Universal Verma modules and the Misra-Miwa Fock space

The Misra-Miwa $v$-deformed Fock space is a representation of the quantized affine algebra of type A. It has a standard basis indexed by partitions and the non-zero 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 the quantum group $U_q(gl_N)$ 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


Introduction
Fock space is an infinite dimensional vector space which is a representation of several important algebras, as described in, for example, [14,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 { |λ | λ is a partition} and F v := F ⊗ Q Q(v). Following Hayashi [11], Misra and Miwa [23] define an action of the quantized universal enveloping algebra U v ( sl ) on F v . The only non-zero matrix elements µ|Fī|λ of the Chevalley generators Fī in terms of the standard basis occur when µ is obtained by adding a singleī-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 ∆(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|Fī|λ is non-zero then, for sufficiently large N , ∆ A (λ) ⊗ A V ⊗ A Q(q) contains a highest weight vector of weight µ. There is a unique such highest weight vector v µ which satisfies a certain triangularity condition with respect to an integral basis of ∆ A (λ) ⊗ A V . We show that the matrix element µ|Fī|λ is equal to v val φ 2 (vµ,vµ) , where (·, ·) is the Shapovalov form and val φ 2 is the valuation at the cyclotomic polynomial φ 2 .
Our proof is computational, making use of the Shapovalov determinant [26,9,20]. 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 [17] and studied in the quantum case by Kamita [15]. The statement for Weyl modules is then a straightforward consequence.
Before beginning, let us discuss some related work. In [19], 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 [6] 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 ∆(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 ∆(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 [12] as extended to the quantum case by Wiesner [28].
Brundan [5] generalized Kleshchev's [19] techniques and used this information to give modular branching rules for Hecke algebras. As discussed in [2,21], 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 [25] recovered the full U v ( sl ) action on Fock space. To complete the picture one should construct a graded category, where multiplication by v in the sl representation corresponds to a grading shift. Recent work of Brundan-Kleshchev [7] and Ariki [1] explains that one solution to this problem is through the representation theory of Khovanov-Lauda-Rouquier algebras [18,24]. 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 ∧ m V ⊗ ∧ n V , 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 [27] 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 Beilinson and Bernstein [4].
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 [8, 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 is a Hopf algebra with coproduct and antipode given by As a Q(q)-vector space, U q (gl N ) has a triangular decomposition where the inverse isomorphism is given by multiplication (see [8,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 ±1 i for i = 1, . . . , N .

The integral quantum group. Let
, and . As discussed in [22,Section 6], this is an integral form in the sense that , where the isomorphism is given by multiplication (see [8,Proposition 9.3.3]). In this case, where E ij is the matrix with 1 in position (i, j) and 0 everywhere else. Let h = span{E 11 , E 22 , . . . , E N N }. Let ε i ∈ h * be the weight of gl N given by ε i (E jj ) = δ i,j . Define Q := span Z (R + ), Q + := span Z ≥0 (R + ), and Q − := span Z ≤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 the negative part of the root lattice, respectively. For

Partitions and Fock space
We now describe the v-deformed Fock space representation of U v ( sl ) constructed by Misra and Miwa [23] following work of Hayashi [11]. Our presentation largely follows [3, Chapter 10].

Partitions.
A partition λ is a finite length non-increasing sequence of positive integers. Associated to a partition is its Ferrers diagram. We draw these diagrams as in Figure 1 so that, if λ = (λ 1 , . . . , λ N ), then λ i is the number of boxes in row i (rows run southeast to northwest ). Say that λ is contained in µ if the diagram for λ fits inside the diagram for µ and let µ/λ be the collection of boxes of µ 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 . 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.
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 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 ∆(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, [  More precisely, U v ( sl ) is the algebra generated by Eī, Fī, K ±1 i , forī ∈ Z/ Z, with relations Our F v is only the charge 0 part of Fock space described in [16]. Fix i ∈ Z/ Z and partitions λ ⊆ µ such that µ/λ is a single box. Define to the right of µ/λ}|−|{b ∈ Aī(λ) : b to the right of µ/λ}| to be the set of addable boxes of colorī, the set of removable boxes of colorī, the left removableaddable difference, and the right removable-addable difference, respectively.
where c(λ/µ) denotes the color of λ/µ and the sum is over partitions µ which differ from λ by removing (respectively adding) a singleī-colored box.
As a U v ( sl )-module, F v is isomorphic to an infinite direct sum of copies of the basic representation V (Λ 0 ). Using the grading of F v where |λ has degree |λ|, the highest weight vectors in F v occur in degrees divisible by , and the number of highest weight vectors in degree k is the number of partitions of k.
, where x k has degree k, and U v ( sl ) acts trivially on the second factor (see [16,Prop. 2.3]). Note that we are working with the 'derived' quantum group U v ( sl ), not the 'full' quantum group U v ( sl ), which is why there are no δ-shifts in the summands of F v .
However, these numbers play a slightly different role in Ariki's work, which is explained by a different choice of conventions.

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 [17] in the classical case, and was studied in the quantum case by Kamita [15].

4.1.
Rational universal Verma modules. Let K := Q(q, z 1 , z 2 , . . . , z N ). This field is isomorphic to the field of fractions of U q (gl N ) 0 via the map 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 The µ-shifted rational universal Verma module µ M is the U q (gl N )-module The universal Verma module µ M is actually a module over U q (gl N ) ⊗ Uq(gl N ) 0 U 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 µ M 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 (that is, z i versus L i ).

4.2.
Integral universal Verma modules. The field K contains an A-subalgebra 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 µ M R is the U A q (gl N )-submodule of µ M generated by v µ+ . By restricting (4.4), There is a surjective U A q (gl N )-module homomorphism "evaluation at λ" For fixed λ, the maps ev R λ and ev λ extend to a map from the subspace of K and µ M = µ M R ⊗ R K respectively where no denominators evaluate to 0. Where it is clear we denote both these extended maps by ev λ .

Weight decompositions.
Let V be a U q (gl N )⊗ A R-module. For each ν ∈ h * Z , we define the ν-weight space of V to be (4.10) The universal Verma module µ M R is a U q (gl N ) ⊗ A R-module, where the second factor acts as multiplication. The weight space µ M η = 0 if and only if η = µ − ν with ν in the positive part Q + of the root lattice. These non-zero weight spaces and the weight decomposition of µ M can be described explicitly by 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 ⊗ A W are We also need the following: Proposition 4.2. The tensor product of a universal Verma module with a Weyl module satisfies Proof. Fix ν ∈ P + . In general, M (λ + µ) ⊗ ∆(ν) has a Verma filtration (see, for example, [13, Theorem 2.2]) and if λ + µ + γ is dominant for all γ such that ∆(ν) γ = 0 then (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.

The Shapovalov form and the Shapovalov determinant
The map ω is also a co-algebra involution. An ω-contravariant form on a U q (gl N )-module V is a symmetric bilinear form (·, ·) such that (5.2) (u, a · v) = (ω(a) · u, v), for u, v ∈ V and a ∈ U q (gl N ).
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 ∆(λ) to ∆ A (λ) is ω-contravariant.

Evaluation at λ gives an
The form (·, ·) µ M R can be extended by linearity to an ω-contravariant form (·, ·) µ M on µ M .

The Shapovalov determinant.
Let V be a (U A q (gl N ) ⊗ A R)-module with a chosen ω-contravariant form. Let B η be an R basis for the η-weight space V η of V . Let det V 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 det V η to mean the determinant calculated on an unspecified basis (det V η which is only defined up to multiplication by unit in R). The Shapovalov determinant is Then p(γ) = dimM (λ) γ+λ for any λ, and η ∈ Q − implies that p(η) = 0 and det M R η = 1.  [20,Theorem 3.4], [26]) For any weight η, The result follows by taking determinants.
In the case when W = µ M R , evaluation of the ω-contravariant form (·, ·) µ M R ⊗ A V at λ gives an ω-contravariant form (·, ·) M A (µ+λ)⊗ A V : for u 1 , u 2 ∈ µ M and v 1 , v 2 ∈ V . As in Section 4.3, this evaluation can be extended to the A-submodule of the rational module where no denominators evaluate to zero.
6. The Misra-Miwa formula for Fī from U A q (gl N ) representation theory Let us prepare the setting for our main result (Theorem 6.1). Fix ≥ 2 and a partition λ. Let N 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 ).
where the sum is over those indices 1 = k 1 < k 2 < · · · < k m λ ≤ N for which λ + ε k j is a partition. For ease of notation let 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 k j , and this is not contained in W k j−1 (since k j > k j−1 ). This implies that there unique singular vector v µ (j) of weight µ (j) in where we recall that U q (gl N ) = U A q (gl N ) ⊗ A Q(q). • There is a unique ω-contravariant form on ∆ A (λ) normalized so that (v λ , v λ ) = 1 and a unique ω-contravariant form on V normalized so that (v 1 , v 1 ) = 1. As in section 5.4, define a ω-contravariant form on ∆ A (λ) ⊗ A V ⊗ A Q(q) by (u 1 ⊗ w 1 , u 2 ⊗ w 2 ) = (u 1 , u 2 )(w 1 , w 2 ). For each 1 ≤ j ≤ m λ , define an element r j (λ) ∈ Q(q) by (6.3) r j (λ) := (v µ (j) , v µ (j) ).
Theorem 6.1. The Misra-Miwa operators Fī from Section 3.3 satisfy ) is the color of box b (j) as in Figure 1, φ 2 is the 2 th cyclotomic polynomial in q and val φ 2 r is the number of factors of φ 2 in the numerator of r minus the number of factors of φ 2 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 0 M R from section 4.2.
Definition 6.2. Recursively define singular weight vectors v ε k + ∈ M R ⊗ A V ⊗ R K and elements s k ∈ K for 1 ≤ k ≤ N by and the factor of K acts by multiplication on M R . Let s k = (v ε k + , v ε k + ).
The s k are quantized versions of the Jantzen numbers first calculated in [12,Section 5] and quantized in [28]. It follows immediately from the definition that s 1 = 1. Lemma 6.3. For any weight η, up to multiplication by a power of q, where, as in Section 5.3, det M R η−ε k 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 det M R η−ε k 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 ) <0 γ . Consider the following three K-bases for ( M R ⊗ A V ) η ⊗ R K: By the definition of the ω-contravariant form on M R ⊗ A V (see Section 4.5), For 1 ≤ k ≤ N , V ε k is one dimensional and det V ε k is a power of q. Hence, up to multiplication by a power of q, (6.7) simplifies to Notice that U A q (gl N ) <0 · v ε k + is isomorphic to ε k M , 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 ε k M η by (6.9) Using (v ε k + , v ε k + ) = s k , where the last equality uses Proposition 5.2. Here, as in Section 5.3, det ε k M R is the Shapovalov determinant calculated with respect to the basis ε k B 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 ⊗ A V ) Aη = det( M R ⊗ A V ) 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, 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, where, for each 1 ≤ x ≤ N , c ε k −εx is a unit in Q(q)[z ±1 1 , . . . , z ±1 N ]. 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 i z −1 j − q 2+2i−2j z −1 i z 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 .
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 have been dropped because they are powers of q. Using the fact that s 1 = 1 and making the change of variables j → x and x → j on the 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.
For k ≥ 2, the lemma now follows by induction. For k = 1 the result simply says that s 1 = 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, , if λ + ε k is a partition, 0, if λ + ε k is not a partition.
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, 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.
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 where the notation is as in Section 3.3. Since [x] is divisible by φ 2 if and only if x is divisible by , and [x] is never divisible by φ 2 2 . The result now follows from Proposition 6.5.