Quantized Weyl algebras, the double centralizer property, and a new first fundamental theorem for Uq(gln)

Let P:=Pm×n denote the quantized coordinate ring of the space of m × n matrices, equipped with natural actions of the quantized enveloping algebras Uq(glm) and Uq(gln) . Let L and R denote the images of Uq(glm) and Uq(gln) in End(P) , respectively. We define a q-analogue of the algebra of polynomial-coefficient differential operators inside End(P) , henceforth denoted by PD , and we prove that L∩PD and R∩PD are mutual centralizers inside PD . Using this, we establish a new First Fundamental theorem of invariant theory for Uq(gln) . We also compute explicit formulas in terms of q-determinants for generators of the algebras Lh∩PD and Rh∩PD , where Lh and Rh denote the images of the Cartan subalgebras of Uq(glm) and Uq(gln) in End(P) , respectively. Our algebra PD and the algebra Pol(Matm,n)q that is defined in (Shklyarov et al 2004 Int. J. Math. 15 855–94) are related by extension of scalars, but we give a new construction of PD using deformed twisted tensor products.


Introduction
The First Fundamental Theorem (FFT) is one of the pinnacles of invariant theory with a history as old as Hermann Weyl's influential book, The Classical Groups [Wy39]. In its original form, the FFT for the group GL n describes the generators of the subalgebra of GL n -invariants in the polynomial algebra P(V ⊕k ⊕ (V * ) ⊕l ), where V := C n denotes the standard GL n -module.
It was pointed out by R. Howe [Ho95,Sec. 2.3] that the FFT has an equivalent formulation as a double centralizer property, which we now recall. Let Mat m×n denote the vector space of complex m × n matrices. Then Mat m×n has a natural GL m × GL n -module structure by left and right matrix multiplication. We equip the algebra P := P(Mat m×n ) of polynomials on Mat m×n and the algebra PD := PD(Mat m×n ) of polynomial-coefficient differential operators on Mat m×n with their canonical GL m × GL n -module structures. Then the (infinitesimal) actions of the Lie algebras gl m and gl n on P are given by certain differential operators of order one, which are usually called polarization operators. It follows that there exists a homomorphism of algebras φ : U m,n → PD, where U m,n := U (gl m ) ⊗ U (gl n ) is the tensor product of the universal enveloping algebras of gl m and gl n , such that the diagram (1) commutes. The operator commutant version of the FFT, according to [Ho95,Thm 2.3.3], states that the subalgebra PD GL m of GL m -invariants in PD is generated by the image of U (gl n ). Since PD GLm = PD U (gl m ) , the latter assertion is equivalent to the following: the images of U (gl m ) and U (gl n ) in PD are mutual centralizers.
In [LZZ11,Sec. 6] the authors extend the original form of the FFT to the quantized enveloping algebra U q (gl n ) by considering a q-analogue of P(V ⊕k ⊕(V * ) l ) that carries a U q (gl n )-action, and then describing the generators of the subalgebra of invariants. It is then natural to ask if the operator commutant version of the FFT also has a q-analogue. It turns out that in the quantized setting, the situation for the operator commutant FFT is more subtle than in the classical case. One major issue is how to quantize the Weyl algebra PD and, more importantly, the map φ : U m,n → PD. Indeed we provide some justification that the latter map cannot be fully quantized (see Remark 4.1.3). Nevertheless, our first main result (Theorem A) is a positive answer to the above question.
From now on let k := C(q) be the field of rational functions in a parameter q. For the operator commutant FFT in the quantized setting we need a quantized Weyl algebra PD := PD m×n . The k-algebra PD that we consider was already introduced in [SSV04,BKV06]. We give a different construction of PD as the deformed twisted tensor product of P := P m×n , the quantized coordinate ring of Mat m×n , and D := D m×n , the quantized algebra of constant-coefficient differential operators on Mat m×n (see Section 3 for precise definitions). The construction of P m×n and D m×n is analogous to the FRT construction [KS97,Sec. 9.1]. Concretely, the algebra PD is generated by 2mn generators t i,j and ∂ i,j , where 1 ≤ i ≤ m and 1 ≤ j ≤ n, modulo the relations that are given in Section 3. From now on we set U L := U q (gl m ) , U R := U q (gl n ) , U LR := U L ⊗ U R .
Both P and PD are U LR -module algebras (the explicit formulas for the U LR -action on generators are given in Remark 3.4.4). Furthermore, P is naturally a PD -module. In particular, we have homomorphisms of associative algebras φ U : U LR → End k (P) and φ P D : PD → End k (P).

Furthermore, we set
(2) L i,j := n r=1 t i,r ∂ j,r for 1 ≤ i, j ≤ m and R i,j := m r=1 t r,i ∂ r,j for 1 ≤ i, j ≤ n.
We adopt the following notation: given an associative algebra X and two subsets Y, Z ⊆ X , we set (3) Y Z := {y ∈ Y : yz = zy for all z ∈ Z}.
Our first main theorem is the following.
Theorem A. Let L , R, L • , and R • be the subalgebras of End k (P) defined above. We identify PD with φ P D (PD ) ⊆ End k (P). Then the following statements hold.
(iii) L • is generated by the L i,j for 1 ≤ i, j ≤ m.
(iv) R • is generated by the R i,j for 1 ≤ i, j ≤ n.
Note that in general we have L • L and R • R. In fact if m ≤ n then the restriction of φ U to U L ⊗ 1 yields an isomorphism U L ∼ = L but one can show that φ −1 U (L • ) is properly contained in the locally finite part of U L (see Proposition 4.1.2 and Example 10.2.8). Furthermore, unlike the classical polarization operators, in general the L i,j (respectively, the R i,j ) are not in the images of the root vectors of U L (respectively, U R ).
Theorem B. The following statements hold.
Let us elucidate the relation between Theorem A and the literature on Howe duality and the FFT in the quantized setting. Quantized analogues of (gl m , gl n )-duality have been established in [Zh02] and [NYM93], but these works do not consider the double centralizer property inside a quantized Weyl algebra. To compare our results with those of Lehrer-Zhang-Zhang [LZZ11], we briefly explain their formulation of the FFT for U q (gl n ). In [LZZ11,Sec. 6] the authors define a q-analogue of the algebra P(V ⊕k ⊕ (V * ) ⊕l ), which they call A k,l . The algebra A k,l is isomorphic to a twisted tensor product of P k×n and D l×n , but the twisting is only with respect to the universal R-matrix of U q (gl n ). In particular, in the special case k = l the relations on the generators of A k,k are not symmetric with respect to the indices of the generators. Because of this asymmetry, A k,k does not appear to be the desired object for proving a double centralizer statement. The twisting that we consider to define PD uses the universal R-matrices of both U L and U R . In addition, unlike A k,l whose relations are homogeneous, the relation (R6) of PD is not homogeneous. From this viewpoint, PD resembles the classical Weyl algebra more than A k,k .
Using Theorem A we can easily prove a variant of the FFT analogous to [LZZ11, Thm 6.10]. For a fixed n ≥ 1, let A k,l for k, l ≥ 1 denote the algebra with kn generators t i,j and ln generators ∂ i ′ ,j , where 1 ≤ i ≤ k, 1 ≤ i ′ ≤ l, and 1 ≤ j ≤ n, that satisfy relations similar to those of PD , i.e., the relations (R1), (R2), (R1 ′ ), (R2 ′ ), and (R3)-(R6). Then A k,l is isomorphic to a U R -invariant subalgebra of PD = PD m×n for m := max{k, l}, and therefore A k,l is a U R -module algebra. The standard degree filtration (corresponding to setting deg t i,j = deg ∂ i ′ ,j = 1) is U R -invariant, hence the associated graded algebra gr(A k,l ) is also a U R -module algebra. Let ǫ R denote the counit of U R and set We define gr(A k,l ) (ǫ R ) similarly. For 1 ≤ i ≤ k and 1 ≤ j ≤ l we define operators L i,j in A k,l and in gr(A k,l ) similar to (2).
Theorem C. The algebras (A k,l ) (ǫ R ) and gr(A k,l ) (ǫ R ) are generated by the L i,j for 1 ≤ i ≤ k and 1 ≤ j ≤ l.
Despite the aforementioned differences between our results and the work of [LZZ11], we borrow at least two key ideas from [LZZ11]. First, we use the natural bialgebra structure of P m×m to define a map Γ onto the subalgebra of U R -invariants in PD (assuming m ≤ n). The map Γ, given in Definition 3.6.1, is essentially the same as the map introduced in [LZZ11, Lem. 6.11]. Second, we define a new product on P m×m such that the map Γ becomes a homomorphism of algebras modulo the filtration of its codomain (see Proposition 7.1.2). A similar product was used in [LZZ11, Lem. 6.13]. However, our product is given by a more complicated (and asymmetric) formula, because it needs to be simutaneously compatible with two universal R-matrices. As a consequence, establishing the desired properties of this product required new ideas.
The results of this paper were obtained as part of a project on Capelli operators for quantum symmetric spaces. From this standpoint, it is natural to ask if one can define quantized Weyl algebras in the latter setting and then realize the action of (a large subalgebra of) the quantized enveloping algebra via elements of this Weyl algebra. We address this question and its connection to Capelli operators in upcoming work [LSS22a,LSS22b].
The structure of this paper is as follows. In Section 2 we review the required background material on Hopf algebras and twisted tensor products. In Section 3 we construct the quantized Weyl algebra PD, the action of U LR on PD , and the map Γ mentioned above. The main goal of Section 4 is to prove that the elements x a and y b , defined in (27), of the Cartan subalgebra of U LR belong to φ −1 (PD). In Section 5 we compute explicit formulas for φ U (x a ) and φ U (y b ). In Section 6 we establish some properties of the map Γ. The proof of Theorem A occupies Sections 7-9. Theorem B and Theorem C are proved in Sections 10 and 11, respectively. Finally, in Sections 12 and 13 we prove two technical statements whose proofs are postponed for the reader's convenience.

Hopf algebras and deformed twisted tensor products
In this section we review some basic notions from Hopf algebras and then define deformed twisted tensor product algebras. Throughout this section K will denote an arbitrary field.
2.1. The locally finite part. Let H be a Hopf algebra over K. If I ⊆ H is a two-sided ideal of H as an associative algebra, then we set E(x, I) := {ad y (x) + I : y ∈ H} for x ∈ H, where ad y (x) := y 1 xS(y 2 ) is the left adjoint action of H. Furthermore, we set Of course for I = 0 this is the locally finite part of H (in the sense of [JL94]), which we will denote by F (H). We have E(xy, I) ⊆ E(x, I)E(y, I), from which it follows that F (H, I) is a subalgebra of H.

2.2.
Matrix coefficients and the finite dual of H. Given a finite dimensional left H-module V , by the right dual of V we mean the dual space V * equipped with the H-action defined by Recall that H • has a canonical Hopf algebra structure. Let ∆ • denote the coproduct of H • , so that The following remark will be used in Section 3. Remark 2.2.1. Let H • ⊆ H • be a sub-bialgebra. Then H • is an H-module algebra with respect to right translation, where the action is defined by x · λ, y := λ, yx for λ ∈ H • and x, y ∈ H. If H is equipped with a linear endomorphism x → x ♮ that yields an isomorphism of Hopf algebras H → H op , then H • has another H-module algebra structure defined by x · λ, y := λ, x ♮ y , which we call left translation. Given any homomorphism of associative algebras τ : H → H, we can define τ -twisted left and right translation actions of H on H • , given respectively by the formulas x · λ, y := λ, τ (x) ♮ y and x · λ, y := λ, yτ (x) .
Furthermore, when H • is equipped with the τ -twisted actions, (i) if τ : H → H is a homomorphism of coalgebras, then H • is an H-module algebra. (ii) if τ : H → H is an anti-homomorphism of coalgebras, then H • is an H cop -module algebra.
2.3. H-invariants and the ǫ-isotypic component. In Section 6 we will need the following general remark about the relation between commutants and trivial H-modules. For any Remark 2.3.1. Let V be an H-module, and let ψ : H → End K (V ) be the algebra homomorphism corresponding to this module structure. One can equip End K (V ) with an H-module structure, defined by h · T := ψ(h 1 )T ψ(S(h 2 )), where ∆(h) = h 1 ⊗ h 2 . By standard abstract arguments one can show that End K (V ) (ǫ) = End K (V ) ψ(H) , where the right hand side is defined as in (3). The inclusion ⊇ follows from and the inclusion ⊆ follows from 2.4. Matrix coefficients of quasitriangular Hopf algebras. In the rest of this section we assume that H has a universal R-matrix, that is, an invertible 2-tensor that satisfies it is well known (see for example [KS97, Sec. 8.1.1]) that Remark 2.4.1. We do not require that R be strictly in H ⊗ H. Indeed in this paper we are primarily interested in the case where H = U LR . It is well known that in this case the universal R-matrix belongs to a suitable completion of H ⊗ H. In this context the universal R-matrix is an infinite formal sum However, we only need to consider how R acts on finite dimensional H-modules that are direct sums of tensor products of modules of type (1, . . . , 1) of U L and U R , where the type of a module is defined as in [Ja96, Sec. 5.2]. On any such module, all but finitely many terms of R vanish. More generally, we can fix a category C of finite dimensional H-modules such that C is closed under taking tensor products and the defining properties (7) of R hold as relations in the endomorphism algebras of objects of C.
Let C be a category of H-modules as in Remark 2.4.1. For V, W ∈ obj(C) we define We also setŘ V, be the dual bases of V * and W * . We denote the matrix entries of R V,W in the basis v i ⊗ w j by the R kl ij , so that The matrix coefficients of objects of C span a sub-bialgebra H •• ⊆ H • . From (9) it follows that 2.5. Twisted tensor products and their deformations. Let A and B be two H-module algebras, with products m A and m B . It is well known (for example see [LZZ11, Thm 2.3]) that the vector space A ⊗ K B can be equipped with an H-module algebra structure with the product We denote this associative algebra by A ⊗ R B. It is sometimes called the twisted tensor product of A and B. Let E A ⊆ A and E B ⊆ B be H-invariant subspaces that generate A and B, respectively. Thus A ∼ = T (E A )/I A and B ∼ = T (E B )/I B , where T (X) denotes the tensor algebra on X, and I A and I B denote the corresponding ideals of relations. We consider the map Then the map E A ⊕ E B → A ⊗ R B given by the assignment a ⊕ b → a ⊗ 1 + 1 ⊗ b yields a canonical isomorphism of associative algebras where I A,B denotes the two-sided ideal generated by I A , I B , and relations of the form where ǫ : H → K denotes the counit of H. Let I A,B,ψ denote the two-sided ideal of T (E A ⊕ E B ) that is generated by I A , I B , and relations of the form For the next proposition, recall that the H-module structure on E A ⊕ E B induces a canonical H-module algebra structure on T (E A ⊕ E B ).
Proposition 2.5.2. The H-module algebra structure on T (E A ⊕ E B ) descends to an H-module algebra structure on A ⊗ R,ψ B.
Proof. It suffices to prove that the relations (14) are preserved under the H-action. Any x ∈ H acts on T 0 (E A ⊕ E B ) ∼ = K by ǫ(x). Thus by (13) for a ∈ E A , b ∈ E B , and x ∈ H we have 3.1. The algebra U q (gl n ). For n ∈ N, the quantized enveloping algebra U q (gl n ) is the k-algebra generated by E i , F i , where 1 ≤ i ≤ n − 1, and K ±1 ε i , where 1 ≤ i ≤ n, that satisfy the relations as well as the quantum Serre relations. For λ := n i=1 m i ε i ∈ Zε 1 + · · · + Zε n we define The Cartan subalgebra of U q (gl n ) is the subalgebra spanned by the K λ for λ ∈ Zε 1 + · · · + Zε n . Following [KS97] for the choice of the coproduct ∆ on U q (gl n ), we set The counit and antipode of U q (gl n ) are given by 3.2. The universal R-matrix of U q (gl n ). Throughout this paper we fix a universal R-matrix for U q (gl n ). For more details see [VY20, Thm 3.108] or [KS97, Sec. 8.3.2].
Definition 3.2.1. Given n ∈ N, the standard root vectors of U q (gl n ) are where 1 ≤ i < j ≤ n and [x, y] q ±1 := xy − q ±1 yx. The universal R-matrix of U q (gl n ) that we use is 3.3. The algebras P n×n and D n×n . From now on, for n ∈ N let ∆ + n := {ε i − ε j : 1 ≤ i < j ≤ n} denote the standard positive system of the root system of U q (gl n ). Let V (n) denote the irreducible U q (gl n )-module of highest weight q −εn (all highest weights are considered with respect to ∆ + n ). Thus V (n) ∼ = k n as a vector space and the homomorphism of algebras U q (gl n ) → End k V (n) is uniquely determined by the assignments where the E i,j are the elementary matrix units associated to the standard basis {e i } n i=1 of V (n) and 1 := n i=1 E i,i . Using (15) the R-matrix of V (n) ⊗ V (n) can be computed directly, and we obtain For 1 ≤ i, j ≤ n let t i,j denote the matrix coefficient m e * i ,e j of V (n) . By (10) the t i,j satisfy the following relations: Similarly, letV (n) denote the irreducible U q (gl n )-module with highest weight q ε 1 . AgainV (n) ∼ = k n as vector spaces but the map U q (gl n ) → End k V (n) is uniquely determined by the assignments The corresponding R-matrix is If ∂ i,j for 1 ≤ i, j ≤ n denotes the matrix coefficient m e * i ,e j ofV (n) , then again from (10) it follows that the ∂ i,j satisfy relations similar to those between the t i,j , with q replaced by q −1 . Equivalently, Let P n×n denote the subalgebra of U q (gl n ) • generated by the t i,j , for 1 ≤ i, j ≤ n. Similarly, let D n×n denote the subalgebra of U q (gl n ) • generated by the ∂ i,j , for 1 ≤ i, j ≤ n. It is well known that the relations (R1)-(R2) yield a description of P n×n by generators and relations (that is, no further relations are required). A similar statement holds for D n×n (see [Ta92] for a proof).
From Section 2 it follows that both P n×n and D n×n are bialgebras with the coproducts satisfying and the counits satisfying t i,j , ∂ i,j → i, j . It is well known (for example see [No96, Sec. 1.4]) that there exists a unique k-linear isomorphism of Hopf algebras Thus according to Remark 2.2.1, the canonical (U q (gl n ), U q (gl n ))-bimodule structure of U q (gl n ) by left and right translation equips both P n×n and D n×n with U q (gl n ) ⊗ U q (gl n )-module algebra structures. Our next goal is to describe the latter actions explicitly. In what follows, all of the actions are from the left side.
Let R D be the action of U q (gl n ) on D n×n by right translation, as in Remark 2.2.1. We have Similarly, let L D be the action of U q (gl n ) on D n×n by left translation. This action satisfies By Remark 2.2.1 both L D and R D equip D n×n with U q (gl n )-module algebra structures.
To define the desired actions on P n×n , first we consider two new actions R ′ D and L ′ D of U q (gl n ) on D n×n , which are in some sense opposite to R D and L D , respectively. More precisely, let R ′ D be the τ -twist of L D , and let L ′ D be the τ -twist of R D , where τ (x) := S −1 (x) ♮ for x ∈ U q (gl n ). By Remark 2.2.1(ii) these two actions equip D n×n with U q (gl n ) cop -module algebra structures. We have There exists a unique isomorphism of bialgebras ι : Any U q (gl n ) cop -module algebra structure on D n×n corresponds to a U q (gl n )-module algebra structure on P n×n , defined by x · u := ι −1 (x · ι(u)) for u ∈ P n×n . Under this correspondence, R ′ D and L ′ D corresponds to actions R P and L P of U q (gl n ) on P n×n , respectively. We have for x ∈ U q (gl n ) and u ∈ P n×n . Furthermore, P n×n is a U q (gl n )-module algebra with respect to both L P and R P .
3.4. The algebras P and D. We define P := P m×n and D := D m×n for 1 ≤ m ≤ n as subalgebras of P n×n and D n×n , respectively. A similar definition can be given when m > n, although we do not need to consider this case because the proofs of Theorems A and B remain the same (see also Subsection 3.7).
Definition 3.4.1. For positive integers m ≤ n, we define P m×n (respectively, D := D m×n ) to be the subalgebra of P n×n (respectively, D n×n ) that is generated by the t i,j (respectively, the ∂ i,j ) where 1 ≤ i ≤ m and 1 ≤ j ≤ n.
The subalgebra of U q (gl n ) generated by E i , F i , K ε j for 1 ≤ i ≤ m − 1 and 1 ≤ j ≤ m is isomorphic to U q (gl m ). By restricting the U q (gl n ) ⊗ U q (gl n )-actions L P ⊗ R P and L D ⊗ R D to the subalgebra we obtain U LR -module algebra structures on P and on D. Explicitly, the actions of x⊗ y ∈ U L ⊗ U R on u ∈ P and on v ∈ D are given by Definition 3.4.2. For any integer partition λ satisfying ℓ(λ) ≤ n, where ℓ(λ) denotes the length of λ, let V λ denote the irreducible finite dimensional U R -module of type (1, . . . , 1) with highest weight q i λ i ε i (with respect to ∆ + n ). For λ such that ℓ(λ) ≤ m we use the same notation V λ to denote the analogously defined module of U L .
The algebras P and D are naturally graded by degree of monomials. For d ≥ 0 let P (d) (respectively, D (d) ) denote the graded component of degree d of P (respectively, D). Furthermore, let Λ m,d be the set of integer partitions λ such that ℓ(λ) ≤ m and |λ| = d, where |λ| denotes the size of λ. The following proposition is well known and its proof can be found for example in [NYM93,Ta92,Zh02].
Proposition 3.4.3. As U LR -modules, Remark 3.4.4. The action of U L ⊗ U R on the generators of P and D can be computed explicitly. For U R , the action is given by where 1 ≤ k ≤ n − 1, 1 ≤ i ≤ m and 1 ≤ j ≤ n. For U L the formulas are similar but the action occurs in the first index (thus, they are obtained by replacing ∂ i,j by ∂ j,i and t i,j by t j,i ).
3.5. The algebras PD gr and PD. Let m ≤ n be positive integers and let U LR be defined as in Section 1. LetR denote the univeral R-matrix of U LR that is defined by Let P ⊗R D denote the twisted tensor product of P and D, defined as in Subsection 2.5, where we set A := P, E A := P (1) , B := D, E B := D (1) , and R :=R. For convenience in notation we set PD gr := P ⊗R D.
Next we define the quantum Weyl algebra PD := PD m×n as a deformation of P ⊗R D, using Definition 2.5.1. Let ψ • : D (1) × P (1) → k be the k-bilinear form that is uniquely defined by Note that ψ • is U LR -invariant. Again for convenience in notation we set Thus PD is the algebra generated by the 2mn generators t i,j and ∂ i,j for 1 ≤ i ≤ m and 1 ≤ j ≤ n, modulo the relations (R1)-(R2), (R1 ′ )-(R2 ′ ), and other mixed relations coming from (14). We now compute the mixed relations between the t i,j and the ∂ i,j explicitly. As a U LR -module, By a direct calculation using Definition 3.2.1 we obtain is similar, with m replaced by n. From here, and by a direct calculation, we obtain the relations (R3)-(R6) given below: Remark 3.5.1. By Proposition 2.5.2, the actions of U LR on P and D carry over to PD , and make the latter a U LR -module algebra. As in Remark 3.4.4, we denote the action of x ∈ U LR on D ∈ PD by x · D.
Remark 3.5.2. Using Bergman's Diamond Lemma and some straightforward (although tedious) computations one can show that PD has a k-basis consisting of monomials of the form This was also pointed out in [SSV04, Sec. 10]. In [LSS22a] we give a more conceptual proof of this assertion using the theory of PBW deformations of quadratic algebras. The analogous statements for P and for D are well known (for example see [NYM93, Thm 1.4]).
Let I denote the left ideal of PD generated by D (1) . By Remark 3.5.2 we have a U LR -invariant decomposition PD ∼ = I ⊕ P. This decomposition equips P ∼ = PD/I with a PD-module structure given by To distinguish the action of PD on P from the product of PD, we denote the action of D ∈ PD on f ∈ P by D · f .
Proof. This follows from Remark 3.5.1 and U LR -invariance of I .
Proof. This is proved in [SSV04, Thm 2.6] using analytic tools. See Section 12 for a purely algebraic proof.
From Remark 3.5.2 it follows that the map is an isomorphism of vector spaces. For r, s ∈ Z ≥0 we set PD (r,s) := P P (r) ⊗ D (s) . By Proposition 3.4.3 and using the coproduct map U LR → U LR ⊗U LR to obtain a U LR -module from a U LR ⊗U LR module, we obtain an isomorphism of U LR -modules Remark 3.5.5. By Remark 2.5.3, the above map P transfers the U LR ⊗ U LR -module structure of P ⊗ D to PD. Note that the latter U LR ⊗ U LR -module structure on PD is compatible with the U LR -module structure of Remark 3.5.1 through the coproduct map U LR → U LR ⊗ U LR .
Remark 3.5.6. The map P does not induce an isomorphism of associative algebras PD gr → PD . However, the products of the domain and the range of P are the same modulo lower order terms.
3.6. The maps Γ and gr Γ n . Recall that 1 ≤ m ≤ n. In this subsection our goal is to define a map Γ : P m×m → PD that is a bijection onto PD R (see Lemma 6.2.2). A similar map was also used in [LZZ11]. To define Γ, first for n ≥ 1 we set where ι : P n×n → D n×n is the anti-isomorphism of bialgebras defined in (17) and as usual ∆ P n×n (u) = u 1 ⊗ u 2 is the coproduct of P n×n in Sweedler's notation. Let us identify P m×m with the subalgebra of P n×n generated by the t i,j satisfying 1 ≤ i, j ≤ m. Similarly, we identify P (respectively, D) with a subalgebra of P n×n (respectively, of D n×n ) as in Definition 3.4.1. By tensoring the embeddings P ֒→ P n×n and D ֒→ D n×n we can consider PD gr as a subspace (although not a subalgebra) of PD gr n×n . It is straightforward to check that gr Γ n (P m×m ) ⊆ PD gr . Thus the following definition is valid.
3.7. Convention. Unless stated otherwise, from now on we assume 1 ≤ m ≤ n. In particular, we prove Theorems A and B under this assumption. Analogous arguments can be given in the case m > n. In our proofs we also need to consider the algebras P m×m , PD m×m , P n×n , D n×n , PD n×n , and PD gr n×n . For further clarity, we do not suppress the subscripts m × m and n × n for these algebras.

Differential operators associated to the K λ
Recall that we assume 1 ≤ m ≤ n. For 1 ≤ a ≤ m and 1 ≤ b ≤ n we set The key result of this section is Proposition 4.2.1, which proves that x a , y b ∈ φ −1 U (PD). We set 4.1. A necessary local finiteness condition forŮ LR . Recall that the adjoint action of U LR is ad y (x) := y 1 xS(y 2 ) for x, y ∈ U LR . We equip End k (P) with the U LR -module structure of Remark 2.3.1. We denote the latter action by x · T for x ∈ U LR and T ∈ End k (P).
This proves that φ U (ad y (x)) = y · φ U (x) and in particular ad y (U LR ) ⊆ U LR .
From Proposition 3.4.3 it follows that the map L P : U L → End k (P) is an injection (the argument is similar to the proof of [KS97, Thm 7.1.5.13]). Let K n denote the kernel of R P : U R → End k (P).
Proof. Since the action of U LR on P and D is degree preserving, it follows that PD is a locally finite U LR -module. The assertions of the proposition follow from the fact that the mapsŮ L Remark 4.1.3. The proof of Proposition 4.1.2 implies the following "no-go theorem": it is impossible to construct a nontrivial quantized Weyl algebra PD such that PD acts locally finitely on P, the action PD ⊗ P → P is U LR -equivariant, and there exists a homomorphism of algebras U LR → PD that is compatible with the action in the sense of the commutative diagram (1). 4.2. U LR as aŮ LR -module. By a slight abuse of notation we consider the x a and the y b as elements of U L and U R , respectively. Proof. We only prove the assertion for x a (for y b the argument is similar). First we verify the case a = m. By Remark 3.4.4 we have x m · t a 1 ,b 1 · · · t ar,br = q 2 r i=1 m,a i t a 1 ,b 1 · · · t ar,br , and it is straightforward to check that the operator To complete the proof, by Lemma 4.1.1 it suffices to verify that for any a < m, the element x a = K λ L,a lies in the ad(U L )-invariant subalgebra of U L that is generated by x a+1 = K λ L,a+1 and x m = K λ L,m . Denoting the standard generators of . By a simple induction we can verify that But the left hand side of (28) is equal to By a similar argument we can prove that F i ∈ A as well, hence A = U L .

Explicit formulas for
In this section we compute explicit formulas for φ U (x a ) and φ U (y b ), where x a and y b are defined in (27). We remark that in Section 10 we prove that the x a and the y b generate φ −1 U (L h,• ) and φ −1 U (R h,• ), respectively. Recall that the action of D ∈ PD on f ∈ P is denoted by D · f .

5.1.
Some technical statements about the action. We begin with the following lemma.
Proof. We use induction on r. For r = 1 the assertion follows from relations (R3)-(R6).Next suppose r > 1. If i = a 1 and j = b 1 then ∂ i,j t a 1 ,b 1 = t a 1 ,b 1 ∂ i,j and we can use the induction hypothesis. If i = a 1 then j ∈ {b 1 , . . . , b r } and we can write and again the induction hypothesis is applicable to each summand on the right hand side. The argument for the case j = b 1 is similar.
In Subsection 5.3 we need more refined information about the monomials in the t i,j that are generated by ∂ i,j · t a 1 ,b 1 · · · t ar,br . To this end we use the following technical lemma, whose proof is postponed until Section 13. Let us call two r-tuples (m 1 , . . . , m r ) and (m ′ 1 , . . . , m ′ r ) of integers order equivalent, if they satisfy the following property: m i < m j if and only if m ′ i < m ′ j for 1 ≤ i, j ≤ r. Lemma 5.1.2. Let 1 ≤ a 1 , . . . , a r ≤ m and 1 ≤ b 1 , . . . , b r ≤ n. We denote the order equivalence classes of (a 1 , . . . , a r ) and (b 1 , . . . , b r ) by a and b, respectively. Then where c a,b (i, j, σ s , τ s ) ∈ k and the inner sum is over all pairs of injective maps In particular, the coefficients c a,b (i, j, σ s , τ s ) only depend on i, j, σ s , τ s , a and b.
We do not know explicit formulas for the coefficients c a,b (i, j, σ s , τ s ). However, for our purposes Lemma 5.1.2 suffices.
Proof. Set f := ∂ ar,br · (t c 1 ,d 1 · · · t ar,br ) and assume that f = 0. Then by Lemma 5.1.1 we have a r ∈ {c 1 , . . . , c r } and b r ∈ {d 1 , . . . , d r }. Lemma 5.1.2 implies that f can be expressed as a linear combination of monomials of the form t c ′ The claim follows by iterating the above reasoning. Lemma 5.1.3. Suppose that f, g ∈ P satisfy ∂ i 1 ,j 1 · · · ∂ ir,jr · f = g for some 1 ≤ i 1 , . . . , i r ≤ m and 1 ≤ j 1 , . . . , j r ≤ n. Then for any Recall that I is the left ideal of PD generated by D (1) (see Subection 3.5). Set f ′ := ∂ ir,jr ·f .
and the sum is over all pairs (i ′ , j ′ ) that satisfy i r ≤ i ′ ≤ m and j r ≤ j ′ ≤ n. In particular j ′ ∈ {j ′ 1 , . . . , j ′ s }, hence by Lemma 5.1.1 we obtain ∂ ir,jr f t i ′ s . The proof is completed by induction on r.

5.2.
Eigenvalues of D n,r and q-factorial Schur polynomials. For any integer partition ν such that ℓ(ν) ≤ n, let s ν denote the q-factorial Schur polynomial in n variables associated to ν, defined by .
In the rest of this section we will need Okounkov's binomial theorem for interpolation Macdonald polynomials [Ok97]. We remark that in [Ok97] the interpolation Macdonald polynomials are defined slightly differently, and are denoted by the P * λ , but one can show that (30) P * λ (x 1 , . . . , x n ; q, t) = R λ (x 1 , x 2 t −1 , x n t −n+1 ; q, t).
Proof. The proof is a straightforward but somewhat tedious calculation based on a general combinatorial formula in [Sah11, Thm 0.8] for the (q, t)-binomial coefficients. We give a brief outline of this calculation. In the notation of [Sah11], the value of (31) can be expressed as a sum of the form T wt(T ), where T is a standard tableau of shape λ\µ. For λ := (1 n ) and µ := (1 r ), there is only one such tableau. By direct calculation one obtains From these, the assertion of the lemma follows immediately.
To complete the proof, we use Lemma 5.2.1.
5.3. The explicit formulas. In this subsection we prove the following statement.
Let us first prove Proposition 5.3.1 in the special case a = b = 1. For 0 ≤ r ≤ m set (33) D r := D n,r = D ′ m,r .
Proposition 5.3.2. Let x a and y b be defined as in (27). Then φ U (x 1 ) = φ U (y 1 ) = m r=0 (q 2 −1) r D r , where D r is defined as in (33).
Proof. We prove that x 1 , y 1 , and m r=0 (q 2 − 1) r D r act by the same scalar on each irreducible U LRsubmodule V * λ ⊗ V * λ of P associated to an integer partition λ satisfying ℓ(λ) ≤ min{m, n}. By Remark 3.4.4, both x 1 and y 1 act on P (d) by the scalar q 2d . Thus by Proposition 3.4.3 it suffices to verify that for every partition λ satisfying ℓ(λ) ≤ min{m, n} and |λ| = d, the restriction of m r=0 (q 2 − 1) r D r to V * λ ⊗ V * λ is multiplication by the scalar q 2d .
Step 2. From now on we assume m < n. From the relations (R3)-(R6) of PD it follows that there exists a unique embedding of algebras η : PD → PD n×n satisfying η(t i,j ) = t i+n−m,j and η(∂ i,j ) = ∂ i+n−m,j . DefineD r ∈ PD n×n to beD r := i,j M i j M i j , where the summation is over all r-tuples i, j ∈ E n 1 (r), so thatD r is the operator D n,r in PD n×n . By Lemma 5.1.1 and Lemma 5.1.2, M i j · (η(P)) = 0 unless i ∈ E n n−m+1 (r). It follows that for every f ∈ P we havẽ D r · η(f ) = η(D r ) · η(f ) when 0 ≤ r ≤ m, andD r · η(f ) = 0 when m < r ≤ n.
Step 3. Recall that I denotes the left ideal of PD that is generated by D (1) . Let I ′ denote the left ideal of PD n×n that is generated by D But also η(D)·η(f )−η(D)η(f ) ∈ I ′ . From the last two relations we obtain η( Step 4. Let f ∈ P (d) . From Step 3 and Step 2 it follows that From Step 1 it follows that n r=0 (q 2 − 1) rD r · η(f ) = q 2d η(f ). Since η is an injection, we obtain m r=0 (q 2 − 1) r D r · f = q 2r f . We are now ready to prove Proposition 5.3.1 in the general case. We give the proof for x a only (the proof for y b is similar). Every element of P is a linear combination of monomials of the form t i 1 ,j 1 · · · t i k ,j k where i 1 ≥ · · · ≥ i k . Choose k ′ ≤ k such that i k ′ ≥ a and i k ′ +1 < a. Then It suffices to prove that Setm := m − a + 1. There exists an embedding of algebras η : PDm ×n → PD that is uniquely defined by η(t i,j ) = t i+a−1,j and η(∂ i,j ) = ∂ i+a−1,j . We defineD r ∈ PDm ×n byD r : where the summation is over i ∈ Em 1 (r) and j ∈ E n 1 (r). Similar to the proof of Proposition 5.3.2 we have η(D r ) = D ′ m−a+1,r and η(D r · f ) = η(D r ) · η(f ) for f ∈ Pm ×n . Thus (35) follows from the assertion of Proposition 5.3.2 for PDm ×n .

Some properties of Γ
The main goal of this section is to relate PD R to the image of Γ. The proofs of Lemma 6.2.1 and Lemma 6.2.2 are similar to some results in [LZZ11].

Invariants and the operators
Lemma 6.1.1. End k (P) L• = End k (P) L and End k (P) R• = End k (P) R .
Proof. We only give the proofs of the two assertions for L . The inclusion End k (P) L• ⊇ End k (P) L is trivial because L • ⊆ L . To prove End k (P) L• ⊆ End k (P) L , choose any T ∈ End k (P) L• . From (32) it follows that T commutes with φ U (K 2εa ⊗ 1) = φ U x −1 a x a+1 for 1 ≤ a ≤ m (we assume x m+1 := 1). From Proposition 3.4.3 it follows that φ U (K ε i ⊗ 1) is a diagonalizable operator whose eigenvalues are powers of q. In particular, the eigenspaces of φ U (K 2ε i ⊗ 1) and φ U (K ε i ⊗ 1) are the same. Thus T also commutes with φ U (K ε i ⊗ 1). Finally, Proposition 4.2.2 implies that T ∈ End k (P) L .
Lemma 6.1.3. L i,j ∈ PD R for 1 ≤ i, j ≤ m and R i,j ∈ PD L for 1 ≤ i, j ≤ n.
Proof. This follows from Lemma 6.1.2 and a straightforward calculation of the action of the standard generators of U L and U R .
Lemma 6.1.4. The U L -submodule of PD that is generated by L m,m contains L i,j for 1 ≤ i, j ≤ m. Similarly, the U R -submodule of PD that is generated by R n,n contains R i,j for 1 ≤ i, j ≤ n.
Proof. We only give the proof for the U L -submodule. Denote this submodule by M . We compute the actions of the standard generators of U L using Remark 3.4.4. By a straightforward computation Using these relations it follows that L i,j ∈ M for j > i. By switching the roles of the E i and the F i we obtain that L i,j ∈ M for i > j. Finally, Since L m,m ∈ M , by an inductive argument it follows that L i,i ∈ M .
Corollary 6.1.5. L i,j ∈ L • for all 1 ≤ i, j ≤ m and R i,j ∈ R • for 1 ≤ i, j ≤ n.
Proof. From the proof of Proposition 4.2.1 we have φ U (x m ) = 1 + (q 2 − 1)L m,m , hence L m,m ∈ L • . Lemma 4.1.1 implies that L • is U L -invariant. Thus by Lemma 6.1.4 we have L i,j ∈ L • . The proof of the inclusion R i,j ∈ R • is similar.
6.2. R-invariants and the image of Γ. Recall the decomposition (25) of PD (r,s) . By Lemma 6.1.2, and since dim Hom U R (V λ , V µ ) = δ λ,µ , we have as U L -modules. In the rest of this section Γ : P m×m → PD is the map introduced in Definition 3.6.1 and we denote the counit of D n×n by ǫ D .
Proof. First we prove that Γ(P m×m ) ⊆ PD R . By definition of Γ it suffices to prove the assertion in the special case m = n. In this case, for x ∈ U R and u ∈ P n×n we have Hence Γ(u) ∈ PD (ǫ R ) = PD R by Lemma 6.1.2.
Since Γ maps P (r) m×m into PD R ∩ PD (r,r) , by Lemma 6.2.1 it suffices to prove that these two vector spaces have equal dimensions for all r ≥ 0. From (36) and Proposition 3.4.3 it follows that both spaces have dimension equal to λ∈Λm,r (dim V λ ) 2 . 7. Proof of Theorem A : the special case m = n Our goal in this section is to prove Theorem A in the special case m = n. Throughout this section we assume that m = n, so that U L ∼ = U R ∼ = U q (gl n ). Recall that P n×n and D n×n are subspaces of the finite dual U q (gl n ) • of U q (gl n ). Henceforth we denote the canonical pairing between U q (gl n ) • and U q (gl n ) by ·, · . We use ·, · for the pairing of U q (gl n ) • ⊗ U q (gl n ) • and U q (gl n ) ⊗ U q (gl n ) too. 7.1. A new product on P n×n . Let us write so that for the universal R-matrixR of (20) we have where we use (8). For the next definition, recall that the action of U LR on P is defined by (18) and (19).
, where x → x ♮ is defined in (16) and the summation ranges over the summands on the right hand sides of (37), ∆ P n×n (u) := u 1 ⊗ u 2 , and ∆ P n×n (v) := v 1 ⊗ v 2 .
For the next proposition, recall that Γ n : P n×n → PD n×n is the map defined in (26).
Proposition 7.1.3. For u ∈ P (r) Proof. The first assertion follows from the fact that u ⋆ v ∈ P (r+s) n×n , which is a direct consequence of Definition 7.1.1. The second assertion follows from Proposition 7.1.2 and Remark 3.5.6. 7.2. The map Υ. Let us now consider the map Υ : P n×n ⊗ P n×n → P n×n ⊗ P n×n defined by , so that if m : P n×n ⊗ P n×n → P n×n denotes the product of the algebra P n×n then Proof. From the defining formula of Υ and the fact that the coproduct of P n×n maps P (r) n×n it follows that Υ leaves P (r) n,n ⊗ P (s) n×n invariant. Since the latter space is finite dimensional, it suffices to prove that Υ is an injection. To this end, we define maps Υ 1 , Υ 2 : P n×n ⊗ P n×n → P n×n ⊗ P n×n , given by Υ 1 (u⊗v) := ι(u 2 )⊗ι(v 2 ), (R R ) −1 21 u 1 ⊗v 1 and Υ 2 (u⊗v) := ((1⊗S(r ♮ L ))·u)⊗(S −1 (r ′ L )⊗1)·v).
Recall that (R R ) −1 21 = r R ⊗ r ′ R . By ι(u 1 ), 1 u 2 = L P (1)u = u and From the coproduct of D n×n and the formulas for A and A ′ we obtain Since (R R ) −1 21 is another universal R-matrix of U R (see for example [KS97,Sec. 8 Next let η : U L → U L be the map x → S(x ♮ ). Then η is an algebra automorphism, so that This completes the proof of injectivity of Υ. 7.3. Completing the proof of Theorem A when m = n. We conclude the proofs of parts (i) and (iii) of Theorem A. Parts (ii) and (iv) follow by symmetry. From L ⊆ End k (P n×n ) R , it follows that L • ⊆ PD R• n×n . Furthermore, Lemma 6.1.1 implies that PD R n×n = PD R• n×n . Let B denote the subalgebra of PD n×n that is generated by the L i,j for 1 ≤ i, j ≤ n. Lemma 6.1.3 implies that B ⊆ PD R• n×n . Since the subspaces PD where B r := Span k {L i 1 ,j 1 · · · L is,js : 1 ≤ s ≤ r and 1 ≤ i u , j u ≤ n for 1 ≤ u ≤ s} . We prove (43) by induction on r. For r = 0, it is trivial. For r = 1, Lemma 6.2.2 implies that PD (1,1) n×n R is spanned by the L i,j = Γ n (t i,j ) for 1 ≤ i, j ≤ n. Recall from Corollary 6.1.5 that L i,j ∈ L • . This completes the proof of (43) for r = 1. Finally, assume r > 1. Then by Proposition 7.2.1 the map is a bijection. From (41) it follows that the map Finally, since Γ n maps P (r) n×n into PD (r,r) n×n R for r ≥ 0, from the induction hypothesis it follows that Γ n (u ′ ) ∈ L • ∩ B 1 , Γ n (u ′′ ) ∈ L • ∩ B r−1 , and D ′ ∈ L • ∩ B r−1 . Consequently, Γ n (u ⋆ u ′ ) ∈ L • ∩ B r .

Proof of Theorem A in the general case: parts (i) and (iii)
The goal of this section is to prove parts (i) and (iii) of Theorem A when m < n. The idea is to reduce the assertions to the case of PD n×n , for which the theorem was already proved in Section 7.
Notation. Recall that when we consider PD n×n we have U L ∼ = U q (gl n ). To avoid confusion between U L in this case and the case of PD = PD m×n (where as before U L ∼ = U q (gl m )), in this section (only) we use U where R (n) is the image of U R in PD n×n . From Remark 3.4.4 it also follows that (45) η(PD ) = {D ∈ PD n×n : a · D = D for all a ∈ S L }.
Recall from Section 7 that PD R (n) n×n is generated by the L i,j for 1 ≤ i, j ≤ n. The products of the form L i 1 ,j 1 · · · L ir,jr are joint eigenvectors for elements of S L . Furthermore from (45) it follows that L i 1 ,j 1 · · · L ir,jr ∈ η(PD ) if and only if i u , j u > n − m for all 1 ≤ u ≤ r. It follows from (44) that PD R is spanned by the products of the form L i 1 ,j 1 · · · L ir,jr where 1 ≤ i u , j u ≤ m for all 1 ≤ u ≤ r. Thus Corollary 6.1.5 implies that PD R ⊆ L • . Since L and R commute, we also have L • ⊆ PD R . Consequently, PD R = L • and L • is generated by the L i,j where 1 ≤ i, j ≤ m.

Proof of Theorem A in the general case: parts (ii) and (iv)
In this section we prove parts (ii) and (iv) of Theorem A when m < n. We remark that the argument of Section 8 is not symmetric with respect to m and n, since it uses the embedding PD ֒→ PD n×n . The proof give n in this section is based on a reduction to PD m×m . This technique is also used in [LZZ11]. However, because of the presence of two R-matrices, in our case the argument is more complicated (in particular, we need Proposition 9.2.1).
Notation. Similar to Section 8, we will work with both PD and PD m×m simultaneously. Thus, to avoid confusion we use U to denote the subalgebras of PD m×m defined analogously to U R , L , R, and R • . 9.1. The embedding of PD m×m into PD . Similar to Section 8, there exists an embedding of associative algebras η ′ : PD m×m → PD that is uniquely defined by In what follows we use the actions of various subalgebras of U LR ⊗ U LR on PD (see Remark 3.5.5).
Let us embed U (m) R into U R as a subalgebra via the homomorphism that is uniquely defined by and then use the latter embedding to identify U (m) with a subalgebra of U LR . With respect to the latter embedding, the map η ′ is U (m) LR -equivariant. Since η ′ is also U L -equivariant, by reasoning similar to (44) we obtain an isomorphism Thus by an argument similar to that of (36), we obtain isomorphisms of U (m) where V (m) λ (respectively, V λ ) denotes the U (m) R -module (respectively, U R -module) associated to λ. Lemma 9.1.1. PD L is generated as a U R ⊗ U R -submodule of PD by PD L ∩ PD (S R ) .
Proof. By (47) it suffices to prove that for λ ∈ Λ m,r we have isomorphisms of U (m) To prove (48), it is sufficient to verify that . These assertions are probably well known. They follow easily from Gelfand-Tsetlin theory for U q (gl n )-modules (see for example [KS97, Sec. 7.3.3]). Also, by considering the standard generators of U R corresponding to the positive system {ε n − ε n−1 , . . . , ε 2 − ε 1 } they can be reduced to [LZZ11, Thm 6.4].
9.2. The U R ⊗ U R -action on PD gr . Recall the map P defined in (24). We set For the next proposition, recall that PD gr is a U R ⊗ U R -module.
Proposition 9.2.1. Let a ⊗ b ∈ U R ⊗ U R . Also, fix a non-negative integer r and suppose that 1 ≤ i u , j u ≤ n are given for 1 ≤ u ≤ r. Then (a ⊗ b) · ( gr R i 1 ,j 1 · · · gr R ir,jr ) is a linear combination of products of the form gr R i ′ Proof. We prove the assertion by induction on r. For r = 0 the assertion is trivial, and for r = 1 it follows from the explicit formulas for the action of U R ⊗ U R on the gr R i,j (see Remark 3.4.4). Next assume r > 1. Set α := gr R i 1 ,j 1 and β := gr R i 2 ,j 2 · · · gr R ir,jr . Then we can express α and β as . (see Remark 9.1.2). Then using formula (11) for the product of PD gr we have To complete the proof, we use the induction hypothesis for ( for all possibilities of a 1 , u b 1 ,a 2 , u ′ b 1 ,a 2 , and b 2 . Since f, ∂, f and ∂ ′ belong to the finite dual of U R , only finitely many such possibilites result in nonzero summands. 9.3. Completing the proofs of parts (ii) and (iv). Let B ′ denote the subalgebra of PD that is generated by the R i,j for 1 ≤ i, j ≤ n. By Lemma 6.1.4 we have B ′ ⊆ PD L . Next we prove the reverse inclusion. As in (36) we have PD (r,s) L = 0 for r = s. Thus, it suffices to show that PD (r,r) L ⊆ B ′ r for r ≥ 0, where B ′ r := Span k {R i 1 ,j 1 · · · R is,js : 0 ≤ s ≤ r and 1 ≤ i u , j u ≤ n for 1 ≤ u ≤ s} .
We prove the latter assertion by induction on r. For r = 0 the assertion is trivially true. For r = 1, by (25) we have where V (n) andV (n) are as in Subsection 3.3. The dimension of the right hand side is n 2 , hence PD (1,1) L is spanned by the R i,j for 1 ≤ i, j ≤ n. Next assume that r > 1. From (46) and the special case of Theorem A for P m×m (see Section 7) it follows that PD L ∩ PD (S R ) is generated by the R i,j for n − m + 1 ≤ i, j ≤ n. In particular, PD L ∩ PD (S R ) ⊆ B ′ . Thus by Lemma 9.1.1 and Proposition 9.2.1 it suffices to prove that B ′ r is U R ⊗ U R -invariant for r ≥ 0. Take an element of B ′ of the form R i 1 ,j 1 · · · R is,js where s ≤ r, and an element a ⊗ b ∈ U R ⊗ U R . By Lemma 6.1.3 we have R i 1 ,j 1 · · · R is,js ∈ PD L . Thus by Remark 3.5.6 and U LR ⊗ U LR -equivariance of P (see Remark 3.5.5) we obtain Since the actions of U L and U R ⊗ U R commute, using Lemma 6.1.2 we obtain Also by the induction hypothesis Consequently, from U LR ⊗ U LR -equivariance of P we obtain (50) (a ⊗ b) · R i 1 ,j 1 · · · R is,js − P (a ⊗ b) · ( gr R i 1 ,j 1 · · · gr R is,js ) ∈ B ′ s−1 . Proposition 9.2.1 implies that (a ⊗ b) · ( gr R i 1 ,j 1 · · · gr R is,js ) is a linear combination of products of the form gr R i ′ 1 ,j ′ 1 · · · gr R i ′ s ,j ′ s . From the proof of (49) and the induction hypothesis that PD (u,u) L ⊆ B ′ u for u < r we obtain From (50) and (51) it follows that B ′ r is U R ⊗ U R -invariant. This completes the proof of B ′ = PD L . It only remains to prove that PD L = R • . To this end, note that by Lemma 6.1.3 we have

Proof of Theorem B
In this section we prove Theorem B(i). The proof of Theorem B(ii) is analogous. As a byproduct, in Corollary 10.5.1 we obtain explicit generators for φ −1 U (L h,• ) and φ −1 U (R h,• ). For convenience, in this section we simplify our notation by writing K λ instead of 1 ⊗ K λ ∈ U h,R .
Step 1. Set D := φ U (x). By Lemma 4.1.1 we have φ U (ad y (x)) = y · D for y ∈ U R . Since the U R -action on PD is locally finite, φ U (ad U R (x)) is a finite dimensional subspace of PD . Furthermore for every f ∈ P, if we set ). Note that the latter upper bound is independnt of f .
Step 2. Set α i := ε i − ε i+1 . It suffices to prove that λ, α i ∈ 2Z ≥0 for λ ∈ I and 1 ≤ i ≤ n − 1. For r ≥ 1 we have Now take a nonzero U h,R -weight vector f ∈ P of weight q −γ for γ := n i=1 γ i ε i , where (γ 1 , . . . , γ n ) is an n-tuple of non-negative integers. Such weight vectors are in the span of the monomials of the form t Step 3. For any λ ∈ I, if λ, α i ∈ 2Z ≥0 then r−1 j=0 1 − q λ,α i −2j = 0 for all sufficiently large r. Thus, if we set I ′ := λ ∈ I : λ, α i ∈ 2Z ≥0 then for all sufficiently large r we have Note that the lower bound on r is independent of γ.
Step 4. Assume that I ′ = ∅. After possibly scaling x, we can assume that the c λ are nonzero polynomials in q. Let λ max denote the maximum of I ′ with respect to <. Let r ∈ N be sufficiently large such that (52) holds. Choose γ as in Lemma 10.1.1. For λ ∈ I ′ let q N (λ) be the lowest power of q that occurs after expanding and simplifying c λ q − λ,γ r−1 and for all other λ ∈ I ′ we have By the choice of γ, for λ ∈ I\{λ max } we have λ max , γ ≥ 1+ λ, γ . Thus λ max , kγ ≥ k+ λ, kγ for all k ∈ N. Consequently, after replacing γ by kγ for k sufficiently large we obtain N (λ max ) < N (λ) for all λ ∈ I ′ \{λ max }. Together with Step 3, this proves that for the latter choice of γ we have λ∈I c λ q − λ,γ r−1 j=0 1 − q λ,α i −2j = 0.
Step 5. In Step 4 we can also choose γ such that γ i −γ i+1 ≥ r+1, or equivalently −γ, α i ≤ −r−1. A standard argument based on representation theory of U q (sl 2 ) implies E r i · f = 0. Since the vectors E s i · f for 0 ≤ s ≤ r have distinct U h,R -weights, they are linearly independent. From Step 2 and Step 4 it follows that the vectors ad E s i (x) · f for 0 ≤ s ≤ r are also linearly independent, hence dim W f ≥ r + 1. Since r can be arbitrarily large, this contradicts Step 1.
10.2. Some technical lemmas. Given any two ordered pairs of integers (i, j) and (i ′ , j ′ ), we set (i, j) ⊳ (i ′ , j ′ ) if i ≤ i ′ and j ≤ j ′ and at least one of the latter inequalities is strict. Let I a,b denote the left ideal of PD that is generated by the ∂ i,j where i ≥ a and j ≥ b. For a ∈ Z we set Lemma 10.2.1. Let a ≥ 0 and let 1 ≤ k ≤ n. Then ∂ 1,k t a+1 1,k = c(a)t a 1,k + D where D ∈ I 1,k . Proof. Follows by induction on a. For a = 0 the assertion follows from the relation Suppose that ∂ 1,k t a 1,k = c(a − 1)t a−1 1,k + D 2 with D 2 ∈ I 1,k . Using (54) we obtain ∂ 1,k t a+1 1,k = 1 + q 2 t 1,k ∂ 1,k + D 1 t a 1,k = (1 + q 2 c(a − 1)t a 1,k + q 2 t 1,k D 2 + D 1 t a 1,k . From Lemma 5.1.1 it follows that D 1 t a 1,k ∈ I 1,k . Consequently, c(a) = 1 + q 2 c(a − 1). Lemma 10.2.2. Let a, b ≥ 0 and let 1 ≤ k ≤ n.
Lemma 10.2.3. Let a, b ≥ 0 and let 1 ≤ k ≤ n. Assume that f ∈ P is a product of the t 1,j for j ≤ k − 1. Then the following hold: The assumption on f and Lemma 5.1.1 imply that Df ∈ I 1,k .
Remark 10.2.4. It is easy to verify that c(a, b) = c(a)c(a − 1) · · · c(a − b) for a ≥ b ≥ 0. We extend the domain of c(a, b) to pairs (a, b) satisfying a, b ≥ −1 by setting c(a, b) = 0 for −1 ≤ a < b and c(a, b) = 1 for a ≥ b = −1. Note that c(a, b) is always a polynomial in q 2 with integer coefficients.
For the next lemma, recall that we denote the action of PD on P by D ⊗ f → D · f . Lemma 10.2.6. Let a := (a 1 , . . . , a n ) and b := (b 1 , . . . , b n ) be n-tuples of non-negative integers. Then the following statements hold.
Remark 10.2.7. By an argument similar to Remark 3.5.2, we can show that PD has a basis consisting of monomials of the form (58) t am,n m,n · · · t a m,1 m,1 · · · t a 1,1 1,n · · · t a 1,1 m,1 · · · ∂ bm,n m,n , a i,j , b i,j ∈ Z ≥0 . Here the ∂ i,j (respectively, the t i,j ) are sorted according to the lexicographic order (respectively, the reverse lexicographic order) on indices. Now let D ∈ PD and let a := (a 1 , . . . , a n ) be an n-tuple of non-negative integers. Recall the notation ∂ a := ∂ a 1 1,1 · · · ∂ an 1,n and t a := t an 1,n · · · t a 1 1,1 for an n-tuple of integers (a 1 , . . . , a n ). Assume that D · t a = ct a for some c ∈ k. We can write D as D = D 1 + D 2 + D 3 where (i) D 1 is a linear combination of basis vectors of the form t b ′ ∂ b ′ where b ′ is an n-tuple of non-negative integers, (ii) D 2 is a linear combination of basis vectors of the form t a ′ ∂ b ′ where a ′ and b ′ are n-tuples of non-negative integers and a ′ = b ′ , and (iii) D 3 is a linear combination of the remaining basis vectors in (58).
Using Lemma 5.1.1 and then Lemma 10.2.6 we obtain D · t a = (D 1 + D 2 ) · t a = D 1 · t a .
Proof. By scaling x by an element of k if necessary, we can assume that the c λ are polynomials in q. Set D := φ U (x), so that D ∈ PD . For a weight γ := n i=1 γ i ε i , we set t γ := t γn 1,n · · · t γ 1 1,1 . Then Write D = D 1 + D 2 + D 3 as in Remark 10.2.7 and suppose that D 1 = a∈Zc (a)t a ∂ a where Z is a finite set of n-tuples of non-negative integers and thec(a) ∈ k. Then by Lemma 10.2.6 we obtain D · t γ = D 1 · t γ = ac (a)φ a (q 2 )t γ where the φ a are polynomials with integer coefficients. Note that thec(a) are independent of γ, but the φ a can depend on γ.
Setλ := λ max where λ max is the maximum of I according to the total order introduced in Subsection 10.1. By Lemma 10.1.1 we can choose γ such that we have λ , γ > µ, γ for all µ ∈ I\{λ}. If the assertion of the proposition is not true, thenλ 1 > 0 and thus by choosing γ 1 sufficiently large we can also assume that λ , γ ≥ 1. Thus for a sufficiently large integer k ≥ 1, the lowest power of q that occurs in λ∈I c λ q − λ,kγ is from the summand cλq − λ ,kγ , and is equal to d − k λ , γ , where d is the lowest power of q that occurs in cλ. By comparing with ac (a)φ a (q 2 ) it follows that d − k λ , γ ≥ − a∈Z degc a (q −1 ). The right hand side is independent of k and γ. However, this is a contradiction since k can be arbitrarily large and λ , γ ≥ 1.
Proof. We assume that the assertion is false, and arrive at a contradiction.
Step 2. By scaling x if necessary, we can assume that the c λ are polynomials in q. Set D := φ U (x) so that D ∈ PD . We keep using the notation t γ for γ := n i=1 γ i ε i from the proof of Proposition 10.3.1. Then D · t γ = λ∈I c λ q − λ,γ t γ . Next we express D as D = D 1 + D 2 + D 3 according to Remark 10.2.7. Suppose that D 1 = b∈Z z(b)t b ∂ b , where Z is a finite set of n-tuples of non-negative integers and the z(b) ∈ k. By scaling x again if necessary, we can assume that the z(b) are also polynomials in q. From Lemma 10.2.6(ii) it follows that Step 3. Letλ ∈ I be such that −λ is the maximum of −I := {−λ : λ ∈ I} with respect to the total order < of Subsection 10.1. Using Lemma 10.1.1 for −I, we can choose γ such that − λ , γ > − µ, γ for all µ ∈ I\{λ}. Sinceλ 1 ∈ {−1, −3, −5, . . .}, by choosing the parity of γ 1 suitably we can also assume that − λ , γ is an odd integer. Then for k ∈ N sufficiently large, the highest power of q that occurs in λ∈I c λ q − λ,kγ is from the summand cλq − λ ,kγ , and is equal to d − k λ , γ , where d := deg cλ.
Note that the map b → λ b is an injection. Let b max ∈ Z be such that λ bmax = max{λ b : b ∈ Z}, where the maximum is taken with respect to <. From (59) and Lemma 10.1.1 applied to the set {λ b : b ∈ Z} it follows that we can choose γ and k in Step 3 such that the highest power of q that occurs in b∈Z z(b)φ b (q 2 ) is from the summand z(b max )φ bmax (q 2 ), and is equal to d ′ + 2 deg φ bmax , where d ′ := deg z(b max ). Note that the φ b depend on γ and k, but the z(b) only depend on x and in particular they are independent of the choices of γ and k.
Step 5. Recall that t γ is an eigenvector of D, hence D · t γ = D 1 · t γ = b∈Z z(b)φ b (q 2 ) by Remark 10.2.7. By comparing the highest power of q in the eigenvalue from Step 3 and Step 4 it follows that Since d ′ is independent of γ and k, the parity of the left hand side of (60) does not change by varying k and γ. However, recall that λ , γ is an odd integer and the only constraint on k is that it should be sufficiently large. Thus, we can choose k such that the parities of the two sides of (60) are different. This is a contradiction.
10.5. Completing the proof of Theorem B. Theorem B(i) is an immediate consequence of the following corollary and Proposition 5.3.1.
Corollary 10.5.1. Let I be a finite subset of Zε 1 + · · · + Zε n . Let x := λ∈I c λ K λ ∈ U R,h where c λ ∈ k for λ ∈ I, and assume that x ∈ φ −1 U (PD ). Then x is in the subalgebra of U h,R that is generated by the y b for 1 ≤ b ≤ n.
Note that Corollary 10.5.1 also describes the generators of φ −1 U (R h,• ). An analogous statement holds for U R : the algebra φ −1 U (R h,• ) is generated by the x a for 1 ≤ a ≤ m.
11. Proof of Theorem C 11.1. Proof for A k,l . The case k = l is a direct consequence of Theorem A. From now on we assume that k < l and assume that m = l (the case k > l is similar). Note that unlike the previous sections it is possible that m > n. There exists an embedding κ k,l : A k,l ֒→ PD = PD l×n , that is uniquely defined by the assignments t i,j → t i+l−k,j and ∂ i,j → ∂ i+l−k,j . Recall from Remark 3.5.5 that PD is a U LR ⊗ U LR -module, and set We remark that U L acts trivially on the subalgebra of A k,l that is generated by the t i,j . The K ε i ∈ U L act diagonally on PD, and (61) κ k,l (A k,l ) = {D ∈ PD : K ε i · D = D for 1 ≤ i ≤ l − k}.
Let D ∈ (A k,l ) (ǫ R ) . Since κ k,l is U R -equivariant, κ k,l (D) ∈ (κ k,l (A k,l )) (ǫ R ) . By Theorem A and Lemma 6.1.2 we can express κ k,l (D) as a linear combination of products of the form L i 1 ,j 1 · · · L ir,jr where 1 ≤ i a , j a ≤ l for 1 ≤ a ≤ r. These products are joint eigenvectors of the K ε i ∈ U L . Moreover, from (61) it follows that L i 1 ,j 1 · · · L ir,jr ∈ κ k,l (A k,l ) if and only if i a > l − k for 1 ≤ a ≤ r. Thus, in expressing κ k,l (D) as a linear combinations of the products L i 1 ,j 1 · · · L ir,jr , only those satisfying i a > l − k for all 1 ≤ a ≤ r can occur with nonzero coefficients. Applying κ −1 k,l concludes the proof.
11.2. Proof for gr(A k,l ). The argument is standard, and is based on reduction to the case A k,l . We assume k ≤ l (the argument for the case k > l is similar). The monomials of the form (22), where a i,j = 0 for i > k and b i,j = 0 for i > l, constitute a basis of A k,l . For d ≥ 0 let V d be the span of such monomials of total degree d (that is, i,j (a i,j + b i,j ) = d). The V d form a U R -invariant splitting of the filtration of A k,l and naturally correspond to an isomorphism of U R -modules F : A k,l → gr(A k,l ). Furthermore, with respect to the vector space grading A k,l = d≥0 V d , the products of A k,l and gr(A k,l ) are the same up to terms of lower degree. Let D ∈ gr(A k,l ) (ǫ R ) and write F −1 (D) = d≥0 D d where D d ∈ V d . From (36) it follows that D d = 0 only if d is even. Set d • := max{d ≥ 0 : D 2d = 0}, so that D 2d• = 0 but D d = 0 for d > 2d • . From Subsection 11.2 it follows that F −1 (D) is a linear combination of products of the form L i 1 ,j 1 · · · L ir,jr . From the proof of Theorem A it also follows that in the latter linear combination only products satisfying r ≤ d • occur. Thus, say F −1 (D) = r≤d• i, j c i, j L i 1 ,j 1 · · · L ir,jr for some c i, j ∈ k.
Setting gr L i,j := F(L i,j ), it follows that F −1 D − r≤d• i, j c i, j gr L i 1 ,j 1 · · · gr L ir,jr ∈ d<2d• V d . Theorem C for gr(A k,l ) follows from iterating the above degree reduction process.

Proof of Proposition 3.5.4
In this section we give a purely algebraic proof that PD acts faithfully on P. Recall that we use the dot symbol to denote the action of PD on P. Set P (≤k) := k i=0 P (i) .
Proof. Follows by induction on k and the mixed relations (R3)-(R6) in Subsection 3.5.
Define a total order ≺ on the set of pairs (i, j) with 1 ≤ i ≤ m and 1 ≤ j ≤ n as follows: we set (i, j) ≺ (i ′ , j ′ ) if either i + j < i ′ + j ′ , or i + j = i ′ + j ′ and i < i ′ .
Proof. We use induction on k. From (i 1 , j 1 ) ≺ (i, j) it follows that either i > i 1 or j > j 1 . If i > i 1 then by the mixed relations (R3) or (R5) we have ∂ i,j t i 1 ,j 1 · · · t i k ,j k = c 1 t i 1 ,j 1 ∂ i,j t i 2 ,j 2 · · · t i k ,j k + δ j,j 1 c 2 j ′ >j t i 1 ,j ′ ∂ i,j ′ t i 2 ,j 2 · · · t i k ,j k , for some c 1 , c 2 ∈ k. The claim now follows from the induction hypothesis, because i + j ′ > i + j and therefore (i, j) ≺ (i, j ′ ). When j > j 1 the argument is similar.
Lemma 12.1.3. Assume that (i r , j r ) ≺ (i, j) for 1 ≤ r ≤ k. Let c(a) be as in (53). Then ∂ i,j · (t a i,j t i 1 ,j 1 · · · t i k ,j k ) = c(a − 1)t a−1 i,j t i 1 ,j 1 · · · t i k ,j k for a ≥ 1.
Since min{i ′ + j, i + j ′ , i ′ + j ′ } > i + j, the assertion follows by induction on a and Lemma 12.1.2.
Remark 12.1.4. Using Remark 3.5.2 and by straightforward arguments based on the relations between the t i,j and the ∂ i,j one can prove that PD has a basis consisting of monomials of the form where the ∂ i,j (respectively, the t i,j ) occur in ascending (respectively, descending) order relative to the total order ≺.  (i, j). In other words, we assume that b := (b 1,1 , b 1,2 , b 2,1 , . . . , b m−1,n , b m,n−1 , b m,n ). Let b := (b i,j ) be the minimum of T in the reverse lexicographic order. Thus, we havẽ b m,n = min{b m,n : (b i,j ) ∈ T }, then alsob m−1,n = min{b m−1,n : (b i,j ) ∈ T and b m,n =b m,n }, and so on. From Lemma 12.1.2 and Lemma 12.1.3 it follows that D d · i,j tb i,j = 0.
13. Proof of Lemma 5.1.2 Define a subset S a of {1, . . . , r} as follows: k ∈ S a if and only if a k > a 1 and for all k ′ < k we have a k = a k ′ . Define S b similarly, relative to (b 1 , . . . , b r ). Note that S a and S b only depend on a and b, respectively. Lemma 5.1.1 and the mixed relations (R3)-(R6) of Subsection 3.5 imply that ∂ a i ,b j · t a 1 ,b 1 · · · t ar,br = q a i ,a 1 + b j ,b 1 t a 1 ,b 1 ∂ a i ,b j · t a 2 ,b 2 · · · t ar,br + a i , a 1 q b j ,b 1 (q − q −1 ) u∈Sa t au,b 1 ∂ au,b j · t a 2 ,b 2 · · · t ar,br t a 1 ,bv ∂ a i ,bv · t a 2 ,b 2 · · · t ar,br (63) + a i , a 1 b j , b 1 (q − q −1 ) 2 u∈Sa,v∈S b t au,bv ∂ au,bv · t a 2 ,b 2 · · · t ar,br + a i , a 1 b j , b 1 t a 2 ,b 2 · · · t ar,br .
We now proceed by induction on r to simplify each line on the right hand side. For the first line, Lemma 5.1.1 implies that ∂ a i ,b j · t a 2 ,b 2 · · · t ar,br = 0 unless both of the following conditions hold: (i) Either i ≥ 2, or i = 1 but a 1 = a i ′ for some i ′ ≥ 2. (ii) Either j ≥ 2, or j = 1 but b 1 = b j ′ for some j ′ ≥ 2.