our Fv is only the charge 0 part of Fock space described in [27]. Fix i̅∈ℤ/ℓℤ and partitions λ⊆μ such that μ/λ is a single box. Define (3.3)Ai̅(λ):={boxesb∣b∉λ,bhascolori̅andλ∪bisapartition},Ri̅(λ):={boxesb∣b∈λ,bhascolori̅andλ∖bisapartition},Ni̅l(μ/λ):=|{b∈Ri̅(λ)∣bistotheleftofμ/λ}|-|{b∈Ai̅(λ)∣bistotheleftofμ/λ}|,Ni̅r(μ/λ):=|{b∈Ri̅(λ)∣bistotherightofμ/λ}|-|{b∈Ai̅(λ)∣bistotherightofμ/λ}|,
to be the set of addable boxes of colori̅, the set of removable boxes of colori̅, the left removable-addable difference, and the right removable-addable difference, respectively.Theorem 3.1 (see [25, Theorem 10.6]).
There is an action ofUv′(𝔰𝔩̂ℓ) on Fv determined by
(3.4)Ei̅|λ〉:=∑c¯(λ/μ)=i̅v-Ni̅r(λ/μ)|μ〉,Fi̅|λ〉:=∑c¯(μ/λ)=i̅vNi̅l(μ/λ)|μ〉,
where c¯(λ/μ) denotes the color of λ/μ and the sum is over partitionsμ which differ from λ by removing (resp. adding) a single i̅-colored box.As aUv′(𝔰𝔩̂ℓ)-module, Fv is isomorphic to an infinite direct sum of copies of the basic representation V(Λ0). Using the grading of Fv where |λ〉 has degree |λ|, the highest weight vectors in Fv occur in degrees divisible by ℓ, and the number of the highest weight vectors in degree ℓk is the number of partitions of k. Then, Fv≅V(Λ0)⊗ℂ[x1,x2,…], where xk has degree ℓk, and Uv′(𝔰𝔩̂ℓ) acts trivially on the second factor (see [27, Proposition 2.3]). Note that we are working with the “derived” quantum group Uv′(𝔰𝔩̂ℓ), not the “full” quantum group Uv(𝔰𝔩̂ℓ), which is why there are no δ-shifts in the summands of Fv.Comment 1.
Comparing with [25, Chapter 10], our Ni̅l(μ/λ) is equal to Ariki's -Ni̅a(μ/λ) and our Ni̅r(μ/λ) is equal to Ariki's -Ni̅b(μ/λ). However, these numbers play a slightly different role in Ariki's work, which is explained by a different choice of conventions.


## 3.1. Partitions

A partitionλ is a finite length  nonincreasing sequence of positive integers. Associated to a partition is its Ferrers diagram. We draw these diagrams as in Figure 1 so that, if λ=(λ1,…,λN), then λi is the number of boxes in row i (rows run southeast to northwest ↖). Say that λ is contained inμ if the diagram for λ fits inside the diagram forμ and let μ/λ be the collection of boxes ofμ that are not in λ. For each box b∈λ, the contentc(b) is the horizontal position of b and the colorc¯(b) is the residue of c(b) modulo ℓ. In Figure 1, the numbers c(b) are listed below the diagram. The size|λ| of a partition λ is the total number of boxes in its Ferrers diagram.Figure 1
The partition(7,6,6,5,5,3,3,1) with each box containing its color for ℓ=3. The content c(b) of a box b is the horizontal position of b reading right to left. The contents of boxes are listed beneath the diagram so that c(b) is aligned with all boxes b of that content.The setP+ of dominant polynomial weights from Section 2.3 is naturally identified with partitions with at most N parts. If λ∈P+, then (3.1)Δ(λ)⊗Δ(ε1)≅⨁1≤k≤Nλ+εk∈P+Δ(λ+εk)
as Uq(𝔤𝔩N)-modules. The diagram of λ+εk is obtained from the diagram of λ by adding a box on row k, and Δ(λ+εk) appears in the sum on the right side of (3.1) if and only if λ+εk is a partition. See, for example, [26, Section 6.1, Formula 6.8] for the classical statement and [23, Proposition 10.1.16] for the quantum case.


## 3.2. The Quantum Affine Algebra

LetUv′(𝔰𝔩̂ℓ) be the quantized universal enveloping algebra corresponding to the ℓ-node Dynkin diagramMore precisely,Uv′(𝔰𝔩̂ℓ) is the algebra generated by Ei̅,Fi̅,Ki̅±1, for i̅∈ℤ/ℓℤ, with relations (3.2)Ki̅Kj̅=Kj̅Ki̅,Ki̅Ki̅-1=Ki̅-1Ki̅=1,Ei̅Fj̅-Fj̅Ei̅=δi̅,j̅Ki̅-Ki̅-1v-v-1,Ki̅Ej̅Ki̅-1={v2Ej̅,ifi̅=j̅,v-1Ej̅,ifi̅=j̅±1,Ej̅,otherwise,Ki̅Fj̅Ki̅-1={v-2Fj̅,ifi̅=j̅,vFj̅,ifi̅=j̅±1,Fj̅,otherwise,Ei̅Ej̅=Ej̅Ei̅,Fi̅Fj̅=Fj̅Fi̅,if|i̅-j̅|≥2,Ei̅2Ej̅-(v+v-1)Ei̅Ej̅Ei̅+Ej̅Ei̅2=Fi̅2Fj̅-(v+v-1)Fi̅Fj̅Fi̅+Fj̅Fi̅2=0,if|i̅-j̅|=1.
See [23, Definition Proposition 9.1.1]. The algebra Uv′(𝔰𝔩̂ℓ) is the quantum group corresponding to the  nontrivial central extension 𝔰𝔩̂ℓ′=𝔰𝔩ℓ[t,t-1]⊕ℂc of the algebra of polynomial loops in 𝔰𝔩ℓ.


## 3.3. Fock Space

Definev-deformed Fock space to be the ℚ(v) vector space Fv with basis {|μ〉∣λisapartition}. Our Fv is only the charge 0 part of Fock space described in [27]. Fix i̅∈ℤ/ℓℤ and partitions λ⊆μ such that μ/λ is a single box. Define (3.3)Ai̅(λ):={boxesb∣b∉λ,bhascolori̅andλ∪bisapartition},Ri̅(λ):={boxesb∣b∈λ,bhascolori̅andλ∖bisapartition},Ni̅l(μ/λ):=|{b∈Ri̅(λ)∣bistotheleftofμ/λ}|-|{b∈Ai̅(λ)∣bistotheleftofμ/λ}|,Ni̅r(μ/λ):=|{b∈Ri̅(λ)∣bistotherightofμ/λ}|-|{b∈Ai̅(λ)∣bistotherightofμ/λ}|,
to be the set of addable boxes of colori̅, the set of removable boxes of colori̅, the left removable-addable difference, and the right removable-addable difference, respectively.Theorem 3.1 (see [25, Theorem 10.6]).
There is an action ofUv′(𝔰𝔩̂ℓ) on Fv determined by
(3.4)Ei̅|λ〉:=∑c¯(λ/μ)=i̅v-Ni̅r(λ/μ)|μ〉,Fi̅|λ〉:=∑c¯(μ/λ)=i̅vNi̅l(μ/λ)|μ〉,
where c¯(λ/μ) denotes the color of λ/μ and the sum is over partitionsμ which differ from λ by removing (resp. adding) a single i̅-colored box.As aUv′(𝔰𝔩̂ℓ)-module, Fv is isomorphic to an infinite direct sum of copies of the basic representation V(Λ0). Using the grading of Fv where |λ〉 has degree |λ|, the highest weight vectors in Fv occur in degrees divisible by ℓ, and the number of the highest weight vectors in degree ℓk is the number of partitions of k. Then, Fv≅V(Λ0)⊗ℂ[x1,x2,…], where xk has degree ℓk, and Uv′(𝔰𝔩̂ℓ) acts trivially on the second factor (see [27, Proposition 2.3]). Note that we are working with the “derived” quantum group Uv′(𝔰𝔩̂ℓ), not the “full” quantum group Uv(𝔰𝔩̂ℓ), which is why there are no δ-shifts in the summands of Fv.Comment 1.
Comparing with [25, Chapter 10], our Ni̅l(μ/λ) is equal to Ariki's -Ni̅a(μ/λ) and our Ni̅r(μ/λ) is equal to Ariki's -Ni̅b(μ/λ). However, these numbers play a slightly different role in Ariki's work, which is explained by a different choice of conventions.


## 4. Universal Verma Modules

The purpose of this section is to construct a family of representations which are universal Verma modules in the sense that each can be “evaluated" to obtain any given Verma module. This notion was defined by Kashiwara [7] in the classical case and was studied in the quantum case by Kamita [8].


### 4.1. Rational Universal Verma Modules

Let𝕂:=ℚ(q,z1,z2,…,zN). This field is isomorphic to the field of fractions of Uq(𝔤𝔩N)0 via the map (4.1)ψ:Uq(glN)0⟶K,definedbyψ(Li±1)=zi±1.
For each μ∈𝔥ℤ*, define a ℚ(q)-linear automorphism σμ:𝕂→𝕂 by (4.2)σμ(zi):=q(μ,εi)zi,for1≤i≤N,
where (·,·) is the inner product on 𝔥ℤ* defined by (εi,εj)=δi,j. Let 𝕂μ=span𝕂{vμ+} be the one-dimensional vector space over 𝕂 with basis vector vμ+ and Uq(𝔤𝔩N)≥0 action given by (4.3)Xi⋅vμ+=0,for1≤i≤N-1,a⋅vμ+=σμ(ψ(a))vμ+,fora∈Uq(glN)0.Theμ-shifted rational universal Verma moduleμM̃ is the Uq(𝔤𝔩N)-module(4.4)μM̃:=Uq(glN)⊗Uq(glN)≥0Kμ.
The universal Verma module μM̃ is actually a module over Uq(𝔤𝔩N)⊗Uq(𝔤𝔩N)0Ũq(𝔤𝔩N)0, where Ũq(𝔤𝔩N)0 is the field of fractions of Uq(𝔤𝔩N)0. However, if we identify Ũq(𝔤𝔩N)0 with 𝕂 using the map ψ, the action of Ũq(𝔤𝔩N)0 on μM̃ is not by multiplication, but rather is twisted by the automorphism σμ. It is to keep track of the difference between the action of Uq(𝔤𝔩N)0 and multiplication that we use different notation for the generators of 𝕂 and Uq(𝔤𝔩N)0 (i.e., zi versus Li).


### 4.2. Integral Universal Verma Modules

The field𝕂 contains an 𝒜-subalgebra (4.5)Rgeneratedbyzi±1,[zi;ck],(1≤i≤N,c∈Z,k∈Z>0),
which is isomorphic to Uq𝒜(𝔤𝔩N)0 via the restriction of the map ψ in (4.1). The integral universal Verma moduleμM̃ℛ is the Uq𝒜(𝔤𝔩N)-submodule of μM̃ generated by vμ+. By restricting (4.4),
(4.6)μM̃R=UqA(glN)⊗UqA(glN)≥0Rμ,
where ℛμ is the ℛ-submodule of 𝕂μ spanned by vμ+. In particular, μM̃ℛ is a free ℛ-module.


### 4.3. Evaluation

Letevλℛ:ℛ→𝒜 be the map defined by
(4.7)evλR(zi)=q(λ,εi),evλR[zi;cn]=[q(λ,εi);cn],
where (·,·) is the inner product on 𝔥* defined by (εi,εj)=δi,j. There is a surjective Uq𝒜(𝔤𝔩N)-module homomorphism “evaluation at λ” (4.8)evλ:μM̃R⟶MA(μ+λ)definedbyevλ(a⋅vμ+):=a⋅vμ+λ,∀a∈UqA(glN).For fixedλ, the maps evλℛ and evλ extend to a map from the subspace of 𝕂 and μM̃=μM̃ℛ⊗ℛ𝕂, respectively, where no denominators evaluate to 0. Where it is clear we denote both these extended maps by evλ.Example 4.1.
Computing the action ofLi on vμ+ and vμ+λ,
(4.9)Li⋅vμ+=q(μ,εi)zivμ+,Li⋅vμ+λ=evλ(q(μ,εi)zi)vμ+λ=q(μ,εi)q(λ,εi)vμ+λ=q(μ+λ,εi)vμ+λ.


### 4.4. Weight Decompositions

LetṼ be a Uq(𝔤𝔩N)⊗𝒜ℛ-module. For each ν∈𝔥Z*, we define the ν-weight space of Ṽ to be (4.10)Ṽν:={v∈Ṽ:Li⋅v=q(ν,εi)ziv}.
The universal Verma module μM̃ℛ is a Uq(𝔤𝔩N)⊗𝒜ℛ-module, where the second factor acts as multiplication. The weight space μM̃η≠0 if and only if η=μ-ν with ν in the positive part Q+ of the root lattice. These  nonzero weight spaces and the weight decomposition of μM̃ can be described explicitly by
(4.11)μM̃μ-νR=UqA(glN)-ν<0⋅Rμ,μM̃R=⨁ν∈Q+μM̃μ-νR.
Here, Uq𝒜(𝔤𝔩N)-ν<0 is defined using the grading of Uq(𝔤𝔩N)<0 with Fi∈Uq(𝔤𝔩N)-(εi-εi+1)<0.


### 4.5. Tensor Products

LetṼ be a Uq𝒜(𝔤𝔩N)⊗𝒜ℛ-module and W a Uq𝒜(𝔤𝔩N)-module. The tensor product Ṽ⊗𝒜W is a Uq𝒜(𝔤𝔩N)⊗𝒜ℛ-module, where the first factor acts via the usual coproduct and the second factor acts by multiplication on Ṽ. In the case when Ṽ and W both have weight space decompositions, the weight spaces of Ṽ⊗𝒜W are
(4.12)(Ṽ⊗AW)ν=⨁γ+η=νṼγ⊗AWη.We also need the following.Proposition 4.2.
The tensor product of a universal Verma module with a Weyl module satisfies(4.13)(μM̃R⊗AΔA(ν))⊗RK≅(⨁γ(μ+γM̃R)⨁dim⁡ΔA(ν)γ)⊗RK.Proof.
Fixν∈P+. In general, M(λ+μ)⊗Δ(ν) has a Verma filtration (see, e.g., [28, Theorem 2.2]) and if λ+μ+γ is dominant for all γ such that Δ(ν)γ≠0 then
(4.14)M(λ+μ)⊗Δ(ν)≅⨁γM(λ+μ+γ)⨁dim⁡Δ(ν)γ,
which can be seen by, for instance, taking central characters. The proposition follows since this is true for a Zariski dense set of weights λ.


## 4.1. Rational Universal Verma Modules

Let𝕂:=ℚ(q,z1,z2,…,zN). This field is isomorphic to the field of fractions of Uq(𝔤𝔩N)0 via the map (4.1)ψ:Uq(glN)0⟶K,definedbyψ(Li±1)=zi±1.
For each μ∈𝔥ℤ*, define a ℚ(q)-linear automorphism σμ:𝕂→𝕂 by (4.2)σμ(zi):=q(μ,εi)zi,for1≤i≤N,
where (·,·) is the inner product on 𝔥ℤ* defined by (εi,εj)=δi,j. Let 𝕂μ=span𝕂{vμ+} be the one-dimensional vector space over 𝕂 with basis vector vμ+ and Uq(𝔤𝔩N)≥0 action given by (4.3)Xi⋅vμ+=0,for1≤i≤N-1,a⋅vμ+=σμ(ψ(a))vμ+,fora∈Uq(glN)0.Theμ-shifted rational universal Verma moduleμM̃ is the Uq(𝔤𝔩N)-module(4.4)μM̃:=Uq(glN)⊗Uq(glN)≥0Kμ.
The universal Verma module μM̃ is actually a module over Uq(𝔤𝔩N)⊗Uq(𝔤𝔩N)0Ũq(𝔤𝔩N)0, where Ũq(𝔤𝔩N)0 is the field of fractions of Uq(𝔤𝔩N)0. However, if we identify Ũq(𝔤𝔩N)0 with 𝕂 using the map ψ, the action of Ũq(𝔤𝔩N)0 on μM̃ is not by multiplication, but rather is twisted by the automorphism σμ. It is to keep track of the difference between the action of Uq(𝔤𝔩N)0 and multiplication that we use different notation for the generators of 𝕂 and Uq(𝔤𝔩N)0 (i.e., zi versus Li).


## 4.2. Integral Universal Verma Modules

The field𝕂 contains an 𝒜-subalgebra (4.5)Rgeneratedbyzi±1,[zi;ck],(1≤i≤N,c∈Z,k∈Z>0),
which is isomorphic to Uq𝒜(𝔤𝔩N)0 via the restriction of the map ψ in (4.1). The integral universal Verma moduleμM̃ℛ is the Uq𝒜(𝔤𝔩N)-submodule of μM̃ generated by vμ+. By restricting (4.4),
(4.6)μM̃R=UqA(glN)⊗UqA(glN)≥0Rμ,
where ℛμ is the ℛ-submodule of 𝕂μ spanned by vμ+. In particular, μM̃ℛ is a free ℛ-module.


## 4.3. Evaluation

Letevλℛ:ℛ→𝒜 be the map defined by
(4.7)evλR(zi)=q(λ,εi),evλR[zi;cn]=[q(λ,εi);cn],
where (·,·) is the inner product on 𝔥* defined by (εi,εj)=δi,j. There is a surjective Uq𝒜(𝔤𝔩N)-module homomorphism “evaluation at λ” (4.8)evλ:μM̃R⟶MA(μ+λ)definedbyevλ(a⋅vμ+):=a⋅vμ+λ,∀a∈UqA(glN).For fixedλ, the maps evλℛ and evλ extend to a map from the subspace of 𝕂 and μM̃=μM̃ℛ⊗ℛ𝕂, respectively, where no denominators evaluate to 0. Where it is clear we denote both these extended maps by evλ.Example 4.1.
Computing the action ofLi on vμ+ and vμ+λ,
(4.9)Li⋅vμ+=q(μ,εi)zivμ+,Li⋅vμ+λ=evλ(q(μ,εi)zi)vμ+λ=q(μ,εi)q(λ,εi)vμ+λ=q(μ+λ,εi)vμ+λ.


## 4.4. Weight Decompositions

LetṼ be a Uq(𝔤𝔩N)⊗𝒜ℛ-module. For each ν∈𝔥Z*, we define the ν-weight space of Ṽ to be (4.10)Ṽν:={v∈Ṽ:Li⋅v=q(ν,εi)ziv}.
The universal Verma module μM̃ℛ is a Uq(𝔤𝔩N)⊗𝒜ℛ-module, where the second factor acts as multiplication. The weight space μM̃η≠0 if and only if η=μ-ν with ν in the positive part Q+ of the root lattice. These  nonzero weight spaces and the weight decomposition of μM̃ can be described explicitly by
(4.11)μM̃μ-νR=UqA(glN)-ν<0⋅Rμ,μM̃R=⨁ν∈Q+μM̃μ-νR.
Here, Uq𝒜(𝔤𝔩N)-ν<0 is defined using the grading of Uq(𝔤𝔩N)<0 with Fi∈Uq(𝔤𝔩N)-(εi-εi+1)<0.


## 4.5. Tensor Products

LetṼ be a Uq𝒜(𝔤𝔩N)⊗𝒜ℛ-module and W a Uq𝒜(𝔤𝔩N)-module. The tensor product Ṽ⊗𝒜W is a Uq𝒜(𝔤𝔩N)⊗𝒜ℛ-module, where the first factor acts via the usual coproduct and the second factor acts by multiplication on Ṽ. In the case when Ṽ and W both have weight space decompositions, the weight spaces of Ṽ⊗𝒜W are
(4.12)(Ṽ⊗AW)ν=⨁γ+η=νṼγ⊗AWη.We also need the following.Proposition 4.2.
The tensor product of a universal Verma module with a Weyl module satisfies(4.13)(μM̃R⊗AΔA(ν))⊗RK≅(⨁γ(μ+γM̃R)⨁dim⁡ΔA(ν)γ)⊗RK.Proof.
Fixν∈P+. In general, M(λ+μ)⊗Δ(ν) has a Verma filtration (see, e.g., [28, Theorem 2.2]) and if λ+μ+γ is dominant for all γ such that Δ(ν)γ≠0 then
(4.14)M(λ+μ)⊗Δ(ν)≅⨁γM(λ+μ+γ)⨁dim⁡Δ(ν)γ,
which can be seen by, for instance, taking central characters. The proposition follows since this is true for a Zariski dense set of weights λ.


## 5. The Shapovalov Form and the Shapovalov Determinant

### 5.1. The Shapovalov Form

TheCartan involutionω:Uq(𝔤𝔩N)→Uq(𝔤𝔩N) is the ℚ(q)-algebra anti-involution of Uq(𝔤𝔩N) defined by (5.1)ω(Li±1)=Li±1,ω(Xi)=YiLiLi+1-1,ω(Yi)=Li-1Li+1Xi.
The map ω is also a coalgebra involution. An ω-contravariant form on a Uq(𝔤𝔩N)-module V is a symmetric bilinear form (·,·) such that (5.2)(u,a⋅v)=(ω(a)⋅u,v),foru,v∈V,a∈Uq(glN).It follows by the same argument used in the classical case [4] that there is an ω-contravariant form (the Shapovalov form) on each Verma module M(λ) and this is unique up to rescaling. The radical of (·,·) is the maximal proper submodule of M(λ), so Δ(λ)=M(λ)/Rad(·,·) for all λ∈P+. In particular, (·,·) descends to an ω-contravariant form on Δ(λ).Sinceω fixes Uq𝒜(𝔤𝔩N)⊆Uq(𝔤𝔩N), there is a well-defined notion of an ω-contravariant form on a Uq𝒜(𝔤𝔩N) module. In particular, the restriction of the Shapovalov form on Δ(λ) to Δ𝒜(λ) is ω-contravariant.


### 5.2. Universal Shapovalov Forms

There are surjective maps of𝒜-algebras p-:Uq𝒜(𝔤𝔩N)<0→ℚ(q) and p+:Uq𝒜(𝔤𝔩N)>0→ℚ(q) defined by p-(Fi)=0 and p+(Ei)=0, for 1≤i≤N. Using the triangular decomposition (2.7), there is an 𝒜-linear surjection(5.3)π0:=p-⊗Id⊗p+:UqA(glN)≅UqA(glN)<0⊗AUqA(glN)0⊗AUqA(glN)>0⟶UqA(glN)0.
The standard universal Shapovalov form is the ℛ-bilinear form (·,·)μM̃ℛ:μM̃ℛ⊗μM̃ℛ→ℛ defined by (5.4)(a1⋅vμ+,a2⋅vμ+)μM̃R=(σμ∘ψ∘π0)(ω(a2)a1)
for all a1,a2∈Uqℛ(𝔤𝔩N)<0. Here, ψ and σμ are as in (4.1) and (4.2). Since (5.5)(a1a2⋅vμ+,a3⋅vμ+)μM̃R=(σμ∘ψ∘π0)(ω(a2)ω(a1)a3)=(a2⋅vμ+,ω(a1)a3⋅vμ+)μM̃R
for a1,a2,a3∈Uq(𝔤𝔩N), the form (·,·)μM̃ℛ is ω-contravariant. As with the usual Shapovalov form, distinct weight spaces are orthogonal, where weight spaces are defined as in Section 4.4.Evaluation atλ gives an 𝒜-valued ω-contravariant form (·,·)M𝒜(μ+λ) on M𝒜(μ+λ) by
(5.6)(evλ(u1),evλ(u2))MA(μ+λ)=evλ((u1,u2)μM̃R)foru1,u2∈μM̃R.
The form (·,·)μM̃ℛ can be extended by linearity to an ω-contravariant form (·,·)μM̃ on μM̃.


### 5.3. The Shapovalov Determinant

LetṼ be a (Uq𝒜(𝔤𝔩N)⊗𝒜ℛ)-module with a chosen ω-contravariant form. Let Bη be an ℛ basis for the η-weight space Ṽη of Ṽ. Let detṼBη be the determinant of the form evaluated on the basis Bη. Changing the basis Bη changes the determinant by a unit in ℛ, and we sometimes write detṼη to mean the determinant calculated on an unspecified basis (detṼη which is only defined up to multiplication by unit in ℛ). The Shapovalov determinant is
(5.7)det⁡⁡M̃ηR:=det⁡⁡((bi,bj)M̃R)bi,bj∈Bη.Define thepartition functionp:𝔥*→ℤ≥0 by (5.8)p(γ)≔dim⁡⁡M(0)γ.
Then, p(γ)=dimM(λ)γ+λ for any λ, and η∉Q- implies that p(η)=0 and detM̃ηℛ=1.Theorem 5.1 (see [5, Proposition 1.9A], [6, Theorem 3.4], [4]).
For any weightη,
(5.9)det⁡⁡M̃ηR=cη∏1≤i<j≤Nm>0(zizj-1-q2m+2i-2jzi-1zj)p(η+mεi-mεj),
where cη is a unit in ℛ⊗𝒜ℚ(q)=Q(q)[z1±1,…,zN±1].Proposition 5.2.
Fixμ,η∈𝔥ℤ* with η-μ∈Q-. Choose an 𝒜-basis Bη-μ for Uq𝒜(𝔤𝔩N)η-μ. Consider the ℛ-bases B̃η-μ:={b·v+∣b∈Bη-μ} for M̃η-μℛ and μB̃η:={b·vμ+∣b∈Bη-μ} for μM̃ηℛ. Then detM̃μ(μB̃η)ℛ=σμ(detM̃B̃η-μℛ).Proof.
Forb,b′∈Bη-μ,
(5.10)(b⋅vμ+,b′⋅vμ+)μM̃R=σμ∘ψ∘π0(ω(b′)b)=σμ((b⋅v0+,b′⋅v0+)M̃R).
The result follows by taking determinants.


### 5.4. Contravariant Forms on Tensor Products

IfV and W are Uq𝒜(𝔤𝔩N)-modules with ω-contravariant forms (·,·)V and (·,·)W, define an 𝒜-bilinear form (·,·)W⊗V by (w1⊗v1,w2⊗v2)W⊗V=(w1,w2)W(v1,v2)V. Similarly, for a Uq𝒜(𝔤𝔩N)⊗𝒜ℛ module W̃ with ℛ-bilinear ω-contravariant form (·,·)W̃, define a ℛ-bilinear form (·,·)W̃⊗ℚ(q)V on W̃⊗ℚ(q)V by
(5.11)(u1⊗v1,u2⊗v2)W̃⨂Q(q)V=(u1,u2)W̃(v1,v2)V.
Since ω is a coalgebra involution (i.e., Δ(ω(a))=(ω⊗ω)Δ(a), for a∈Uq(𝔤𝔩N)), the forms (·,·)V⊗W and (·,·)μM̃⊗Q(q)V are ω-contravariant.In the case whenW̃=μM̃ℛ, evaluation of the ω-contravariant form (·,·)μM̃ℛ⊗𝒜V at λ gives an ω-contravariant form (·,·)M𝒜(μ+λ)⊗𝒜V: (5.12)(u1⊗v1,u2⊗v2)MA(μ+λ)⨂AV=evλ((u1⊗v1,u2⊗v2)μM̃⨂AV)=(evλ(u1)⊗v1,evλ(u2)⊗v2)M(μ+λ)⨂AV,
for u1,u2∈μM̃ and v1,v2∈V. As in Section 4.3, this form can be extended to the 𝒜-submodule of the rational module where no denominators evaluate to zero.


## 5.1. The Shapovalov Form

TheCartan involutionω:Uq(𝔤𝔩N)→Uq(𝔤𝔩N) is the ℚ(q)-algebra anti-involution of Uq(𝔤𝔩N) defined by (5.1)ω(Li±1)=Li±1,ω(Xi)=YiLiLi+1-1,ω(Yi)=Li-1Li+1Xi.
The map ω is also a coalgebra involution. An ω-contravariant form on a Uq(𝔤𝔩N)-module V is a symmetric bilinear form (·,·) such that (5.2)(u,a⋅v)=(ω(a)⋅u,v),foru,v∈V,a∈Uq(glN).It follows by the same argument used in the classical case [4] that there is an ω-contravariant form (the Shapovalov form) on each Verma module M(λ) and this is unique up to rescaling. The radical of (·,·) is the maximal proper submodule of M(λ), so Δ(λ)=M(λ)/Rad(·,·) for all λ∈P+. In particular, (·,·) descends to an ω-contravariant form on Δ(λ).Sinceω fixes Uq𝒜(𝔤𝔩N)⊆Uq(𝔤𝔩N), there is a well-defined notion of an ω-contravariant form on a Uq𝒜(𝔤𝔩N) module. In particular, the restriction of the Shapovalov form on Δ(λ) to Δ𝒜(λ) is ω-contravariant.


## 5.2. Universal Shapovalov Forms

There are surjective maps of𝒜-algebras p-:Uq𝒜(𝔤𝔩N)<0→ℚ(q) and p+:Uq𝒜(𝔤𝔩N)>0→ℚ(q) defined by p-(Fi)=0 and p+(Ei)=0, for 1≤i≤N. Using the triangular decomposition (2.7), there is an 𝒜-linear surjection(5.3)π0:=p-⊗Id⊗p+:UqA(glN)≅UqA(glN)<0⊗AUqA(glN)0⊗AUqA(glN)>0⟶UqA(glN)0.
The standard universal Shapovalov form is the ℛ-bilinear form (·,·)μM̃ℛ:μM̃ℛ⊗μM̃ℛ→ℛ defined by (5.4)(a1⋅vμ+,a2⋅vμ+)μM̃R=(σμ∘ψ∘π0)(ω(a2)a1)
for all a1,a2∈Uqℛ(𝔤𝔩N)<0. Here, ψ and σμ are as in (4.1) and (4.2). Since (5.5)(a1a2⋅vμ+,a3⋅vμ+)μM̃R=(σμ∘ψ∘π0)(ω(a2)ω(a1)a3)=(a2⋅vμ+,ω(a1)a3⋅vμ+)μM̃R
for a1,a2,a3∈Uq(𝔤𝔩N), the form (·,·)μM̃ℛ is ω-contravariant. As with the usual Shapovalov form, distinct weight spaces are orthogonal, where weight spaces are defined as in Section 4.4.Evaluation atλ gives an 𝒜-valued ω-contravariant form (·,·)M𝒜(μ+λ) on M𝒜(μ+λ) by
(5.6)(evλ(u1),evλ(u2))MA(μ+λ)=evλ((u1,u2)μM̃R)foru1,u2∈μM̃R.
The form (·,·)μM̃ℛ can be extended by linearity to an ω-contravariant form (·,·)μM̃ on μM̃.


## 5.3. The Shapovalov Determinant

LetṼ be a (Uq𝒜(𝔤𝔩N)⊗𝒜ℛ)-module with a chosen ω-contravariant form. Let Bη be an ℛ basis for the η-weight space Ṽη of Ṽ. Let detṼBη be the determinant of the form evaluated on the basis Bη. Changing the basis Bη changes the determinant by a unit in ℛ, and we sometimes write detṼη to mean the determinant calculated on an unspecified basis (detṼη which is only defined up to multiplication by unit in ℛ). The Shapovalov determinant is
(5.7)det⁡⁡M̃ηR:=det⁡⁡((bi,bj)M̃R)bi,bj∈Bη.Define thepartition functionp:𝔥*→ℤ≥0 by (5.8)p(γ)≔dim⁡⁡M(0)γ.
Then, p(γ)=dimM(λ)γ+λ for any λ, and η∉Q- implies that p(η)=0 and detM̃ηℛ=1.Theorem 5.1 (see [5, Proposition 1.9A], [6, Theorem 3.4], [4]).
For any weightη,
(5.9)det⁡⁡M̃ηR=cη∏1≤i<j≤Nm>0(zizj-1-q2m+2i-2jzi-1zj)p(η+mεi-mεj),
where cη is a unit in ℛ⊗𝒜ℚ(q)=Q(q)[z1±1,…,zN±1].Proposition 5.2.
Fixμ,η∈𝔥ℤ* with η-μ∈Q-. Choose an 𝒜-basis Bη-μ for Uq𝒜(𝔤𝔩N)η-μ. Consider the ℛ-bases B̃η-μ:={b·v+∣b∈Bη-μ} for M̃η-μℛ and μB̃η:={b·vμ+∣b∈Bη-μ} for μM̃ηℛ. Then detM̃μ(μB̃η)ℛ=σμ(detM̃B̃η-μℛ).Proof.
Forb,b′∈Bη-μ,
(5.10)(b⋅vμ+,b′⋅vμ+)μM̃R=σμ∘ψ∘π0(ω(b′)b)=σμ((b⋅v0+,b′⋅v0+)M̃R).
The result follows by taking determinants.


## 5.4. Contravariant Forms on Tensor Products

IfV and W are Uq𝒜(𝔤𝔩N)-modules with ω-contravariant forms (·,·)V and (·,·)W, define an 𝒜-bilinear form (·,·)W⊗V by (w1⊗v1,w2⊗v2)W⊗V=(w1,w2)W(v1,v2)V. Similarly, for a Uq𝒜(𝔤𝔩N)⊗𝒜ℛ module W̃ with ℛ-bilinear ω-contravariant form (·,·)W̃, define a ℛ-bilinear form (·,·)W̃⊗ℚ(q)V on W̃⊗ℚ(q)V by
(5.11)(u1⊗v1,u2⊗v2)W̃⨂Q(q)V=(u1,u2)W̃(v1,v2)V.
Since ω is a coalgebra involution (i.e., Δ(ω(a))=(ω⊗ω)Δ(a), for a∈Uq(𝔤𝔩N)), the forms (·,·)V⊗W and (·,·)μM̃⊗Q(q)V are ω-contravariant.In the case whenW̃=μM̃ℛ, evaluation of the ω-contravariant form (·,·)μM̃ℛ⊗𝒜V at λ gives an ω-contravariant form (·,·)M𝒜(μ+λ)⊗𝒜V: (5.12)(u1⊗v1,u2⊗v2)MA(μ+λ)⨂AV=evλ((u1⊗v1,u2⊗v2)μM̃⨂AV)=(evλ(u1)⊗v1,evλ(u2)⊗v2)M(μ+λ)⨂AV,
for u1,u2∈μM̃ and v1,v2∈V. As in Section 4.3, this form can be extended to the 𝒜-submodule of the rational module where no denominators evaluate to zero.


## 6. The Misra-Miwa Formula forFi̅ from Uq𝒜(𝔤𝔩N) Representation Theory

Let us prepare the setting for our main result (Theorem6.1). Fix ℓ≥2 and a partition λ. Let N be a positive integer greater than the number of parts of λ. All calculations below are in terms of representations of Uq𝒜(𝔤𝔩N).(1)
LetV=Δ𝒜(ε1) be the standard N-dimensional module. Since Δ𝒜(λ)⊗𝒜ℚ(q)=Δ(λ), (3.1) implies(6.1)(ΔA(λ)⊗AV)⊗AQ(q)≃⨁ΔA(λ+εkj)⊗AQ(q),where the sum is over those indices1=k1<k2<⋯<kmλ≤N for which λ+εkj is a partition. For ease of notation, let μ(j)=λ+εkj.(2)
Fix an𝒜-basis {v1,…,vN} of V where vk has weight εk and Yi(vk)=δi,kvk+1. Recursively, define singular weight vectors vμ(j) in (Δ𝒜(λ)⊗V)⊗𝒜ℚ(q) by
(i)
vμ(1)=vλ⊗v1(ii)
for eachk, the submodule Wk of (Δ(λ)⊗𝒜V)⊗𝒜ℚ(q) generated by {vλ⊗vi∣1≤i≤k} contains all weight vectors of (Δ(λ)⊗𝒜V)⊗𝒜ℚ(q) of weight greater than or equal to λ+εk. Thus, using (6.1), for each 1≤j≤mλ there is a one-dimensional space of singular vectors of weight μ(j) in Wkj, and this is not contained in Wkj-1 (since kj>kj-1). This implies that there unique singular vector vμ(j) of weight μ(j) in
(6.2)vλ⊗vkj+⨁1≤i<jUq(glN)vμ(i)⊆(ΔA(λ)⊗AV)⊗AQ(q),
where we recall that Uq(𝔤𝔩N)=Uq𝒜(𝔤𝔩N)⊗𝒜ℚ(q).(3)
There is a uniqueω-contravariant form on Δ𝒜(λ) normalized so that (vλ,vλ)=1 and a unique ω-contravariant form on V normalized so that (v1,v1)=1. As in Section 5.4, define a ω-contravariant form on (Δ𝒜(λ)⊗𝒜V)⊗𝒜ℚ(q) by (u1⊗w1,u2⊗w2)=(u1,u2)(w1,w2). For each 1≤j≤mλ, define an element rj(λ)∈ℚ(q) by
(6.3)rj(λ):=(vμ(j),vμ(j)).Theorem 6.1.
The Misra-Miwa operatorsFi̅ from Section 3.3 satisfy
(6.4)Fi̅|λ〉=∑c̅(b(j))=i̅vvalϕ2lrj(λ)|μ(j)〉,
where b(j) is the box μ(j)/λ, c̅(b(j)) is the color of box b(j) as in Figure 1, ϕ2ℓ is the 2ℓth cyclotomic polynomial in q, and valϕ2ℓr is the number of factors of ϕ2ℓ in the numerator of r minus the number of factors of ϕ2ℓ in the denominator of r.The proof of Theorem6.1 will occupy the rest of this section. We will first prove a similar statement, Proposition 6.6, where the role of the Weyl modules is played by the universal Verma modules from Section 4. For ease of notation, let M̃ℛ denote the module 0M̃ℛ from Section 4.2.Definition 6.2.
Recursively define singular weight vectorsvεk+∈(M̃ℛ⊗𝒜V)⊗ℛ𝕂  and elements  sk∈𝕂 for  1≤k≤N  by(i)
vε1+=v+⊗v1,(ii)
since{v+⊗vj∣1≤j≤N} generates M̃ℛ⊗𝒜V as a Uq𝒜(𝔤𝔩N)≤0 module, Proposition 4.2 implies that, for each 1≤k≤N, there is a unique singular vector vεk+ in v+⊗vk+⊕1≤j<kUq𝕂(𝔤𝔩N)vεj+⊆(M̃ℛ⊗𝒜V)⊗ℛ𝕂, where Uq𝕂(𝔤𝔩N):=Uq(𝔤𝔩N)⊗(q)𝕂 and the factor of 𝕂 acts by multiplication on M̃ℛ.
Letsk=(vεk+,vεk+).Thesk are quantized versions of the Jantzen numbers first calculated in [11, Section 5] and quantized in [12]. It follows immediately from the definition that s1=1.Lemma 6.3.
For any weightη, up to multiplication by a power of  q,
(6.5)∏1≤k≤Nskp(η-εk)=∏1≤k≤Ndet⁡⁡M̃η-εkRσεkdet⁡⁡M̃η-εkR,
where, as in Section 5.3, detM̃η-εkℛ is the determinant of the Shapovalov form evaluated on an ℛ-basis for the η-εk weight space of M̃ℛ.Comment 2.
In order for Lemma6.3 to hold as stated, for each 1≤k≤N, one must calculate the detM̃η-εkℛ in the numerator and denominator with respect to the same ℛ-basis. The power of q which appears depends on this choice of ℛ-bases.Proof of Lemma6.3.
For eachγ∈spanℤ≤0(R+) fix an ℛ-basis Bγ for Uqℛ(𝔤𝔩N)γ<0. Consider the following three 𝕂-bases for ((M̃ℛ⊗𝒜V)η)⊗ℛ𝕂:
(6.6)Aη:={(b⋅v+)⊗vk∣b∈Bη-εk,1≤k≤N},Cη:={b⋅(v+⊗vk)∣b∈Bη-εk,1≤k≤N},Dη:={b⋅vεk+∣b∈Bη-εk,1≤k≤N}.
Let det(M̃ℛ⊗𝒜V)B denote the determinant of (·,·)(M̃ℛ⊗𝒜V)η calculated on B, where B is one of Aη,Cη, or Dη. Let detM̃νBη-νℛ denote detM̃νηℛ calculated with respect to the basis Bη-ν·vν+.
By the definition of theω-contravariant form on M̃ℛ⊗𝒜V (see Section 4.5),(6.7)det⁡⁡(M̃R⊗V)Aη=∏k=1N(det⁡⁡M̃Bη-εkR)dim⁡Vεk(det⁡Vεk)dim⁡M̃η-εkR.
For 1≤k≤N, Vεk is one dimensional and detVεk is a power of q. Hence, up to multiplication by a power of q, (6.7) simplifies to
(6.8)det⁡⁡(M̃R⊗AV)Aη=∏k=1Ndet⁡⁡M̃Bη-εkR.
Notice thatUq𝒜(𝔤𝔩N)<0·vεk+ is isomorphic to εkM̃, and Dη is the union of ℛ-bases for each of these submodules. For each 1≤k≤N, and each η∈𝔥Z* define an ℛ basis of εkM̃η by (6.9)εkB̃η:={b⋅vεk+∣b∈Bη-εk}.
Using (vεk+,vεk+)=sk,
(6.10)det⁡⁡(M̃R⊗V)Dη=∏k=1Nskdim⁡(εkM̃ηR)det⁡M̃εk(εkB̃η)R=∏k=1Nskp(η-εk)σεk(det⁡⁡M̃B̃η-εkR),
where the last equality uses Proposition 5.2. Here, as in Section 5.3, detM̃εk(εkB̃η)ℛ is the Shapovalov determinant calculated with respect to the basis εkB̃η.
The change of basis fromAη to Cη is unitriangular and the change of basis from Cη to Dη is unitriangular. Thus, det(M̃ℛ⊗𝒜V)Aη=det(M̃ℛ⊗𝒜V)Dη, and so the right sides of (6.8) and (6.10) are equal. The lemma follows from this equality by rearranging.Lemma 6.4.
Up to multiplication by a power ofq,
(6.11)sk=∏1≤j<k(zjzk-1-q2+2j-2kzj-1zkσεj(zjzk-1-q2+2j-2kzj-1zk)).Proof.
Fix1≤k≤N. Setting η=εk in Lemma 6.3 and applying Theorem 5.1 we see that, up to multiplication by a power of q,
(6.12)∏1≤x≤Nsxp(εk-εx)=∏1≤x≤Ndet⁡⁡M̃εk-εxRσεxdet⁡⁡M̃εk-εxR=∏1≤x≤N∏1≤i<j≤Nm>0(cεk-εx(zizj-1-q2m+2i-2jzi-1zj)σεx(cεk-εx)σεx(zizj-1-q2m+2i-2jzi-1zj))p(εk-εx+mεi-mεj),
where, for each 1≤x≤N, cεk-εx is a unit in ℚ(q)[z1±1,…,zN±1]. The value p(εk-εx+mεi-mεj) is 0 unless m=1 and x≤i<j≤k. If i>x, then σεx acts as the identity on zizj-1-q2+2i-2jzi-1zj, so the corresponding factors in the numerator and denominator cancel. Hence, we need only consider factors on the right hand side where m=1, i=x, and x<j≤k. If x>k, then εk-εx∉Q-, and hence p(εk-εx)=0, so on the left hand since we only need to consider those factors where 1≤x≤k. Up to multiplication by a power of q, the expression reduces to
(6.13)∏1≤x≤ksxp(εk-εx)=∏1≤x<k(cεk-εxσεx(cεk-εx))p(εk-εj)∏x<j≤k(zxzj-1-q2+2x-2jzx-1zjσεx(zxzj-1-q2+2x-2jzx-1zj))p(εk-εj)=∏1<j≤k(∏1≤x<jzxzj-1-q2+2x-2jzx-1zjσεx(zxzj-1-q2+2x-2jzx-1zj))p(εk-εj).
The last two expressions are equal because they are each a product over pairs (x,j) with 1≤x<j≤k, and the factors of cεk-εx/(σεx(cεk-εx)) have been dropped because they are powers of q. Using the fact that s1=1 and making the change of variables j→x and x→j on the right side, (6.13) becomes
(6.14)∏1<x≤ksxp(εk-εx)=∏1<x≤k(∏1≤j<xzjzx-1-q2+2j-2xzj-1zxσεj(zjzx-1-q2+2j-2xzj-1zx))p(εk-εx).
For k≥2, the lemma now follows by induction. For k=1, the result simply says that s1=1, which we already know.Proposition 6.5.
Letλ be a partition. Let A(λ,<k) (resp. R(λ,<k)) be the set of boxes which can be added to (resp. removed from) λ on rows λj with j<k such that the result is still a partition. Let b=(λ+εk)/λ and let c(·) be as in Figure 1. Then, up to multiplication by a power of q,
(6.15)evλ(sk)={∏r∈R(λ,<k)[c(r)-c(b)]∏a∈A(λ,<k)[c(a)-c(b)],ifλ+εkisapartition,0,ifλ+εkisnotapartition.Proof.
For1≤j≤N, let gj be the last box in row j of λ. By Lemma 6.4, up to multiplication by a power of q,
(6.16)evλ(sk)=evλ(∏1≤j<kzjzk-1-q2+2j-2kzj-1zkσεj(zjzk-1-q2+2j-2kzj-1zk))=∏1≤j<k[c(gj)-c(b)][c(gj)-c(b)+1],
where the last equality is a simple calculation from definitions. The denominator on the right side is never zero, and the numerator is zero exactly when λk=λk-1, so that λ+εk is no longer a partition. If λj=λj+1 for any j<k, then there is cancellation, giving (6.15). See Figure 2.Figure 2
The partition enclosed by the thick lines isλ=(10,10,8,8,8,6,6,6,6,1,1). If k=6 then A(λ,<6)={a1,a3}, R(λ,<6)={g2,g5}, andevλ(s6)=([2]/[3])([3]/[4])([4]/[5])([7]/[8])([8]/[9])=([2]/[5])([7]/[9])=([c(g5)-c(b)][c(g2)-c(b)])/([c(a3)-c(b)][c(a1)-c(b)]). The factors in the numerator of the first expression are displayed. These are the q-integers corresponding to the hook lengths of the boxes in the same column as the addable box b in row 6.Proposition 6.6.
LetNj̅l(μ/λ) be as in Section 3.3. For any partition  λ,
(6.17)valϕ2levλ(sk)=Ni̅l(μ/λ),ifμ=λ+εkisapartition,andμ/λisani̅coloredbox,evλ(sk)=0,otherwise.Proof.
By Proposition6.5, evλ(sk)=0 if λ+εk is not a partition. If λ+εk is a partition, then
(6.18){b∈A(λ,<k):c̅(b)=c̅(μ/λ)}={b∈Ai̅(λ)∣bistotheleftofμ/λ},{b∈R(λ,<k):c̅(b)=c̅(μ/λ)}={b∈Ri̅(λ)∣bistotheleftofμ/λ},
where the notation is as in Section 3.3. Since
(6.19)[x]=qx-q-xq-q-1=q-x(q-q-1)-1∏d|2xϕd,[x] is divisible by ϕ2ℓ if and only if x is divisible by ℓ, and [x] is never divisible by ϕ2ℓ2. The result now follows from Proposition 6.5.Proof of Theorem6.1.
Fixλ and 1≤k≤mλ. From definitions, (evλ⊗1)vεkj+=vμ(j). Thus, using (5.12),
(6.20)rj(λ)=(vμ(j),vμ(j))=((evλ⊗1)vεkj+,(evλ⊗1)vεkj+)=evλ(vεkj+,vεkj+)=evλ(skj).
The result now follows from Proposition 6.6.



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