Universal Verma Modules and the Misra-Miwa Fock Space

Abstract

The Misra-Miwa 𝑣-deformed Fock space is a representation of the quantized affine algebra 𝑈_𝑣(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 𝑈_𝑞(𝔤𝔩_𝑁) 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.

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 𝐅_𝑣. The space 𝐅 has a standard ℚ-basis {|𝜇⟩∣𝜆isapartition} and 𝐅_𝑣∶=𝐅⊗_ℚℚ(𝑣). Following Hayashi [2], Misra and Miwa [3] define an action of the quantized universal enveloping algebra 𝑈_𝑣(𝔰𝔩_ℓ) on 𝐅_𝑣. The only nonzero matrix elements ⟨𝜇|𝐹_𝑖|𝜆⟩ of the Chevalley generators 𝐹_𝑖 in terms of the standard basis occur when𝜇 is obtained by adding a single 𝑖-colored box to 𝜆, and these are powers of 𝑣.

We show that these powers of 𝑣 also appear naturally in the following context: partitions with at most 𝑁 parts index polynomial Weyl modules Δ(𝜆) for the integral quantum group 𝑈^𝒜_𝑞(𝔤𝔩_𝑁). Let 𝑉 be the standard 𝑁 dimensional representation of 𝑈^𝒜_𝑞(𝔤𝔩_𝑁). If the matrix element ⟨𝜇|𝐹_𝑖|𝜆⟩ is nonzero then, for sufficiently large 𝑁, (Δ^𝒜(𝜆)⊗_𝒜𝑉)⊗_𝒜ℚ(𝑞) contains the highest weight vector of weight𝜇. There is a unique such highest weight vector 𝑣_𝜇 which satisfies a certain triangularity condition with respect to an integral basis of Δ^𝒜(𝜆)⊗_𝒜𝑉. We show that the matrix element ⟨𝜇|𝐹_𝑖|𝜆⟩ is equal to 𝑣^{val𝜙2ℓ(𝑣𝜇,𝑣𝜇)}, where (⋅,⋅) is the Shapovalov form and val_𝜙2ℓ is the valuation at the cyclotomic polynomial 𝜙_2ℓ.

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 𝔤𝔩_𝑁−1 highest weight vectors in a Weyl module for 𝔤𝔩_𝑁 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 𝔤𝔩_𝑁−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 𝔤𝔩_𝑁. 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 𝔰𝔩_𝑝 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 𝑈_𝑣(𝔰𝔩_ℓ) action on Fock space. To complete the picture, one should construct a graded category, where multiplication by 𝑣 in the 𝔰𝔩_ℓ 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 𝑈_𝜀(𝔤𝔩_𝑁). 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. Sections 2 and 3 are background on the quantum group 𝑈_𝑞(𝔤𝔩_𝑁) and the Fock space 𝐅_𝑣. 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 Group 𝑈_𝑞(𝔤𝔩_𝑁) and Its Integral Form 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)

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

𝑈_𝑞(𝔤𝔩_𝑁) is the associative algebra over the field of rational functions ℚ(𝑞) generated by 𝑋₁,…,𝑋_𝑁−1,𝑌₁,…,𝑌_𝑁−1,𝐿₁^±1,…,𝐿_𝑁^±1, (2.1) with relations 𝐿_𝑖𝐿_𝑗=𝐿_𝑗𝐿_𝑖,𝐿_𝑖𝐿_𝑖⁻¹=𝐿_𝑖⁻¹𝐿_𝑖=1,𝑋_𝑖𝑌_𝑗−𝑌_𝑗𝑋_𝑖=𝛿_𝑖,𝑗𝐿_𝑖𝐿⁻¹_𝑖+1−𝐿_𝑖+1𝐿_𝑖⁻¹𝑞−𝑞⁻¹,𝐿_𝑖𝑋_𝑗𝐿_𝑖⁻¹=𝑞𝑋_𝑗𝑞,if𝑖=𝑗,₋₁𝑋_𝑗𝑋,if𝑖=𝑗+1,_𝑗𝐿,otherwise,^𝑖𝑌_𝑗𝐿_𝑖⁻¹=𝑞⁻¹𝑌_𝑗,if𝑖=𝑗,𝑞𝑌_𝑗𝑌,if𝑖=𝑗+1,_𝑗𝑋,otherwise,_𝑖𝑋_𝑗=𝑋_𝑗𝑋_𝑖,𝑌_𝑖𝑌_𝑗=𝑌_𝑗𝑌_𝑖𝑋,if𝑖−𝑗≥2,_2𝑖𝑋_𝑗−𝑞+𝑞⁻¹𝑋_𝑖𝑋_𝑗𝑋_𝑖+𝑋_𝑗𝑋²_𝑖=𝑌²_𝑖𝑌_𝑗−𝑞+𝑞⁻¹𝑌_𝑖𝑌_𝑗𝑌_𝑖+𝑌_𝑗𝑌²_𝑖=0,if𝑖−𝑗=1. (2.2) The algebra 𝑈_𝑞(𝔤𝔩_𝑁) is a Hopf algebra with coproduct and antipode given by Δ𝐿_𝑖=𝐿_𝑖⊗𝐿_𝑖,𝑆𝐿_𝑖=𝐿_𝑖⁻¹,Δ𝑋_𝑖=𝑋_𝑖⊗𝐿_i𝐿⁻¹_𝑖+1+1⊗𝑋_𝑖,𝑆𝑋_𝑖=−𝑋_𝑖𝐿_𝑖⁻¹𝐿_𝑖+1,Δ𝑌_𝑖=𝑌_𝑖⊗1+𝐿_𝑖⁻¹𝐿_𝑖+1⊗𝑌_𝑖,𝑆𝑌_𝑖=−𝐿_𝑖𝐿⁻¹_𝑖+1𝑌_𝑖, (2.3) respectively, (see [23, Section 9.1]).

As a ℚ(𝑞)-vector space, 𝑈_𝑞(𝔤𝔩_𝑁) has a triangular decomposition 𝑈_𝑞𝑔𝔩_𝑁≅𝑈_𝑞𝑔𝔩_𝑁^<0⊗𝑈_𝑞𝑔𝔩_𝑁⁰⊗𝑈_𝑞𝑔𝔩_𝑁^>0, (2.4) where the inverse isomorphism is given by multiplication (see [23, Proposition 9.1.3]). Here 𝑈_𝑞(𝔤𝔩_𝑁)^<0 is the subalgebra generated by the 𝑌_𝑖 for 𝑖=1,…,𝑁−1, 𝑈_𝑞(𝔤𝔩_𝑁)^>0 is the subalgebra generated by the 𝑋_𝑖 for 𝑖=1,…,𝑁−1, and 𝑈_𝑞(𝔤𝔩_𝑁)⁰ is the subalgebra generated by the 𝐿_𝑖^±1 for 𝑖=1,…,𝑁.

2.2. The Integral Quantum Group

Let 𝒜=ℤ[𝑞,𝑞⁻¹]. For 𝑛,𝑘∈ℤ_>0 and 𝑐∈ℤ, let [𝑛]𝑞∶=^𝑛−𝑞^−𝑛𝑞−𝑞⁻¹,𝑥^(𝑘)𝑥∶=^𝑘[𝑘]⋯[21],][𝑘−1][𝑘𝑥;𝑐∶=^𝑘_𝑠=1𝑥𝑞^{𝑐+1−𝑠}−𝑥⁻¹𝑞^{𝑠−1−𝑐}𝑞^𝑠−𝑞^−𝑠, (2.5) in ℚ(𝑞,𝑥). The restricted integral form 𝑈^𝒜_𝑞(𝔤𝔩_𝑁) is the 𝒜-subalgebra of 𝑈_𝑞(𝔤𝔩_𝑁) generated by 𝑋_𝑖^(𝑘), 𝑌_𝑖^(𝑘), 𝐿_𝑖^±1 and 𝐿_𝑖𝑘;𝑐 for 1≤𝑖≤𝑁,𝑐∈ℤ,𝑘∈ℤ_>0. As discussed in [24, Section 6], this is an integral form in the sense that 𝑈^𝒜_𝑞𝔤𝔩_𝑁⊗_𝒜ℚ𝑞=𝑈_𝑞𝔤𝔩_𝑁. (2.6) As with 𝑈_𝑞(𝔤𝔩_𝑁), the algebra 𝑈^𝒜_𝑞(𝔤𝔩_𝑁) has a triangular decomposition 𝑈^𝒜_𝑞𝔤𝔩_𝑁≅𝑈^𝒜_𝑞𝔤𝔩_𝑁^<0⊗𝑈^𝒜_𝑞𝔤𝔩_𝑁⁰⊗𝑈^𝒜_𝑞𝔤𝔩_𝑁^>0, (2.7) where the isomorphism is given by multiplication (see [23, Proposition 9.3.3]). In this case, 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)^<0 is the subalgebra generated by the 𝑌_𝑖^(𝑘), 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)^>0 is the subalgebra generated by the 𝑋_𝑖^(𝑘), and 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)⁰ is generated by 𝐿_𝑖^±1 and 𝐿_𝑖𝑘;𝑐 for 1≤𝑖≤𝑁, 𝑐∈ℤ, and 𝑘∈ℤ_>0.

2.3. Rational Representations

The Lie algebra 𝔤𝔩_𝑁=𝑀_𝑁(ℂ) of 𝑁×𝑁 matrices has standard basis {𝐸_𝑖𝑗∣1≤𝑖,𝑗≤𝑁}, where 𝐸_𝑖𝑗 is the matrix with 1 in position (𝑖,𝑗) and 0 everywhere else. Let 𝔥=span{𝐸₁₁,𝐸₂₂,…,𝐸_𝑁𝑁}. Let 𝜀_𝑖∈𝔥^∗ be the weight of 𝔤𝔩_𝑁 given by 𝜀_𝑖(𝐸_𝑗𝑗)=𝛿_𝑖,𝑗. Define 𝔥^∗_ℤ∶=𝜆=𝜆₁𝜀₁+𝜆₂𝜀₂+⋯+𝜆_𝑁𝜀_𝑁∈𝔥^∗∣𝜆₁,…,𝜆_𝑁,𝔥∈𝑍_∗ℤ+∶=𝜆=𝜆₁𝜀₁+𝜆₂𝜀₂+⋯+𝜆_𝑁𝜀_𝑁∈𝔥^∗_ℤ∣𝜆₁≥𝜆₂≥⋯≥𝜆_𝑁,𝑃⁺∶=𝜆=𝜆₁𝜀₁+𝜆₂𝜀₂+⋯+𝜆_𝑁𝜀_𝑁∈𝔥^∗_ℤ⁺∣𝜆_𝑁,𝑅≥0₊∶=𝜀_𝑖−𝜀_𝑗∣1≤𝑖<𝑗≤𝑁,𝑄∶=span_ℤ𝑅⁺,𝑄⁺∶=span_ℤ≥0𝑅⁺,𝑄⁻∶=span_ℤ≤0𝑅⁺ (2.8) 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 weight 𝜆=𝜆₁𝜀₁+⋯+𝜆_𝑁𝜀_𝑁, the Verma module 𝑀(𝜆) for 𝑈_𝑞(𝔤𝔩_𝑁) of the highest weight 𝜆 is 𝑀(𝜆)∶=𝑈_𝑞𝔤𝔩_𝑁⊗_{𝑈𝑞(𝔤𝔩𝑁)}^≥0ℚ𝑞_𝜆, (2.9) where ℚ(𝑞)_𝜆=span_ℚ(𝑞){𝑣_𝜆} is the one dimensional vector space over ℚ(𝑞) with 𝑈_𝑞(𝔤𝔩_𝑁)^≥0 action given by 𝑋_𝑖⋅𝑣_𝜆=0,𝐿_𝑗⋅𝑣_𝜆=𝑞^𝜆𝑗𝑣_𝜆,for1≤𝑖≤𝑁−1,1≤𝑗≤𝑁. (2.10)

Theorem 2.1 (see [23, Chapter 10.1]). If 𝜆∈(𝔥^∗_ℤ)⁺ then 𝑀(𝜆) has a unique finite dimensional quotient Δ(𝜆) and the map 𝜆↦Δ(𝜆) is a bijection between (𝔥^∗_ℤ)⁺, and the set of irreducible finite dimensional 𝑈_𝑞(𝔤𝔩_𝑁)-modules.

A singular vector in a representation of 𝑈_𝑞(𝔤𝔩_𝑁) is a vector 𝑣 such that 𝑋_𝑖⋅𝑣=0 for all i.

2.4. Integral Representations

The integral Verma module 𝑀^𝒜(𝜆) is the 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)-submodule of 𝑀(𝜆) generated by 𝑣_𝜆. The integral Weyl module Δ^𝒜(𝜆) is the 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)-submodule of Δ(𝜆) generated by 𝑣_𝜆. Using (2.6) and (2.4), 𝑀^𝒜(𝜆)⊗_𝒜ℚ𝑞=𝑀(𝜆),Δ^𝒜(𝜆)⊗_𝒜ℚ𝑞=Δ(𝜆). (2.11) In general, Δ^𝒜(𝜆) is not irreducible as a 𝑈^𝒜_𝑞(𝔤𝔩_𝑁) module.

3. Partitions and Fock Space

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

3.1. Partitions

A partition 𝜆 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,…,𝜆_𝑁), then 𝜆_𝑖 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 content 𝑐(𝑏) is the horizontal position of 𝑏 and the color 𝑐(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 set 𝑃⁺ of dominant polynomial weights from Section 2.3 is naturally identified with partitions with at most 𝑁 parts. If 𝜆∈𝑃⁺, then Δ(𝜆)⊗Δ𝜀₁≅_{1≤𝑘≤𝑁𝜆+𝜀𝑘∈𝑃+}Δ𝜆+𝜀_𝑘 (3.1) as 𝑈_𝑞(𝔤𝔩_𝑁)-modules. The diagram of 𝜆+𝜀_𝑘 is obtained from the diagram of 𝜆 by adding a box on row 𝑘, and Δ(𝜆+𝜀_𝑘) appears in the sum on the right side of (3.1) if and only if 𝜆+𝜀_𝑘 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

Let 𝑈^′_𝑣(𝔰𝔩_ℓ) be the quantized universal enveloping algebra corresponding to the ℓ-node Dynkin diagram

More precisely, 𝑈^′_𝑣(𝔰𝔩_ℓ) is the algebra generated by 𝐸_𝑖,𝐹_𝑖,𝐾^±1_𝑖, for 𝑖∈ℤ/ℓℤ, with relations 𝐾_𝑖𝐾_𝑗=𝐾_𝑗𝐾_𝑖,𝐾_𝑖𝐾⁻¹_𝑖=𝐾⁻¹_𝑖𝐾_𝑖=1,𝐸_𝑖𝐹_𝑗−𝐹_𝑗𝐸_𝑖=𝛿_𝑖,𝑗𝐾_𝑖−𝐾⁻¹_𝑖𝑣−𝑣⁻¹,𝐾_𝑖𝐸_𝑗𝐾⁻¹_𝑖=𝑣²𝐸_𝑗,if𝑖=𝑣𝑗,₋₁𝐸_𝑗,if𝑖=𝐸𝑗±1,_𝑗𝐾,otherwise,^𝑖𝐹_𝑗𝐾⁻¹_𝑖=𝑣⁻²𝐹_𝑗,if𝑖=𝑗,𝑣𝐹_𝑗,if𝑖=𝐹𝑗±1,_𝑗𝐸,otherwise,_𝑖𝐸_𝑗=𝐸_𝑗𝐸_𝑖,𝐹_𝑖𝐹_𝑗=𝐹_𝑗𝐹_𝑖,if𝑖−𝑗𝐸≥2,_2𝑖𝐸_𝑗−𝑣+𝑣⁻¹𝐸_𝑖𝐸_𝑗𝐸_𝑖+𝐸_𝑗𝐸²_𝑖=𝐹²_𝑖𝐹_𝑗−𝑣+𝑣⁻¹𝐹_𝑖𝐹_𝑗𝐹_𝑖+𝐹_𝑗𝐹²_𝑖=0,if𝑖−𝑗=1. (3.2) See [23, Definition Proposition 9.1.1]. The algebra 𝑈^′_𝑣(𝔰𝔩_ℓ) is the quantum group corresponding to the nontrivial central extension 𝔰𝔩^′_ℓ=𝔰𝔩_ℓ[𝑡,𝑡⁻¹]⊕ℂ𝑐 of the algebra of polynomial loops in 𝔰𝔩_ℓ.

3.3. Fock Space

Define 𝑣-deformed Fock space to be the ℚ(𝑣) vector space 𝐅_𝑣 with basis {|𝜇⟩∣𝜆isapartition}. Our 𝐅_𝑣 is only the charge 0 part of Fock space described in [27]. Fix 𝑖∈ℤ/ℓℤ and partitions 𝜆⊆𝜇 such that 𝜇/𝜆 is a single box. Define 𝐴_𝑖(𝜆)∶=boxes𝑏∣𝑏∉𝜆,𝑏hascolor,𝑅𝑖and𝜆∪𝑏isapartition_𝑖(𝜆)∶=boxes𝑏∣𝑏∈𝜆,𝑏hascolor,𝑁𝑖and𝜆⧵𝑏isapartition_𝑙𝑖𝜇/𝜆∶=𝑏∈𝑅_𝑖(𝜆)∣𝑏istotheleftof𝜇/𝜆−𝑏∈𝐴_𝑖(𝜆),𝑁∣𝑏istotheleftof𝜇/𝜆_𝑟𝑖𝜇/𝜆∶=𝑏∈𝑅_𝑖(𝜆)∣𝑏istotherightof𝜇/𝜆−𝑏∈𝐴_𝑖(𝜆)∣𝑏istotherightof𝜇/𝜆, (3.3) to be the set of addable boxes of color 𝑖, the set of removable boxes of color 𝑖, the left removable-addable difference, and the right removable-addable difference, respectively.

Theorem 3.1 (see [25, Theorem 10.6]). There is an action of 𝑈^′_𝑣(𝔰𝔩_ℓ) on 𝐅_𝑣 determined by 𝐸_𝑖|𝜆⟩∶=_{𝑐=𝜆/𝜇𝑖}𝑣^{−𝑁𝑟𝑖(𝜆/𝜇)}𝜇,𝐹_𝑖|𝜆⟩∶=_{𝑐=𝜇/𝜆𝑖}𝑣^{𝑁𝑙𝑖(𝜇/𝜆)}𝜇, (3.4) 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 𝑖-colored box.

As a 𝑈^′_𝑣(𝔰𝔩_ℓ)-module, 𝐅_𝑣 is isomorphic to an infinite direct sum of copies of the basic representation 𝑉(Λ₀). Using the grading of 𝐅_𝑣 where |𝜆⟩ has degree |𝜆|, the highest weight vectors in 𝐅_𝑣 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, 𝐅_𝑣≅𝑉(Λ₀)⊗ℂ[𝑥₁,𝑥₂,…], where 𝑥_𝑘 has degree ℓ𝑘, and 𝑈^′_𝑣(𝔰𝔩_ℓ) acts trivially on the second factor (see [27, Proposition 2.3]). Note that we are working with the “derived” quantum group 𝑈^′_𝑣(𝔰𝔩_ℓ), not the “full” quantum group 𝑈_𝑣(𝔰𝔩_ℓ), which is why there are no 𝛿-shifts in the summands of 𝐅_𝑣.

Comment 1. Comparing with [25, Chapter 10], our 𝑁^𝑙_𝑖(𝜇/𝜆) is equal to Ariki's −𝑁^𝑎_𝑖(𝜇/𝜆) and our 𝑁^𝑟_𝑖(𝜇/𝜆) is equal to Ariki's −𝑁^𝑏_𝑖(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

Let 𝕂∶=ℚ(𝑞,𝑧₁,𝑧₂,…,𝑧_𝑁). This field is isomorphic to the field of fractions of 𝑈_𝑞(𝔤𝔩_𝑁)⁰ via the map 𝜓∶𝑈_𝑞𝔤𝔩_𝑁⁰⟶𝕂,definedby𝜓𝐿_𝑖^±1=𝑧_𝑖^±1. (4.1) For each 𝜇∈𝔥^∗_ℤ, define a ℚ(𝑞)-linear automorphism 𝜎_𝜇∶𝕂→𝕂 by 𝜎_𝜇𝑧_𝑖∶=𝑞^{(𝜇,𝜀𝑖)}𝑧_𝑖,for1≤𝑖≤𝑁, (4.2) where (⋅,⋅) is the inner product on 𝔥^∗_ℤ defined by (𝜀_𝑖,𝜀_𝑗)=𝛿_𝑖,𝑗. Let 𝕂_𝜇=span_𝕂{𝑣_𝜇+} be the one-dimensional vector space over 𝕂 with basis vector 𝑣⁺_𝜇 and 𝑈_𝑞(𝔤𝔩_𝑁)^≥0 action given by 𝑋_𝑖⋅𝑣_𝜇+=0,for1≤𝑖≤𝑁−1,𝑎⋅𝑣_𝜇+=𝜎_𝜇𝜓(𝑎)𝑣_𝜇+,for𝑎∈𝑈_𝑞𝔤𝔩_𝑁⁰. (4.3)

The𝜇-shifted rational universal Verma module ^𝜇𝑀 is the 𝑈_𝑞(𝔤𝔩_𝑁)-module^𝜇𝑀∶=𝑈_𝑞𝔤𝔩_𝑁⊗_{𝑈𝑞(𝔤𝔩𝑁)}^≥0𝕂_𝜇. (4.4) The universal Verma module ^𝜇𝑀 is actually a module over 𝑈_𝑞(𝔤𝔩_𝑁)⊗_{𝑈𝑞(𝔤𝔩𝑁)}⁰𝑈_𝑞(𝔤𝔩_𝑁)⁰, where 𝑈_𝑞(𝔤𝔩_𝑁)⁰ is the field of fractions of 𝑈_𝑞(𝔤𝔩_𝑁)⁰. However, if we identify 𝑈_𝑞(𝔤𝔩_𝑁)⁰ with 𝕂 using the map 𝜓, the action of 𝑈_𝑞(𝔤𝔩_𝑁)⁰ on ^𝜇𝑀 is not by multiplication, but rather is twisted by the automorphism 𝜎_𝜇. It is to keep track of the difference between the action of 𝑈_𝑞(𝔤𝔩_𝑁)⁰ and multiplication that we use different notation for the generators of 𝕂 and 𝑈_𝑞(𝔤𝔩_𝑁)⁰ (i.e., 𝑧_𝑖 versus 𝐿_𝑖).

4.2. Integral Universal Verma Modules

The field 𝕂 contains an 𝒜-subalgebra ℛgeneratedby𝑧_𝑖^±1,𝑧_𝑖𝑘,;𝑐1≤𝑖≤𝑁,𝑐∈ℤ,𝑘∈ℤ_>0, (4.5) which is isomorphic to 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)⁰ via the restriction of the map 𝜓 in (4.1). The integral universal Verma module ^𝜇𝑀^ℛ is the 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)-submodule of ^𝜇𝑀 generated by 𝑣_𝜇+. By restricting (4.4), ^𝜇𝑀^ℛ=𝑈^𝒜_𝑞𝔤𝔩_𝑁⊗_𝑈^𝒜_{𝑞(𝔤𝔩𝑁)}^≥0ℛ_𝜇, (4.6) where ℛ_𝜇 is the ℛ-submodule of 𝕂_𝜇 spanned by 𝑣_𝜇+. In particular, ^𝜇𝑀^ℛ is a free ℛ-module.

4.3. Evaluation

Let ev^ℛ_𝜆∶ℛ→𝒜 be the map defined by ev^ℛ_𝜆𝑧_𝑖=𝑞^{(𝜆,𝜀𝑖)},ev^ℛ_𝜆𝑧_𝑖𝑛=;𝑐𝑞^{(𝜆,𝜀𝑖)}𝑛;𝑐, (4.7) where (⋅,⋅) is the inner product on 𝔥^∗ defined by (𝜀_𝑖,𝜀_𝑗)=𝛿_𝑖,𝑗. There is a surjective 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)-module homomorphism “evaluation at 𝜆” ev_𝜆∶^𝜇𝑀^ℛ⟶𝑀^𝒜𝜇+𝜆definedbyev_𝜆𝑎⋅𝑣_𝜇+∶=𝑎⋅𝑣_𝜇+𝜆,∀a∈𝑈^𝒜_𝑞𝔤𝔩_𝑁. (4.8)

For fixed 𝜆, the maps ev^ℛ_𝜆 and ev_𝜆 extend to a map from the subspace of 𝕂 and ^𝜇𝑀=^𝜇𝑀^ℛ⊗_ℛ𝕂, respectively, where no denominators evaluate to 0. Where it is clear we denote both these extended maps by ev_𝜆.

Example 4.1. Computing the action of 𝐿_𝑖 on 𝑣_𝜇+ and 𝑣_𝜇+𝜆, 𝐿_𝑖⋅𝑣_𝜇+=𝑞^{(𝜇,𝜀𝑖)}𝑧_𝑖𝑣_𝜇+,𝐿_𝑖⋅𝑣_𝜇+𝜆=ev_𝜆𝑞^{(𝜇,𝜀𝑖)}𝑧_𝑖𝑣_𝜇+𝜆=𝑞^{(𝜇,𝜀𝑖)}𝑞^{(𝜆,𝜀𝑖)}𝑣_𝜇+𝜆=𝑞^{(𝜇+𝜆,𝜀𝑖)}𝑣_𝜇+𝜆. (4.9)

4.4. Weight Decompositions

Let 𝑉 be a 𝑈_𝑞(𝔤𝔩_𝑁)⊗_𝒜ℛ-module. For each 𝜈∈𝔥^∗_𝑍, we define the 𝜈-weight space of 𝑉 to be 𝑉_𝜈∶=𝑣∈𝑉∶𝐿_𝑖⋅𝑣=𝑞^{(𝜈,𝜀𝑖)}𝑧_𝑖𝑣. (4.10) The universal Verma module ^𝜇𝑀^ℛ is a 𝑈_𝑞(𝔤𝔩_𝑁)⊗_𝒜ℛ-module, where the second factor acts as multiplication. The weight space ^𝜇𝑀_𝜂≠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 ^𝜇𝑀 can be described explicitly by ^𝜇𝑀^ℛ_𝜇−𝜈=𝑈^𝒜_𝑞𝔤𝔩_𝑁^<0_−𝜈⋅ℛ_𝜇,^𝜇𝑀^ℛ=_𝜈∈𝑄+^𝜇𝑀^ℛ_𝜇−𝜈. (4.11) Here, 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)^<0_−𝜈 is defined using the grading of 𝑈_𝑞(𝔤𝔩_𝑁)^<0 with 𝐹_𝑖∈𝑈_𝑞(𝔤𝔩_𝑁)^<0_{−(𝜀𝑖−𝜀𝑖+1)}.

4.5. Tensor Products

Let 𝑉 be a 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)⊗_𝒜ℛ-module and 𝑊 a 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)-module. The tensor product 𝑉⊗_𝒜𝑊 is a 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)⊗_𝒜ℛ-module, where the first factor acts via the usual coproduct and the second factor acts by multiplication on 𝑉. In the case when 𝑉 and 𝑊 both have weight space decompositions, the weight spaces of 𝑉⊗_𝒜𝑊 are 𝑉⊗_𝒜𝑊_𝜈=_{𝛾+𝜂=𝜈}𝑉_𝛾⊗_𝒜𝑊_𝜂. (4.12)

We also need the following.

Proposition 4.2. The tensor product of a universal Verma module with a Weyl module satisfies ^𝜇𝑀^ℛ⊗_𝒜Δ^𝒜(𝜈)⊗_ℛ𝕂≅_𝛾(^𝜇+𝛾𝑀^ℛ)^{⨁dimΔ𝒜(𝜈)𝛾}⊗_ℛ𝕂. (4.13)

Proof. Fix 𝜈∈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 𝑀𝜆+𝜇⊗Δ(𝜈)≅_𝛾𝑀𝜆+𝜇+𝛾^{⨁dimΔ(𝜈)𝛾}, (4.14) 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

The Cartan involution 𝜔∶𝑈_𝑞(𝔤𝔩_𝑁)→𝑈_𝑞(𝔤𝔩_𝑁) is the ℚ(𝑞)-algebra anti-involution of 𝑈_𝑞(𝔤𝔩_𝑁) defined by 𝜔𝐿_𝑖^±1=𝐿_𝑖^±1,𝜔𝑋_𝑖=𝑌_𝑖𝐿_𝑖𝐿⁻¹_𝑖+1,𝜔𝑌_𝑖=𝐿_𝑖⁻¹𝐿_𝑖+1𝑋_𝑖. (5.1) The map 𝜔 is also a coalgebra involution. An 𝜔-contravariant form on a 𝑈_𝑞(𝔤𝔩_𝑁)-module 𝑉 is a symmetric bilinear form (⋅,⋅) such that (𝑢,𝑎⋅𝑣)=(𝜔(𝑎)⋅𝑢,𝑣),for𝑢,𝑣∈𝑉,𝑎∈𝑈_𝑞𝔤𝔩_𝑁. (5.2)

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).

Since 𝜔 fixes 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)⊆𝑈_𝑞(𝔤𝔩_𝑁), there is a well-defined notion of an 𝜔-contravariant form on a 𝑈^𝒜_𝑞(𝔤𝔩_𝑁) module. In particular, the restriction of the Shapovalov form on Δ(𝜆) to Δ^𝒜(𝜆) is 𝜔-contravariant.

5.2. Universal Shapovalov Forms

There are surjective maps of 𝒜-algebras 𝑝₋∶𝑈^𝒜_𝑞(𝔤𝔩_𝑁)^<0→ℚ(𝑞) and 𝑝₊∶𝑈^𝒜_𝑞(𝔤𝔩_𝑁)^>0→ℚ(𝑞) defined by 𝑝₋(𝐹_𝑖)=0 and 𝑝₊(𝐸_𝑖)=0, for 1≤𝑖≤𝑁. Using the triangular decomposition (2.7), there is an 𝒜-linear surjection𝜋₀∶=𝑝₋⊗Id⊗𝑝₊∶𝑈^𝒜_𝑞𝔤𝔩_𝑁≅𝑈^𝒜_𝑞𝔤𝔩_𝑁^<0⊗_𝒜𝑈^𝒜_𝑞𝔤𝔩_𝑁⁰⊗_𝒜𝑈^𝒜_𝑞𝔤𝔩_𝑁^>0⟶𝑈^𝒜_𝑞𝔤𝔩_𝑁⁰. (5.3) The standard universal Shapovalov form is the ℛ-bilinear form (⋅,⋅)_𝜇𝑀^ℛ∶^𝜇𝑀^ℛ⊗^𝜇𝑀^ℛ→ℛ defined by 𝑎₁⋅𝑣_𝜇+,𝑎₂⋅𝑣_𝜇+𝜇𝑀^ℛ=𝜎_𝜇∘𝜓∘𝜋₀𝜔𝑎₂𝑎₁ (5.4) for all 𝑎₁,𝑎₂∈𝑈^ℛ_𝑞(𝔤𝔩_𝑁)^<0. Here, 𝜓 and 𝜎_𝜇 are as in (4.1) and (4.2). Since 𝑎₁𝑎₂⋅𝑣_𝜇+,𝑎₃⋅𝑣_𝜇+𝜇𝑀^ℛ=𝜎_𝜇∘𝜓∘𝜋₀𝜔𝑎₂𝜔𝑎₁𝑎₃=𝑎₂⋅𝑣_𝜇+,𝜔𝑎₁𝑎₃⋅𝑣_𝜇+𝜇𝑀^ℛ (5.5) for 𝑎₁,𝑎₂,𝑎₃∈𝑈_𝑞(𝔤𝔩_𝑁), the form (⋅,⋅)_𝜇𝑀^ℛ 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 at 𝜆 gives an 𝒜-valued 𝜔-contravariant form (⋅,⋅)_𝑀^𝒜_(𝜇+𝜆) on 𝑀^𝒜(𝜇+𝜆) by ev_𝜆𝑢₁,ev_𝜆𝑢_{2𝑀𝒜(𝜇+𝜆)}=ev_𝜆𝑢₁,𝑢_2𝜇𝑀^ℛfor𝑢₁,𝑢₂∈^𝜇𝑀^ℛ. (5.6) The form (⋅,⋅)_𝜇𝑀^ℛ can be extended by linearity to an 𝜔-contravariant form (⋅,⋅)_𝜇𝑀 on ^𝜇𝑀.

5.3. The Shapovalov Determinant

Let 𝑉 be a (𝑈^𝒜_𝑞(𝔤𝔩_𝑁)⊗_𝒜ℛ)-module with a chosen 𝜔-contravariant form. Let 𝐵_𝜂 be an ℛ basis for the 𝜂-weight space 𝑉_𝜂 of 𝑉. Let det𝑉_𝐵𝜂 be the determinant of the form evaluated on the basis 𝐵_𝜂. Changing the basis 𝐵_𝜂 changes the determinant by a unit in ℛ, and we sometimes write det𝑉_𝜂 to mean the determinant calculated on an unspecified basis (det𝑉_𝜂 which is only defined up to multiplication by unit in ℛ). The Shapovalov determinant is det𝑀^ℛ_𝜂∶=det𝑏_𝑖,𝑏_𝑗𝑀^ℛ_{𝑏𝑖,𝑏𝑗∈𝐵𝜂}. (5.7)

Define the partition function 𝑝∶𝔥^∗→ℤ_≥0 by 𝑝𝛾∶=dim𝑀(0)_𝛾. (5.8) Then, 𝑝(𝛾)=dim𝑀(𝜆)_𝛾+𝜆 for any 𝜆, and 𝜂∉𝑄⁻ implies that 𝑝(𝜂)=0 and det𝑀^ℛ_𝜂=1.

Theorem 5.1 (see [5, Proposition 1.9A], [6, Theorem 3.4], [4]). For any weight 𝜂, det𝑀^ℛ_𝜂=𝑐_{𝜂1≤𝑖<𝑗≤𝑁𝑚>0}𝑧_𝑖𝑧_𝑗⁻¹−𝑞^{2𝑚+2𝑖−2𝑗}𝑧_𝑖⁻¹𝑧_𝑗^{𝑝(𝜂+𝑚𝜀𝑖−𝑚𝜀𝑗)}, (5.9) where 𝑐_𝜂 is a unit in ℛ⊗_𝒜ℚ(𝑞)=𝑄(𝑞)[𝑧₁^±1,…,𝑧_𝑁^±1].

Proposition 5.2. Fix 𝜇,𝜂∈𝔥^∗_ℤ with 𝜂−𝜇∈𝑄⁻. Choose an 𝒜-basis 𝐵_𝜂−𝜇 for 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)_𝜂−𝜇. Consider the ℛ-bases 𝐵_𝜂−𝜇∶={𝑏⋅𝑣₊∣𝑏∈𝐵_𝜂−𝜇} for 𝑀^ℛ_𝜂−𝜇 and ^𝜇𝐵_𝜂∶={𝑏⋅𝑣_𝜇+∣𝑏∈𝐵_𝜂−𝜇} for ^𝜇𝑀^ℛ_𝜂. Then det^𝜇𝑀^ℛ_(𝜇𝐵_𝜂)=𝜎_𝜇(det𝑀𝐵^ℛ_𝜂−𝜇).

Proof. For 𝑏,𝑏′∈𝐵_𝜂−𝜇, (𝑏⋅𝑣_𝜇+,𝑏′⋅𝑣_𝜇+)_𝜇𝑀^ℛ=𝜎_𝜇∘𝜓∘𝜋₀𝜔𝑏^′𝑏=𝜎_𝜇𝑏⋅𝑣₀₊,𝑏^′⋅𝑣₀₊𝑀^ℛ. (5.10) The result follows by taking determinants.

5.4. Contravariant Forms on Tensor Products

If 𝑉 and 𝑊 are 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)-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 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)⊗_𝒜ℛ module 𝑊 with ℛ-bilinear 𝜔-contravariant form (⋅,⋅)_𝑊, define a ℛ-bilinear form (⋅,⋅)_{𝑊⊗ℚ(𝑞)𝑉} on 𝑊⊗_ℚ(𝑞)𝑉 by 𝑢₁⊗𝑣₁,𝑢₂⊗𝑣_{2𝑊⨂ℚ(𝑞)𝑉}=𝑢₁,𝑢_2𝑊𝑣₁,𝑣_2𝑉. (5.11) Since 𝜔 is a coalgebra involution (i.e., Δ(𝜔(𝑎))=(𝜔⊗𝜔)Δ(𝑎), for 𝑎∈𝑈_𝑞(𝔤𝔩_𝑁)), the forms (⋅,⋅)_𝑉⊗𝑊 and (⋅,⋅)_𝜇𝑀_{⊗𝑄(𝑞)𝑉} are 𝜔-contravariant.

In the case when 𝑊=^𝜇𝑀^ℛ, evaluation of the 𝜔-contravariant form (⋅,⋅)_𝜇𝑀^ℛ⊗_𝒜^𝑉 at 𝜆 gives an 𝜔-contravariant form (⋅,⋅)_𝑀^𝒜_{(𝜇+𝜆)⊗𝒜𝑉}: 𝑢₁⊗𝑣₁,𝑢₂⊗𝑣_{2𝑀𝒜(𝜇+𝜆)⨂𝒜𝑉}=ev_𝜆𝑢₁⊗𝑣₁,𝑢₂⊗𝑣_2𝜇𝑀_⨂𝒜^𝑉=ev_𝜆𝑢₁⊗𝑣₁,ev_𝜆𝑢₂⊗𝑣_{2⨂𝑀(𝜇+𝜆)𝒜𝑉}, (5.12) for 𝑢₁,𝑢₂∈^𝜇𝑀 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 for 𝐹_𝑖 from 𝑈^𝒜_𝑞(𝔤𝔩_𝑁) Representation Theory

Let us prepare the setting for our main result (Theorem 6.1). 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 𝑈^𝒜_𝑞(𝔤𝔩_𝑁).(1)Let 𝑉=Δ^𝒜(𝜀₁) be the standard 𝑁-dimensional module. Since Δ^𝒜(𝜆)⊗_𝒜ℚ(𝑞)=Δ(𝜆), (3.1) implies Δ^𝒜(𝜆)⊗_𝒜𝑉⊗_𝒜ℚ𝑞≃Δ^𝒜𝜆+𝜀_𝑘𝑗⊗_𝒜ℚ𝑞, (6.1)where the sum is over those indices 1=𝑘₁<𝑘₂<⋯<𝑘_𝑚𝜆≤𝑁 for which 𝜆+𝜀_𝑘𝑗 is a partition. For ease of notation, let 𝜇^(𝑗)=𝜆+𝜀_𝑘𝑗.(2)Fix an 𝒜-basis {𝑣₁,…,𝑣_𝑁} of 𝑉 where 𝑣_𝑘 has weight 𝜀_𝑘 and 𝑌_𝑖(𝑣_𝑘)=𝛿_𝑖,𝑘𝑣_𝑘+1. Recursively, define singular weight vectors 𝑣_𝜇(𝑗) in (Δ^𝒜(𝜆)⊗𝑉)⊗_𝒜ℚ(𝑞) by (i)𝑣_𝜇(1)=𝑣_𝜆⊗𝑣₁(ii)for each 𝑘, the submodule 𝑊_𝑘 of (Δ(𝜆)⊗_𝒜𝑉)⊗_𝒜ℚ(𝑞) generated by {𝑣_𝜆⊗𝑣_𝑖∣1≤𝑖≤𝑘} contains all weight vectors of (Δ(𝜆)⊗_𝒜𝑉)⊗_𝒜ℚ(𝑞) of weight greater than or equal to 𝜆+𝜀_𝑘. Thus, using (6.1), for each 1≤𝑗≤𝑚_𝜆 there is a one-dimensional space of singular vectors of weight 𝜇^(𝑗) in 𝑊_𝑘𝑗, and this is not contained in 𝑊_𝑘𝑗−1 (since 𝑘_𝑗>𝑘_𝑗−1). This implies that there unique singular vector 𝑣_𝜇(𝑗) of weight 𝜇^(𝑗) in 𝑣_𝜆⊗𝑣_𝑘𝑗+_{1≤𝑖<𝑗}𝑈_𝑞𝔤𝔩_𝑁𝑣_𝜇(𝑖)⊆Δ^𝒜(𝜆)⊗_𝒜𝑉⊗_𝒜ℚ𝑞, (6.2) where we recall that 𝑈_𝑞(𝔤𝔩_𝑁)=𝑈^𝒜_𝑞(𝔤𝔩_𝑁)⊗_𝒜ℚ(𝑞). (3)There is a unique 𝜔-contravariant form on Δ^𝒜(𝜆) normalized so that (𝑣_𝜆,𝑣_𝜆)=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 (Δ^𝒜(𝜆)⊗_𝒜𝑉)⊗_𝒜ℚ(𝑞) by (𝑢₁⊗𝑤₁,𝑢₂⊗𝑤₂)=(𝑢₁,𝑢₂)(𝑤₁,𝑤₂). For each 1≤𝑗≤𝑚_𝜆, define an element 𝑟_𝑗(𝜆)∈ℚ(𝑞) by 𝑟_𝑗(𝜆)∶=𝑣_𝜇(𝑗),𝑣_𝜇(𝑗). (6.3)

Theorem 6.1. The Misra-Miwa operators 𝐹_𝑖 from Section 3.3 satisfy 𝐹_𝑖|𝜆⟩=_{𝑐𝑏(𝑗)=𝑖}𝑣^{val𝜙2ℓ𝑟𝑗(𝜆)}𝜇^(𝑗), (6.4) where 𝑏^(𝑗) is the box 𝜇^(𝑗)/𝜆, 𝑐(𝑏^(𝑗)) is the color of box 𝑏^(𝑗) as in Figure 1, 𝜙_2ℓ is the 2ℓth cyclotomic polynomial in 𝑞, and val_𝜙2ℓ𝑟 is the number of factors of 𝜙_2ℓ in the numerator of 𝑟 minus the number of factors of 𝜙_2ℓ in the denominator of 𝑟.

The proof of Theorem 6.1 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 𝑀^ℛ denote the module ⁰𝑀^ℛ from Section 4.2.

Definition 6.2. Recursively define singular weight vectors  𝑣_𝜀𝑘+∈(𝑀^ℛ⊗_𝒜𝑉)⊗_ℛ𝕂  and elements  𝑠_𝑘∈𝕂 for  1≤𝑘≤𝑁  by(i)𝑣_𝜀1+=𝑣₊⊗𝑣₁, (ii)since {𝑣₊⊗𝑣_𝑗∣1≤𝑗≤𝑁} generates 𝑀^ℛ⊗_𝒜𝑉 as a 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)^≤0 module, Proposition 4.2 implies that, for each 1≤𝑘≤𝑁, there is a unique singular vector 𝑣_𝜀𝑘+ in 𝑣₊⊗𝑣_𝑘+⊕_{1≤𝑗<𝑘}𝑈^𝕂_𝑞(𝔤𝔩_𝑁)𝑣_𝜀𝑗+⊆(𝑀^ℛ⊗_𝒜𝑉)⊗_ℛ𝕂, where 𝑈^𝕂_𝑞(𝔤𝔩_𝑁)∶=𝑈_𝑞(𝔤𝔩_𝑁)⊗_(𝑞)𝕂 and the factor of 𝕂 acts by multiplication on 𝑀^ℛ.
Let 𝑠_𝑘=(𝑣_𝜀𝑘+,𝑣_𝜀𝑘+).

The 𝑠_𝑘 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 weight 𝜂, up to multiplication by a power of 𝑞, _{1≤𝑘≤𝑁}𝑠^{𝑝(𝜂−𝜀𝑘)}_𝑘=_{1≤𝑘≤𝑁}det𝑀^ℛ_{𝜂−𝜀𝑘}𝜎_𝜀𝑘det𝑀^ℛ_{𝜂−𝜀𝑘}, (6.5) where, as in Section 5.3, det𝑀^ℛ_{𝜂−𝜀𝑘} is the determinant of the Shapovalov form evaluated on an ℛ-basis for the 𝜂−𝜀_𝑘 weight space of 𝑀^ℛ.

Comment 2. In order for Lemma 6.3 to hold as stated, for each 1≤𝑘≤𝑁, one must calculate the det𝑀^ℛ_{𝜂−𝜀𝑘} 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 Lemma 6.3. For each 𝛾∈span_ℤ≤0(𝑅⁺) fix an ℛ-basis 𝐵_𝛾 for 𝑈^ℛ_𝑞(𝔤𝔩_𝑁)_𝛾^<0. Consider the following three 𝕂-bases for ((𝑀^ℛ⊗_𝒜𝑉)_𝜂)⊗_ℛ𝕂: 𝐴_𝜂∶=𝑏⋅𝑣₊⊗𝑣_𝑘∣𝑏∈𝐵_{𝜂−𝜀𝑘},𝐶,1≤𝑘≤𝑁_𝜂∶=𝑏⋅𝑣₊⊗𝑣_𝑘∣𝑏∈𝐵_{𝜂−𝜀𝑘},𝐷,1≤𝑘≤𝑁_𝜂∶=𝑏⋅𝑣_𝜀𝑘+∣𝑏∈𝐵_{𝜂−𝜀𝑘},1≤𝑘≤𝑁. (6.6) Let det(𝑀^ℛ⊗_𝒜𝑉)_𝐵 denote the determinant of (⋅,⋅)₍𝑀^ℛ⊗_𝒜^𝑉)_𝜂 calculated on 𝐵, where 𝐵 is one of 𝐴_𝜂,𝐶_𝜂, or 𝐷_𝜂. Let det^𝜈𝑀^ℛ_{𝐵𝜂−𝜈} denote det^𝜈𝑀^ℛ_𝜂 calculated with respect to the basis 𝐵_𝜂−𝜈⋅𝑣_𝜈+.
By the definition of the 𝜔-contravariant form on 𝑀^ℛ⊗_𝒜𝑉 (see Section 4.5),det𝑀^ℛ⊗𝑉_𝐴𝜂=^𝑁_𝑘=1det𝑀^ℛ_{𝐵𝑘𝜂−𝜀}^{dim𝑉𝜀𝑘}det𝑉_𝜀𝑘^dim𝑀^ℛ_{𝑘𝜂−𝜀}. (6.7) For 1≤𝑘≤𝑁, 𝑉_𝜀𝑘 is one dimensional and det𝑉_𝜀𝑘 is a power of 𝑞. Hence, up to multiplication by a power of 𝑞, (6.7) simplifies to det𝑀^ℛ⊗_𝒜𝑉_𝐴𝜂=^𝑁_𝑘=1det𝑀^ℛ_{𝐵𝑘𝜂−𝜀}. (6.8)
Notice that 𝑈^𝒜_𝑞(𝔤𝔩_𝑁)^<0⋅𝑣_𝜀𝑘+ is isomorphic to ^𝜀𝑘𝑀, and 𝐷_𝜂 is the union of ℛ-bases for each of these submodules. For each 1≤𝑘≤𝑁, and each 𝜂∈𝔥^∗_𝑍 define an ℛ basis of ^𝜀𝑘𝑀_𝜂 by ^𝜀𝑘𝐵_𝜂∶=𝑏⋅𝑣_𝜀𝑘+∣𝑏∈𝐵_{𝜂−𝜀𝑘}. (6.9) Using (𝑣_𝜀𝑘+,𝑣_𝜀𝑘+)=𝑠_𝑘, det𝑀^ℛ⊗𝑉_𝐷𝜂=^𝑁_𝑘=1𝑠^{dim(𝜀𝑘ℛ}𝑀_𝜂)𝑘det^𝜀𝑘𝑀^ℛ_(𝜀𝑘𝐵_𝜂⁾=^𝑁_𝑘=1𝑠^{𝑝(𝜂−𝜀𝑘)}_𝑘𝜎_𝜀𝑘det𝑀𝐵^ℛ_{𝑘𝜂−𝜀}, (6.10) where the last equality uses Proposition 5.2. Here, as in Section 5.3, det^𝜀𝑘𝑀^ℛ_(𝜀𝑘𝐵_𝜂) is the Shapovalov determinant calculated with respect to the basis ^𝜀𝑘𝐵_𝜂.
The change of basis from 𝐴_𝜂 to 𝐶_𝜂 is unitriangular and the change of basis from 𝐶_𝜂 to 𝐷_𝜂 is unitriangular. Thus, det(𝑀^ℛ⊗_𝒜𝑉)_𝐴𝜂=det(𝑀^ℛ⊗_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 of 𝑞, 𝑠_𝑘=_{1≤𝑗<𝑘}𝑧_𝑗𝑧_𝑘⁻¹−𝑞^{2+2𝑗−2𝑘}𝑧_𝑗⁻¹𝑧_𝑘𝜎_𝜀𝑗𝑧_𝑗𝑧_𝑘⁻¹−𝑞^{2+2𝑗−2𝑘}𝑧_𝑗⁻¹𝑧_𝑘. (6.11)

Proof. Fix 1≤𝑘≤𝑁. Setting 𝜂=𝜀_𝑘 in Lemma 6.3 and applying Theorem 5.1 we see that, up to multiplication by a power of 𝑞, _{1≤𝑥≤𝑁}𝑠^{𝑝(𝜀𝑘−𝜀𝑥)}_𝑥=_{1≤𝑥≤𝑁}det𝑀^ℛ_{𝜀𝑘−𝜀𝑥}𝜎_𝜀𝑥det𝑀^ℛ_{𝜀𝑘−𝜀𝑥}=_{1≤𝑥≤𝑁}𝑐1≤𝑖<𝑗≤𝑁𝑚>0^{𝜀𝑘−𝜀𝑥}𝑧_𝑖𝑧_𝑗⁻¹−𝑞^{2𝑚+2𝑖−2𝑗}𝑧_𝑖⁻¹𝑧_𝑗𝜎_𝜀𝑥(𝑐_{𝜀𝑘−𝜀𝑥})𝜎_𝜀𝑥𝑧_𝑖𝑧_𝑗⁻¹−𝑞^{2𝑚+2𝑖−2𝑗}𝑧_𝑖⁻¹𝑧_𝑗^{𝑝(𝜀𝑘−𝜀𝑥+𝑚𝜀𝑖−𝑚𝜀𝑗)}, (6.12) where, for each 1≤𝑥≤𝑁, 𝑐_{𝜀𝑘−𝜀𝑥} is a unit in ℚ(𝑞)[𝑧₁^±1,…,𝑧_𝑁^±1]. The value 𝑝(𝜀_𝑘−𝜀_𝑥+𝑚𝜀_𝑖−𝑚𝜀_𝑗) is 0 unless 𝑚=1 and 𝑥≤𝑖<𝑗≤𝑘. If 𝑖>𝑥, then 𝜎_𝜀𝑥 acts as the identity on 𝑧_𝑖𝑧_𝑗⁻¹−𝑞^{2+2𝑖−2𝑗}𝑧_𝑖⁻¹𝑧_𝑗, 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 𝜀_𝑘−𝜀_𝑥∉𝑄⁻, and hence 𝑝(𝜀_𝑘−𝜀_𝑥)=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 _{1≤𝑥≤𝑘}𝑠^{𝑝(𝜀𝑘−𝜀𝑥)}_𝑥=_{1≤𝑥<𝑘}𝑐_{𝜀𝑘−𝜀𝑥}𝜎_𝜀𝑥(𝑐_{𝜀𝑘−𝜀𝑥})^{𝑝(𝜀𝑘−𝜀𝑗)}_{𝑥<𝑗≤𝑘}𝑧_𝑥𝑧_𝑗⁻¹−𝑞^2+2x−2𝑗𝑧_𝑥⁻¹𝑧_𝑗𝜎_𝜀𝑥𝑧_𝑥𝑧_𝑗⁻¹−𝑞^{2+2𝑥−2𝑗}𝑧_𝑥⁻¹𝑧_𝑗^{𝑝(𝜀𝑘−𝜀𝑗)}=_{1<𝑗≤𝑘1≤𝑥<𝑗}𝑧_𝑥𝑧_𝑗⁻¹−𝑞^{2+2𝑥−2𝑗}𝑧_𝑥⁻¹𝑧_𝑗𝜎_𝜀𝑥𝑧_𝑥𝑧_𝑗⁻¹−𝑞^{2+2𝑥−2𝑗}𝑧_𝑥⁻¹𝑧_𝑗^{𝑝(𝜀𝑘−𝜀𝑗)}. (6.13) 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 𝑐_{𝜀𝑘−𝜀𝑥}/(𝜎_𝜀𝑥(𝑐_{𝜀𝑘−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 _{1<𝑥≤𝑘}𝑠^{𝑝(𝜀𝑘−𝜀𝑥)}_𝑥=_{1<𝑥≤𝑘1≤𝑗<𝑥}𝑧_𝑗𝑧_𝑥⁻¹−𝑞^{2+2𝑗−2𝑥}𝑧_𝑗⁻¹𝑧_𝑥𝜎_𝜀𝑗𝑧_𝑗𝑧_𝑥⁻¹−𝑞^{2+2𝑗−2𝑥}𝑧_𝑗⁻¹𝑧_𝑥^{𝑝(𝜀𝑘−𝜀𝑥)}. (6.14) 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. Let 𝜆 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 𝜆_𝑗 with 𝑗<𝑘 such that the result is still a partition. Let 𝑏=(𝜆+𝜀_𝑘)/𝜆 and let 𝑐(⋅) be as in Figure 1. Then, up to multiplication by a power of 𝑞, ev_𝜆𝑠_𝑘=∏_{𝑟∈𝑅(𝜆,<𝑘)}[𝑐(𝑟)−𝑐(𝑏)]∏_{𝑎∈𝐴(𝜆,<𝑘)}[𝑐(𝑎)−𝑐(𝑏)],if𝜆+𝜀_𝑘isapartition,0,if𝜆+𝜀_𝑘isnotapartition. (6.15)

Proof. For 1≤𝑗≤𝑁, let 𝑔_𝑗 be the last box in row 𝑗 of 𝜆. By Lemma 6.4, up to multiplication by a power of 𝑞, ev_𝜆𝑠_𝑘=ev_{𝜆1≤𝑗<𝑘}𝑧_𝑗𝑧_𝑘⁻¹−𝑞^{2+2𝑗−2𝑘}𝑧_𝑗⁻¹𝑧_𝑘𝜎_𝜀𝑗𝑧_𝑗𝑧_𝑘⁻¹−𝑞^{2+2𝑗−2𝑘}𝑧_𝑗⁻¹𝑧_𝑘=_{1≤𝑗<𝑘}𝑐𝑔_𝑗−𝑐(𝑏)𝑐𝑔_𝑗−𝑐(𝑏)+1, (6.16) 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 𝜆_𝑘=𝜆_𝑘−1, so that 𝜆+𝜀_𝑘 is no longer a partition. If 𝜆_𝑗=𝜆_𝑗+1 for any 𝑗<𝑘, then there is cancellation, giving (6.15). See Figure 2.

Figure 2 

The partition enclosed by the thick lines is 𝜆=(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. Let 𝑁^𝑙_𝑗(𝜇/𝜆) be as in Section 3.3. For any partition 𝜆, 𝑣𝑎𝑙_𝜙2ℓev_𝜆𝑠_𝑘=𝑁^𝑙_𝑖𝜇/𝜆,if𝜇=𝜆+𝜀_𝑘isapartition,and𝜇/𝜆isan𝑖coloredbox,ev_𝜆𝑠_𝑘=0,otherwise. (6.17)

Proof. By Proposition 6.5, ev_𝜆(𝑠_𝑘)=0 if 𝜆+𝜀_𝑘 is not a partition. If 𝜆+𝜀_𝑘 is a partition, then 𝑏∈𝐴(𝜆,<𝑘)∶𝑐(𝑏)=𝑐𝜇/𝜆=𝑏∈𝐴_𝑖(𝜆),()∶∣𝑏istotheleftof𝜇/𝜆𝑏∈𝑅𝜆,<𝑘𝑐(𝑏)=𝑐𝜇/𝜆=𝑏∈𝑅_𝑖(𝜆)∣𝑏istotheleftof𝜇/𝜆, (6.18) where the notation is as in Section 3.3. Since [𝑥]=𝑞^𝑥−𝑞^−𝑥𝑞−𝑞⁻¹=𝑞^−𝑥𝑞−𝑞⁻¹⁻¹_𝑑|2𝑥𝜙_𝑑, (6.19)[𝑥] is divisible by 𝜙_2ℓ if and only if 𝑥 is divisible by ℓ, and [𝑥] is never divisible by 𝜙²_2ℓ. The result now follows from Proposition 6.5.

Proof of Theorem 6.1. Fix 𝜆 and 1≤𝑘≤𝑚_𝜆. From definitions, (ev_𝜆⊗1)𝑣_{𝜀𝑘𝑗+}=𝑣_𝜇(𝑗). Thus, using (5.12), 𝑟_𝑗(𝜆)=𝑣_𝜇(𝑗),𝑣_𝜇(𝑗)=ev_𝜆⊗1𝑣_{𝜀𝑘𝑗+},ev_𝜆⊗1𝑣_{𝜀𝑘𝑗+}=ev_𝜆𝑣_{𝜀𝑘𝑗+},𝑣_{𝜀𝑘𝑗+}=ev_𝜆𝑠_𝑘𝑗. (6.20) The result now follows from Proposition 6.6.

Acknowledgments

The authors thank M. Kashiwara, A. Kleshchev, T. Tanisaki, R. Virk, and B. Webster for helpful discussions. A. Ram was partly supported by the NSF Grant no. DMS-0353038 and the Australian Research Council Grants nos. DP0986774 and DP0879951. P. Tingley was partly supported by the Australia Research Council Grant no. DP0879951 and the NSF Grant no. DMS-0902649.