Quantum deformation and tableaux realization of simple dense 𝔤l(n, ℂ)-modules

Volodymyr Mazorchuk

Max-Planck-Institut für Mathematik, Bonn, Germany^{∗}^{†}

Running title: Quantum deformation of simple dense modules

Abstract

We show that all simple dense 𝔤l(n, ℂ)-modules with finite-dimensional weight spaces admit quantum deformation to the corresponding Drinfeld-Jimbo quantum group. We also realize almost all such modules via Gelfand-Zetlin construction, presenting them by a basis and precise formulae for the action of generators of 𝔤l(n, ℂ). This construction has a direct analogue on the quantum level, hence for such modules the main result becomes very easy.

1 Introduction

Let 𝔤 be a simple finite-dimensional Lie algebra and 𝔥 be its Cartan subalgebra. A 𝔤module, M, is called weight if the action of 𝔥 on M is diagonalizable. If M is a weight module and λ ∈ 𝔥^{∗}, then the weight space M_{λ} of M is the collection of all v ∈ M, on which all h ∈ 𝔥 act via λ(h). The set of all λ such that M_{λ} is non-zero is called the support of M and is denoted by supp(M). Let Δ ⊂ 𝔥^{∗} be the root system of 𝔤 with respect to 𝔥 and Q be the abelian subgroup of 𝔥^{∗} generated by Δ. A weight module, M, is called dense provided supp(M) equals a coset from 𝔥^{∗}/Q. According to Fernando-Futorny’s Theorem, [Fe, Fu], any simple weight 𝔤-module is either dense or induced from a dense module over a parabolic subalgebra of 𝔤. This, in particular, reduces the classification of simple weight modules to that of dense modules. Under the assumption that the weight spaces of M are finite dimensional, all simple dense modules were recently classified by Mathieu, [M]. In particular, such modules exist only if 𝔤 = 𝔰l(n, ℂ) or 𝔤 is of type C_{n} (this result itself belongs to Fernando, [Fe], but in [M] the reader can find short and easy proof). As in this paper we will work mostly with weight modules having finite-dimensional weight spaces, in what follows by weight we will mean weight with finite-dimensional weight spaces.

∗on leave from: Chalmers Tekniska Högskola och Göteborgs Universitet, Göteborg, Sweden, e-mail:

volody@math.chalmers.se

†Junior Associate of the ICTP

The aim of this paper is to show that all 𝔰l(n, ℂ)-modules, constructed by Mathieu, admit true quantum deformations to the corresponding Drinfeld-Jimbo quantum group. The way Mathieu constructed his module will allow us to avoid the usual 𝔸-form technique used for such kind of problems, see [L, K, FM], which also usually involves some extra integrality conditions. Moreover, it happens that almost all simple weight modules can be constructed via Gelfand-Zetlin method, see [Ma1] for the overview and references therein for original results, which provides a special basis in the module and gives precise formulae for the action of generators of 𝔤. This is similar to tableaux realization of weight modules in the sense of [Ma2]. This construction has a straightforward quantum analogue by [MT] and thus gives us an easy way to deform corresponding simple weight modules. We also note that this approach was used by Jimbo, [Ji], to deform finite-dimensional 𝔰l(n, ℂ)-module before the general result of Lusztig, [L].

Our realization of simple dense modules via Gelfand-Zetlin construction is very natural from the point of view of parameterization of simple dense modules. The series of such modules (called coherent families in [M]) over 𝔰l(n, ℂ) are parameterized by special central characters, which are realized as n + 1-tuples of complex parameters. Each tuple in an obvious way defines a Gelfand-Zetlin tableau, i.e. an element from the distinguished basis, which extends to a complete basis of the module. However, in each coherent family there are several modules (the corresponding parameters are defined by some non-trivial linear equations), such that functions, appearing in Gelfand-Zetlin formulae, have singularities on certain basis vectors of such modules. This is precisely the case, when we do not know how to realize the corresponding module via Gelfand-Zetlin construction. But, as we have mentioned, the parameters of such modules satisfy some non-trivial linear equations and hence they really form a small set (countable union of hyperplanes in the variety of all parameters), so we will call them “degenerate” to emphasize the fact the Gelfand-Zetlin formulae degenerate on these modules. However, for pointed coherent families, i.e. those with one dimensional weight subspaces, the functions in Gelfand-Zetlin formulae have removable singularities and thus our construction can be continued even on degenerate module. Here we have to remark that for coherent families the notions of pointed, i.e. having a one-dimensional weight space, and completely pointed, i.e. whose all weight spaces are finite-dimensional, coincide. So, in what follows we will use the first one, which is shorter.

In case of “degenerate” simple dense 𝔰l(n, ℂ)-modules (i.e. those modules, which can not be constructed via Gelfand-Zetlin method) we adjust Mathieu’s technique of localization of the universal enveloping algebra for the quantum group case and use it to construct simple dense modules starting from another simple dense module. In this way we do not have a nice realization of obtained simple dense modules, but comparison of parameters allow us to claim that they tend to the necessary classical simple dense modules under the classical limit. It happens that the set of simple dense modules, which can be realized via Gelfand-Zetlin construction, is big enough to obtain all other simple dense modules applying some twists in the localized algebra. This technique automatically guarantees that the dimension of the weight spaces is preserved during the deformation process, and hence the obtained modules are true quantum deformations of the classical modules we started with.

As a byproduct of our Gelfand-Zetlin realization, we construct several new families of simple weight 𝔰l(n, ℂ)-modules with infinite-dimensional weight spaces, in particular, new simple Harish-Chandra module with respect to a well-embedded 𝔰l(k, ℂ)-subalgebra.

Since the Gelfand-Zetlin method is better adjusted to the reductive algebra 𝔤l(n, ℂ) we will work with it. Using the canonical inclusion 𝔰l(n, ℂ) ⊂ 𝔤l(n, ℂ) one easily translates all the results on 𝔰l(n, ℂ)-case.

In the case of Lie algebras of type C_{n} the situation with Gelfand-Zetlin approach is more complicated. Although the case of finite-dimensional modules has been recently handled by Molev, [Mo], it is not clear so far how to extend his method for infinite-dimensional modules. Although most of the localization technique can be easily translated on this case, it is not straightforward to construct quantum deformation for a big enough class of modules on the first step, which can be done via Gelfand-Zetlin method for A_{n}. This is why this case will be treated in a separate paper.

Let us briefly describe the structure of the paper. In the next section we collect necessary preliminaries about U_{q}(𝔤l(n, ℂ)), Gelfand-Zetlin construction and Mathieu’s classification. In Section 3 we construct tableaux realization for simple dense modules over 𝔤l(n, ℂ). This works for almost all modules, but, in the case of pointed modules, a variation of our construction works always. This is the content of Section 4. Section 5 is a quantum version of two preceding sections. In Section 6 we show how one can use Gelfand-Zetlin tableaux to construct new simple modules over 𝔤l(n, ℂ) and U_{q}(𝔤l(n, ℂ)). Finally, in Section 7 we complete the picture with constructing quantum deformations for all simple dense 𝔤l(n, ℂ)-modules.

2 Preliminaries

2.1 Quantum algebra U_{q}(𝔤l(n, ℂ))

Here we give the basic definitions for quantum groups and refer the reader to [Ja, Jo, KS] for details.

Let 𝔤 = 𝔤l(n, ℂ), 𝔥 be the standard Cartan subalgebra of 𝔤l(n, ℂ), consisting of all diagonal matrices, and Δ be the corresponding root system of 𝔤. We fix the standard basis π of Δ and denote by α_{i} the root corresponding to the matrix unit e_{i,i+1}, i = 1, …, n − 1. Then W is the Weyl group and (·, ·) is the standard W -invariant form on 𝔥^{∗}. Choose q ∈ ℂ such that q ≠ 0 and q^{l} ≠ 1, l ∈ ℕ. Fix some h ∈ ℂ such that q = exp(h) and for x ∈ ℂ set q^{x} = exp(hx). Denote by 1(q) the set of all x ∈ ℂ such that q^{x} = 1. For x ∈ ℂ we put

[x]_{q} = \frac{q^{x} − q^{x−1}}{q − q^{−1}} . Clearly, [x]_{q} = 0 if and only if 2x ∈ 1(q).

The quantized universal enveloping algebra U_{q} = U_{q}(𝔤l(n, ℂ)) is defined as the unital associative ℂ-algebra with generators E_{i}, F_{i}, i = 1, …, n−1; K_{i}, K_{i}^{−1}, i = 1, …, n, subject to the following relations (here indices of K run through {1, …, n} and indices of E, F run through {1, …, n − 1}):

K_{i}K_{i}^{−1} = K_{i}^{−1}K_{i} = 1, K_{i}^{±1}K_{j}^{±1} = K_{j}^{±1}K_{i}^{±1},
K_{i}E_{j} = q^{δ_{i,j}−δ_{i,j+1}}E_{j}K_{i}, K_{i}F_{j} = q^{−δ_{i,j}+δ_{i,j+1}}F_{j}K_{i},

E_{i}F_{j} − F_{j}E_{i} = δ_{i,j} \frac{K_{i}K_{i+1}^{−1} − K_{i}^{−1}K_{i+1}}{q − q^{−1}} , [E_{i}, E_{j}] = [F_{i}, F_{j}] = 0, |i − j| ≥ 2,

E_{i}^{2}E_{i±1} − (q + q^{−1})E_{i}E_{i±1}E_{i} + E_{i±1}E_{i}^{2} = 0, F_{i}^{2}F_{i±1} − (q + q^{−1})F_{i}F_{i±1}F_{i} + F_{i±1}F_{i}^{2} = 0.

A weight U_{q}-module is a module, on which the action of the subalgebra K, generated by all K_{i}, is diagonalizable. Let V be a weight U(𝔤)-module. There is a natural correspondence between weight spaces with respect to 𝔥 and weight spaces with respect to K given by 𝔥^{∗} ∋ λ ↦ q^{(λ,·)}, the last being evaluated on K_{α}^{±1} as q^{±(λ,α)}. However, the inequality λ ≠ μ ∈ 𝔥^{∗} does not guarantee q^{±(λ,α)} ≠ q^{±(μ,α)} for some α ∈ π. At the same time, as q is a non-zero non-root of unity, for any non-zero μ ∈ Q there always exists α ∈ π such that q^{±(λ,α)} ≠ q^{±(λ+μ,α)}. Then a true quantum deformation of a weight 𝔤-module, V, is a module, V_{q}, over U_{q}, such that V is obtained from V_{q} by taking the limit q → 1 and the dimension of the corresponding weight spaces in V and V_{q} is the same.

2.2 Tableaux and Gelfand-Zetlin construction for 𝔤l(n, ℂ) and U_{q}(𝔤l(n, ℂ))

The Gelfand-Zetlin method of constructing simple finite-dimensional 𝔰l(n, ℂ)-modules is, in fact, a restriction from the 𝔤l(n, ℂ) case. Denote by e_{i,j}, 1 ≤ i, j ≤ n, the matrix units. By a tableau we will mean a doubly-indexed complex vector, [l] = (l_{i,j})_{i=1,...,n}^{j=1,...,i}. Let [δ^{i,j}] denote the Kronecker tableau, i.e. δ_{i,j}^{i,j} = 1 and δ_{k,l}^{i,j} = 0 if k ≠ i or l ≠ j.

Simple finite-dimensional 𝔤l(n, ℂ) modules are parameterized by complex vectors, m = (m_{1}, …, m_{n}), satisfying m_{i} −m_{i+1} ∈ ℤ_{+}. The corresponding module V (m) can be realized as a space with the basis B(m) = {[l]|l_{n,i} = m_{i}; l_{i,j} −l_{i−1,j} ∈ ℤ_{+}; l_{i−1,j} −l_{i,j+1} ∈ ℤ_{+}, 1 ≤ j ≤ i ≤ n} on which the action of generators e_{i,i}, i = 1, … n and e_{i,i+1}, e_{i+1,i}, i = 1, …, n − 1, is given by the following Gelfand-Zetlin formulae:

e_{i,i+1}[l] = − ∑_{j=1}^{i} \frac{∏_{k=1}^{i+1}(l_{i,j} − l_{i+1,k})}{∏_{k≠j}(l_{i,j} − l_{i,k})}[l + δ^{i,j}], e_{i+1,i}[l] = ∑_{j=1}^{i} \frac{∏_{k=1}^{i−1}(l_{i,j} − l_{i−1,k})}{∏_{k≠j}(l_{i,j} − l_{i,k})}[l − δ^{i,j}],

e_{i,i}[l] = (∑_{j=1}^{i} l_{i,j} − ∑_{j=1}^{i−1} l_{i−1,j}) [l].

Restricting V (m) to the canonical copy of 𝔰l(n, ℂ) inside 𝔤l(n, ℂ) we get a simple 𝔰l(n, ℂ)- module, which we will denote also by V (m). Moreover, all simple finite-dimensional 𝔰l(n, ℂ)-modules are obtained in this way. Two modules V (m^{(1)}) and V (m^{(2)}) are isomorphic as 𝔤l(n, ℂ)-modules if and only if m^{(1)} = m^{(2)}, and are isomorphic as 𝔰l(n, ℂ)-modules if and only if there exists c ∈ ℂ such that m_{i}^{(1)} = m_{i}^{(2)} + c for all i. In particular, for finitedimensional modules over 𝔰l(n, ℂ) it is always sufficient to assume that all entries of m are integers.

In [Ji] it is noticed that this construction has a straightforward analogue for U_{q}(𝔰l(n, ℂ)). Indeed, for 𝔰l(n, ℂ) we can assume that all entries of m are integers. Then the quantum deformation V_{q}(m) of V (m) has the same basis B(m) and the action of generators of U_{q}(𝔤l(n, ℂ)) is given by the following quantum Gelfand-Zetlin formulae.

E_{i}[l] = − ∑_{j=1}^{i} \frac{∏_{k=1}^{i+1}[l_{i,j} − l_{i+1,k}]_{q}}{∏_{k≠j}[l_{i,j} − l_{i,k}]_{q}} [l + δ^{i,j}], F_{i}[l] = ∑_{j=1}^{i} \frac{∏_{k=1}^{i−1}[l_{i,j} − l_{i−1,k}]_{q}}{∏_{k≠j}[l_{i,j} − l_{i,k}]_{q}} [l − δ^{i,j}],

K_{i}[l] = q(∑_{j=1}^{i} l_{i,j} − ∑_{j=1}^{i−1} l_{i−1,j})[l].

It is obvious that quantum Gelfand-Zetlin formulae tend to usual Gelfand-Zetlin formulae if q → 1 and thus the above construction defines a true quantum deformation of simple finite-dimensional 𝔤l(n, ℂ)-modules (and hence 𝔰l(n, ℂ)-modules as well).

The idea how one can use these formulae to construct new (infinite-dimensional) weight 𝔤l(n, ℂ)- (resp. U_{q}(𝔤l(n, ℂ))-) modules goes back to Drozd, Futorny and Ovsienko, see [DFO] and references therein (resp. [MT]). For this we start with a tableau, [l], satisfying l_{i,j} −l_{i,k} ∉ ℤ for all i and all j ≠ k. Set B([l]) = {[t]|l_{i,j} −t_{i,j} ∈ ℤ, l_{n,i} = t_{n,i}, 1 ≤ j ≤ i ≤ n}. Then (quantum) Gelfand-Zetlin formulae define on the linear span V ([l]) of B([l]) the structure of an 𝔤l(n, ℂ)- (resp. U_{q}(𝔤l(n, ℂ))-) module of finite length.

Sometimes it is also useful to know that there is a special subalgebra, Γ, of U(𝔤) which separates the elements of the Gelfand-Zetlin basis B(m), resp. B([l]). This one is constructed as follows (see [DFO]): we fix the standard inclusions of 𝔤l(i, ℂ) to 𝔤l(i + 1, ℂ) with respect to the upper left corner. Then the commutative subalgebra Γ of U(𝔤l(n, ℂ)), generated by the centers of all U(𝔤l(i, ℂ)), i = 1, …, n, is called the Gelfand-Zetlin subalgebra of U(𝔤l(n, ℂ)). All elements from B(m) and B([l]) above are eigenvectors with respect to Γ, moreover, for 1 ≤ i ≤ n there is a basis in the center of U(𝔤l(i, ℂ)), such that the eigenvalues of the action of these basis elements on [t] are precisely elementary symmetric functions in t_{i,j}, j = 1, …, i. In particular, if [t] and [s] are tableaux and there exists i such that (t_{i,1}, …, t_{i,i}) can not be obtained from (s_{i,1}, …, s_{i,i}) by permutation of components, then the action of Γ separates [t] and [s].

2.3 Mathieu’s classification of simple dense modules

The Mathieu’s approach to the classification of simple dense modules is based on the notion of coherent family. Denote by U(𝔤)_{0} the centralizer of 𝔥 in U(𝔤). Then a coherent family is a weight 𝔤-module, M, satisfying the following two conditions:

1. dim(M_{λ}) = dim(M_{μ}) for all λ, μ ∈ 𝔥^{∗};

2. the function λ ↦ T r(u)|_{λ} is polynomial in λ for all u ∈ U(𝔤)_{0}.

One of the main steps in the Mathieu’s classification is that each simple dense module is a direct summand of some coherent family. The next step of reduction is that each coherent family contains an admissible simple highest weight module as a subquotient. Here by admissible we mean a module, whose dimensions of weight spaces are uniformly bounded. So the irreducible coherent families can be classified by admissible simple highest weight module they contain, which is a relatively easy technical problem. It turns out that infinitedimensional admissible simple highest weight modules, and hence coherent families, exist only if 𝔤 is of type A_{n} or C_{n}. Moreover, if 𝔤 = 𝔰l(n, ℂ), then irreducible coherent families are in bijection with tuples (m_{1}, …, m_{n}) satisfying ∑_{i=1}^{n} m_{i} = 0, and m_{i} − m_{i+1} ∈ ℕ, i = 2, …, n − 1. A certain permutation of entries in (m_{1}, …, m_{n}) defines the (shifted) highest weight of a simple admissible highest weight module, occurring in this coherent family. However, the central character of this module is computed by certain invariant function in entries of m and hence does not change under permutations.

We will need (a restricted version of) one more technical tool used in [M], which we will call Mathieu’s localization of U(𝔤). Fix any Weyl-Chevalley basis, X_{α}, α ∈ Δ, and H_{α}, α ∈ π, in 𝔤. Let α ∈ Δ. As X_{−α} is locally ad-nilpotent on U(𝔤), the multiplicative set {X_{−α}^{i}|i ∈ ℕ} satisfies Ore’s condition for localizability in U(𝔤) and we can form the corresponding localized algebra U_{α}. On U_{α} there exists a unique 1-parameter family of automorphisms θ_{x} : U_{α} → U_{α} satisfying

1. θ_{x}(u) = X_{−α}^{x}uX_{−α}^{−x}, x ∈ ℤ;

2. the map x ↦ θ_{a}(u) is polynomial in x for all u ∈ U(𝔤).

We refer the reader to [M] for details.

3 Tableaux realization of dense 𝔤l(n, ℂ)-modules

In this section we realize almost all simple dense 𝔤 = 𝔤l(n, ℂ)-modules via Gelfand-Zetlin construction and then transfer this to the quantum case. As quantum Gelfand-Zetlin formulae tend to classical Gelfand-Zetlin formulae in the classical limit, we automatically get that our construction gives us a true quantum deformation of simple dense modules we consider (see Section 5).

Let m = (m_{1}, …, m_{n}) ∈ ℂ^{n} be a tuple satisfying the following condition: m_{i} − m_{i+1} ∈ ℤ_{+}, i = 2, …, n−1. We note that the parameterization we will work with differs from one, used by Mathieu, on the shift by vector (0, 1, 2, …, n − 1). Choose x = (x_{1}, …, x_{n−1}) ∈ ℂ^{n−1} such that x_{i} − m_{2} ∉ ℤ, i = 1, …, n − 1, and consider the set B(m, x) which consists of all tableaux [l] satisfying the following conditions:

1. l_{n,j} = m_{j}, j = 1, …, n;

2. l_{i,1} − x_{i} ∈ ℤ, i = 1, …, n − 1;

3. l_{i,j} − l_{i−1,j} ∈ ℤ_{+}, i = 3, …, n, j = 2, …, i;

4. l_{i−1,j} − l_{i,j+1} ∈ ℤ_{+}, i = 3, …, n, j = 2, …, i.

Denote by M(m, x) the ℂ-vectorspace, spanned by B(m, x).

Lemma 1. The Gelfand-Zetlin formulae define on M(m, x) the structure of a 𝔤-module.

Proof. Let u = 0 be a relation in U(𝔤). It is sufficient to prove that u[l] = 0 for any [l] ∈ B(m, x). We use Gelfand-Zetlin formulae and write u[l] = ∑_{[t]∈I(u,[l])} f(u, [l], [t])[l+t]. The coefficients f(u, [l], [t]) can be considered as rational functions in entries of [l]. Write u in U(𝔤) as a linear combination of monomials in {e_{i,i+1}, e_{i,i}} and let N be the length of the longest monomial occurring (it may be different for different decomposition of u, but we fix one of them). Define m̂ as follows: m̂_{j} = m_{j}, j > 1, and m̂_{1} − m̂_{2} is a very big positive integer, say is bigger than 3N^{n} + 1, and consider the finite-dimensional simple 𝔤-module V (m̂). Take any tableau, [l′], from the Gelfand-Zetlin basis for V (m̂) satisfying l′_{i,j} = l_{i,j}, i = 2, …, n, j = 2, …, i; l′_{1,1} > l_{n,2} + N and l′_{i,1} > l′_{i−1,1} + N, i = 2, …, n. Then our choice of m̂_{1} guarantees that u[l′] = ∑_{[t]∈I(u,[l′])} f(u, [l′], [t])[l + t] and I(u, [l′]) = I(u, [l]). In particular, we get f(u, [l′], [t]) = 0 as rational functions in l′_{i,j}, evaluated in entries of [l′]. As there exists infinitely many independent ways to choose l′_{i,1}, i = 1, …, n, and f(u, [l′], [t]) are rational functions, we derive that f(u, [l′], [t]) does not depend on l′_{i,1} and hence f(u, [l], [t]) does not depend on l_{i,1}. In particular, f(u, [l], [t]) = f(u, [l′], [t]) = 0 since l′_{i,j} = l_{i,j}, i, j > 1. This implies u[l] = 0 and completes the proof.

Define φ(m) = (m_{2}, …, m_{n}). From our assumptions about m it follows that φ(m) is a parameter of some simple finite-dimensional 𝔤l(n − 1, ℂ)-module.

Lemma 2. M(m, x) is a weight, dense 𝔤l(n, ℂ)-module and all non-trivial weight subspaces of M(m, x) have dimension dim(V (φ(m))) (the last is the dimension of the simple finite-dimensional 𝔤l(n − 1, ℂ)-module V (φ(m))).

Proof. The first assertion (M(m, x) is weight and dense) is obvious by construction of M(m, x). To prove the second we fix [l] ∈ B(m, x). By Gelfand-Zetlin formulae, [l] ∈ M(m, x)_{λ}, for some λ ∈ 𝔥^{∗} (it can be precisely computed, but we do not need this). But from Gelfand-Zetlin formulae it also follows that another element, [t] ∈ M(m, x)_{λ}, belongs to M(m, x)_{λ} if and only if for any i = 1, …, n − 1 there holds ∑_{j=1}^{i} l_{i,j} = ∑_{j=1}^{i} t_{i,j}. Now it is easy to construct a bijection between Gelfand-Zetlin basis of V (φ(m)) and the subset S of B(m, x), which consists of all tableaus, lying in M(m, x)_{λ}. If [s] ∈ B(φ(m)) we define [t^{s}] ∈ S as follows: t_{i,j}^{s} = s_{i−1,j−1}, i = 2, …, n, j = 2, …, i; t_{i,1}^{s} = ∑_{j=1}^{i} l_{i,j} − ∑_{j=2}^{i} t_{i,j}, i = 1, …, n. Hence dim(M(m, x)_{λ}) = dim(V (φ(m))).

Corollary 1. M(m, x) has finite length.

Proof. Follows from Lemma 2 and [Fe, Section 4].

We note that from the definition of B(m, x) it follows that Γ separates the elements in B(m, x).

Lemma 3. M(m, x) is simple if and only if x_{i}−x_{i+1} ∉ ℤ, i = 1, …, n−2, and x_{n−1}−m_{1} ∉ ℤ.

Proof. By [Fe, Section 4], the simplicity of M(m, x) is equivalent to the fact that all e_{i,i+1} and all e_{i+1,i} act injectively (hence bijectively) on M(m, x). Fix i and let v = ∑ a_{[l]}[l] ∈ M(m, x). Then Gelfand-Zetlin formulae and the fact that Γ separates the elements of B(m, x) imply that e_{i,i+1}v = 0 (resp. e_{i+1,i}v = 0) is equivalent to e_{i,i+1}[l] = 0 (resp. e_{i+1,i}[l] = 0) for all [l] such that a_{[l]} ≠ 0. Therefore the injectivity of e_{i,i+1} (resp. e_{i+1,i}) should be checked only on elements from B(m, x). But, by Gelfand-Zetlin formulae, e_{i,i+1}[l] = 0 if and only if l_{i,j} = l_{i+1,j} for all j = 1, …, i and e_{i+1,i}[l] = 0 if and only if l_{i,j} = l_{i−1,j} for all j = 1, …, i − 1. According to the definition of B(m, x), the nonexistence of such [l] is obviously equivalent to the conditions of the lemma.

Our choice of m above uniquely defines a coherent family, M = M(m). All simple dense modules in M have form M(λ) = ⊕_{μ∈λ+Q}M_{μ} for some λ ∈ 𝔥^{∗} (not λ ∈ 𝔥^{∗} define simple modules). So, now we can formulate our main result about the realization of M(λ) via Gelfand-Zetlin construction. Take some μ ∈ ℂ and consider the following system of linear equations with indeterminates y_{i}, i = 1, …, n − 1:

\{ \begin{matrix}y_{1} & = & μ(e_{1,1}) \\ −y_{1} + y_{2} & = & μ(e_{2,2}) − m_{2} \\ … & … & … \\ −y_{i−1} + y_{i} & = & μ(e_{i,i}) − m_{i} \\ … & … & … \\ −y_{n−2} + y_{n−1} & = & μ(e_{n−1,n−1}) − m_{n−1} \\ −y_{n−1} & = & μ(e_{n,n}) − m_{1} − m_{n} \end{matrix} (∗)

Clearly, (∗) has a (unique) solution if and only if M_{μ} ≠ 0. So, we assume that M_{μ} ≠ 0 and denote by y(μ) the solution of (∗). It follows from Gelfand-Zetlin formulae that, taking x = y(μ), the Gelfand-Zetlin tableau [l], defined by l_{i,1} = x_{i}, i = 1, …, n − 1; l_{n,1} = m_{1} and l_{i,j} = m_{j}, i, j > 1, represents a vector of weight μ.

Theorem 1. Let λ ∈ 𝔥^{∗} and x = y(λ) does exist. Then the module M(λ) is simple if and only if M(m, x) is simple. Moreover, if M(λ) is simple then M(λ) ≃ M(m, x).

Proof. After the previous results about M(m, x) this is an easy corollary of Mathieu’s classification. Indeed, M(m, x) is a dense module with finite-dimensional weight spaces and has, by construction, the same central character as M. Moreover, M(m, x)_{λ} ≠ 0 by the choice of λ. As the coherent family, corresponding to m is unique, the one, generated by M(m, x) is precisely M. Therefore the simple subquotients of finite length modules M(m, x) and M_{λ} coincide, in particular these modules are isomorphic as soon as one of them is simple. This completes the proof.

This theorem defines almost all simple M_{λ} via its tableaux realization. Those, which can not be constructed in this way correspond to λ such that there exist i ∈ {1, …, n − 1} having the property that y_{i} − m_{2} ∈ ℤ. So these modules are defined by some linear equations in the variety of all parameters. It the next section we show that in the case dim(V (φ(m))) = 1 even these degenerate cases can be covered.

4 Tableaux realization of pointed 𝔤l(n, ℂ)-modules

In this section we extend the above realization of simple dense 𝔤 = 𝔤l(n, ℂ)-modules to all pointed modules, i.e. ones with one-dimensional weight spaces.

Let m = (m_{1}, …, m_{n}) ∈ ℂ^{n} be a tuple satisfying the following condition: m_{i} = m_{i+1} = c, i = 2, …, n − 1 and x = (x_{1}, …, x_{n−1}) ∈ ℂ^{n−1}. Consider the set B(m, x) which consists of all tableaux [l] satisfying the following conditions:

1. l_{n,1} = m_{1};

2. l_{i,1} − x_{i} ∈ ℤ, i = 1, …, n − 1;

3. l_{i,j} = m_{j} = c, i = 2, …, n, j = 2, …, i.

Denote by M(m, x) the ℂ-vectorspace, spanned by B(m, x). For [l] ∈ B(m, x) define the action of the generators of 𝔤l(n, ℂ) on [l] by the following pointed Gelfand-Zetlin formulae:

e_{i,i+1}[l] = −(l_{i,1} − l_{i+,1})[l + δ^{i,1}], i > 1
e_{i+1,i}[l] = (l_{i,1} − l_{i−1,1})[l − δ^{i,1}], i > 1
e_{1,2}[l] = −(l_{1,1} − l_{2,1})(l_{1,1} − l_{2,1})[l + δ^{1,1}],
e_{2,1}[l] = [l − δ^{1,1}],
e_{i,i}[l] = (l_{i,1} − l_{i−1,1} + c)[l].

Let M(m, x) denote the vectorspace spanned by B(m, x).

Lemma 4. The pointed Gelfand-Zetlin formulae define on M(m, x) the structure of pointed 𝔤l(n, ℂ)-module. Moreover, M(m, x) is simple if and only if x_{1} − c ∉ ℤ, x_{n−1} − m_{1} ∉ ℤ and x_{i} − x_{i+1} ∉ ℤ for all i = 1, …, n − 2.

Proof. We will only prove that M(m, x) is a 𝔤l(n, ℂ)-module. Everything else follows from pointed Gelfand-Zetlin formulae as in Section 3.

Clearly, we have only to check Serre’s relations, applying them to arbitrary [l] ∈ B(m, x). Some of them are obvious, e.g. that e_{i,i+1} and e_{j,j+1} (resp. e_{i+1,i} and e_{j+1,j}) commute for |i − j| > 2 or that e_{i,i+1} commute with e_{j+1,j} for |i − j| > 2. It is also clear that all e_{i,i} commute. So, we start with e_{i,i+1}e_{i+1,i} − e_{i+1,i}e_{i,i+1} = e_{i,i} − e_{i−1,i−1}. If i = 1, pointed version of Gelfand-Zetlin formulae coincides with the classical one and hence this relation is correct. So, we may assume i > 1. Applying the left-hand side to [l] we get the following coefficient by the resulting [l]: −(l_{i,1} − 1 − l_{i+1,1})(l_{i,1} − l_{i−1,1})+ (l_{i,1}+1−l_{i−1,1})(l_{i,1}−l_{i+1,1}) = l_{i,1}−l_{i−1,1}+l_{i,1}−l_{i+1,1} and this coincides with the corresponding coefficient for the right-hand side.

Next we consider

[e_{i,i}, e_{j,j+1}] = \begin{cases}e_{j,j+1}, & j = i \\ −e_{j,j+1}, & j + 1 = i \\ 0, & otherwise \end{cases} , [e_{i,i}, e_{j+1,j}] = \begin{cases}e_{j,j+1}, & j + 1 = i \\ −e_{j,j+1}, & j = i \\ 0, & otherwise \end{cases} .

The last cases are obvious and for all others the calculations are the same, so we consider only [e_{i,i}, e_{i,i+1}] = e_{i,i+1}. For i = 1 this follows from classical Gelfand-Zetlin formulae. For i > 1 we take any [l] and obtain the following coefficient by [l + δ^{i,1}] after applying the left-hand side: −(l_{i,1} + 1 − l_{i−1,1} + c)(l_{i,1} − l_{i+1,1})+ (l_{i,1} − l_{i+1,1})(l_{i,1} + −l_{i−1,1} + c) = −(l_{i,1} − l_{i+1,1}), which again coincides with the right-hand-side.

Next we prove that e_{i,i+1}e_{i+2,i+1} = e_{i+2,i+1}e_{i,i+1}. If i = 1 we get the following coefficients by [l + δ^{1,1} − δ^{2,1}]: −(l_{1,1} − l_{2,1} + 1)(l_{1,1} − l_{2,2})(l_{2,1} − l_{1,1}) for the left-hand side and −(l_{2,1} − l_{1,1}−1)(l_{1,1}−l_{2,1})(l_{1,1}−l_{2,2}) for the right-hand one. They clearly coincide. For i > 1 we have −(l_{i,1}−l_{i+1,1}+1)(l_{i+1,1}−l_{i,1}) for the right-hand side and −(l_{i+1,1}−l_{i,1}−1)(l_{i,1}−l_{i+1,1}) for the left-hand one, which is again the same. Analogously we get e_{i+1,i+2}e_{i,i−1} = e_{i,i−1}e_{i+1,i+2}.

Finally we consider e_{i+1,i+2}^{2}e_{i,i+1}− 2e_{i+1,i+2}e_{i,i+1}e_{i+1,i+2}+ e_{i,i+1}e_{i+1,i+2}^{2} = 0 (all other relations of this kind can be checked analogously). Again we have to distinguish two cases: i = 1 and i > 1. The first case differs from the second one by additional factor (l_{1,1} − l_{2,2}), which occurs in all summands and hence does not effect on the zero result. So, we can assume i > 1 and have (up to a sign) the following coefficient by [l + 2δ^{i+1,1} + δ^{i,1}]: (l_{i+1,1}+1−l_{i+2,1})(l_{i+1,1}−l_{i+2,1})(l_{i,1}−l_{i+1,1})− 2(l_{i+1,1}+1−l_{i+2,1})(l_{i,1}−l_{i+1,1}−1)(l_{i+1,1}−l_{i+2,1})+ (l_{i,1} −l_{i+1,1} −2)(l_{i+1,1} +1−l_{i+2,1})(l_{i+1,1} −l_{i+2,1}). We can erase (l_{i+1,1} −l_{i+2,1}) everywhere and the expression reduces to (l_{i,1}−l_{i+1,1})− 2(l_{i,1}−l_{i+1,1})+ 2(l_{i+1,1}−l_{i+2,1})+2− 2(l_{i+1,1}−l_{i+2,1})+ (l_{i,1} − l_{i+1,1}) − 2 = 0. This completes the proof.

Let M = M(m) be the coherent family defined by our choice of m. Take some μ ∈ 𝔥^{∗} such that M_{μ} ≠ 0 and consider the system (∗) of linear equations with indeterminates y_{i}, i = 1, …, n − 1. It is easy to see that this system has a unique solution, which we will denote by ŷ(μ).

Theorem 2. Let λ ∈ 𝔥^{∗} and x = ŷ(λ). Then the module M(λ) is simple if and only if M(m, x) is simple. Moreover, if M(λ) is simple then M(λ) ≃ M(m, x).

Proof. The same as for Theorem 1.

As now we did not have any restriction on x, Theorem 2 gives a realization for all pointed simple dense 𝔰l(n, ℂ)-modules.

5 Quantum deformation of dense modules with tableaux realization

Theorem 3. Let m, x, B(m, x) and M(m, x) be as in Section 3 and, additionally, 2(x_{i} − m_{2}) ∉ 1(q) + 2ℤ, i = 2, …, n − 1. Then quantum Gelfand-Zetlin formulae define on M(m, x) the structure of an U_{q}(𝔤l(n, ℂ))-module, which we will denote by M_{q}(m, x). Moreover, M_{q}(m, x) is a true quantum deformation of M(m, x).

Proof. The condition 2(x_{i}−m_{2}) ∉ 1(q)+2ℤ, i = 2, …, n−1, guarantees that all coefficients in quantum Gelfand-Zetlin formulae are well-defined. Now the proof for the first part is the same as that of Lemma 1, with substitution of l_{i,j} by q^{l_{i,j}}, see also proof of [MT, Theorem 2].

Comparing the classical and quantum Gelfand-Zetlin formulae one gets that if an element, [l] ∈ B(m, x), has 𝔥-weight λ in M(m, x) then it has K-weight q^{(λ,·)} in M_{q}(m, x). As both M(m, x) and M_{q}(m, x) have B(m, x) as a basis, we get that the dimensions of the corresponding weight spaces are the same. And, finally, the remark that quantum Gelfand-Zetlin formulae tend to classical ones under q → 1 completes the proof.

Now we deform the pointed modules. Let m and x be as in Section 4. For [l] ∈ B(m, x) define the action of the generators of U_{q}(𝔤l(n, ℂ)) on [l] by the following quantum pointed Gelfand-Zetlin formulae:

E_{i}[l] = −[l_{i,1} − l_{i+,1}]_{q}[l + δ^{i,1}], i > 1
F_{i}[l] = [l_{i,1} − l_{i−1,1}]_{q}[l − δ^{i,1}], i > 1
E_{1}[l] = −[l_{1,1} − l_{2,1}]_{q}[l_{1,1} − l_{2,1}]_{q}[l + δ^{1,1}],
F_{1}[l] = [l − δ^{1,1}],
K_{i}[l] = q^{(l_{i,1}−l_{i−1,1}+c)}[l].

Theorem 4. Let m, x, B(m, x) and M(m, x) be as in Section 4. Then quantum pointed Gelfand-Zetlin formulae define on M(m, x) the structure of an U_{q}(𝔤l(n, ℂ))-module, which we will denote by M_{q}(m, x). Moreover, M_{q}(m, x) is a true quantum deformation of M(m, x).

Proof. Analogous to that of Theorem 3.

There is one interesting effect appearing in the above theorems in the case of nonintegral weights. It happens that for a fixed q the module M_{q}(m, x) may be reducible even if the original module M(m, x) was simple. Why this may happen is explained in the following simplicity criterion for M_{q}(m, x).

Lemma 5. 1. Let m, x, and B(m, x) be as in Section 3. Then M_{q}(m, x) is simple if and only if 2(x_{i} − x_{i+1}) ∉ 1(q) + 2ℤ, i = 1, …, n − 1, and 2(x_{n−1} − m_{1}) ∉ 1(q) + 2ℤ.

2. Let m, x, c, and B(m, x) be as in Section 4. Then M_{q}(m, x) is simple if and only if 2(x_{1} − c) ∉ 1(q) + 2ℤ, 2(x_{n−1} − m_{1}) ∉ 1(q) + 2ℤ and 2(x_{i} − x_{i+1}) ∉ 1(q) + 2ℤ for all i = 1, …, n − 2.

Proof. We prove the first statement and the second one can by done by analogous arguments. Using the quantum Gelfand-Zetlin formulae and the quantum analogue of Γ, [MT, Section 3], by arguments, analogous to that of Lemma 3, we get that M_{q}(m, x) is not simple if and only if there exists [l] ∈ B(m, x) annihilated by some E_{α} or F_{α}. The last condition is equivalent to the fact that certain coefficients in quantum Gelfand-Zetlin formulae are zero. In other words that [x_{i} − x_{i+1}]_{q} = 0 or [x_{n−1} − m_{1}]_{q} = 0. But q^{x} − q^{−x} = 0 is equivalent to 2x ∈ 1(q). So the reducibility of M_{q}(m, x) is equivalent to the failure of the conditions of our lemma.

6 New simple U(𝔤l(n, ℂ))- and U_{q}(𝔤l(n, ℂ))-modules

The above technique of constructing simple dense modules suggests the following two general constructions, which give new classes of simple modules over U(𝔤l(n, ℂ)) (U(𝔰l(n, ℂ))) and their quantum analogs. In these cases it is not clear what a true quantum deformation is, because these modules, although being weight, have infinite-dimensional weight spaces.

The first construction is a more or less direct generalization of the one from Section 3. Fix 1 < k < n and a tableau, [t], satisfying the following conditions:

1. t_{n,j} − t_{n,j+1} ∈ ℤ_{+}, k ≤ j < n;

2. t_{i,j} − t_{i−1,j} ∈ ℤ_{+}, k ≤ j ≤ i;

3. t_{i−1,j} − t_{i,j+1} ∈ ℤ_{+}, k ≤ j ≤ i;

4. t_{i,j} − t_{i,s} ∉ ℤ, 2 ≤ i ≤ n − 1, 1 ≤ j < k, 1 ≤ s ≤ n.

Consider the set B([t]), which consists of all tableaux [l], satisfying the following conditions:

1. l_{n,j} = t_{n,j}, 1 ≤ j ≤ n;

2. t_{i,j} − l_{i,j} ∈ ℤ, 1 ≤ j ≤ i ≤ n − 1;

3. l_{i,j} − l_{i−1,j} ∈ ℤ_{+}, k ≤ j ≤ i;

4. l_{i−1,j} − l_{i,j+1} ∈ ℤ_{+}, k ≤ j ≤ i.

Let M([t]) denote the ℂ-vector space with B([t]) as a basis. We note that the case k = 2 coincides with the situation considered in Section 3.

Theorem 5. Gelfand-Zetlin formulae define on M([t]) the structure of an U(𝔤l(n, ℂ))- module, which is weight and has finite length. M([t]) is simple if and only if t_{i,j} −t_{i+1,s} ∉ ℤ for all i = 1, …, n − 1 and j, s < k. If k > 2, all weight spaces of M([t]) are infinitedimensional.

Proof. The U(𝔤l(n, ℂ))-module structure follows by the same arguments as Lemma 1. The simplicity criterion follows by the same arguments as Lemma 3. The infinite-dimensionality of the weight spaces is obtained by the same arguments as Lemma 2. The finite length arguments are the same as in [Ma3, Section 7.4] and can be sketched as follows: The algebra Γ separates the elements of B([t]) by construction of B([t]), so any simple subquotient, N, of M([t]) comes with a subset of B([t]), which is a basis of N. Now we can draw a non-oriented graph G with vertices B([t]) and two vertices, [l] and [l + δ^{i,j}], are connected if and only if [l + δ^{i,j}] (resp. [l]) appears with non-zero coefficient in decomposition of e_{i,i+1}[l] (resp. e_{i+1,i}[l + δ^{i,j}]). It is obvious that simple subquotients of B([t]) correspond to connected components of G and now one easily derives from Gelfand-Zetlin formulae that the number of such components is finite.

As in Section 5 the modules above admit a straightforward quantum deformation, which can be considered as a true quantum deformation in the sense that both U(𝔤) and U_{q}(𝔤) actually act on the same space M([t]) the second action tends to the first one under q → 1.

Theorem 6. Assume that 2(t_{i,j} − t_{i,s}) ∉ 1(q) + 2ℤ, 2 ≤ i ≤ n − 1, 1 ≤ j < k, 1 ≤ s ≤ n. Then quantum Gelfand-Zetlin formulae define on M([t]) the structure of a U_{q}(𝔤l(n, ℂ))- module, which is weight and has finite length. We will denote this module by M_{q}([t]). M_{q}([t]) is simple if and only if 2(t_{i,j} − t_{i+1,s}) ∉ 1(q) + 2ℤ for all i = 1, …, n − 1 and j, s < k. If k > 2, all weight spaces of M([t]) are infinite-dimensional.

Proof. Everything but finite-length is analogous to Theorem 3 and Lemma 5. The finite length arguments are the same as in [MT, Theorem 2].

The second construction, which follows, was inspired by the first one during some discussions with L.Turowska. We fix k < n. Choose a tableaux, [t], satisfying the conditions

1. t_{i,j} − t_{i,s} ∉ ℤ, k < i < n, 1 ≤ j < s ≤ i;

2. t_{i,j} − t_{i−1,j} ∈ ℤ_{+}, 2 ≤ i ≤ k, 1 ≤ i ≤ i − 1;

3. t_{i−1,j} − t_{i,j+1} ∈ ℤ_{+}, 2 ≤ i ≤ k, 1 ≤ i ≤ i − 1.

Denote by C([t]) the set of all tableaux [l] satisfying the following conditions:

1. l_{n,j} = t_{n,j}, 1 ≤ j ≤ n;

2. l_{i,j} − t_{i,j} ∈ ℤ, 1 ≤ j ≤ i ≤ n − 1;

3. l_{i,j} − l_{i−1,j} ∈ ℤ_{+}, 2 ≤ i ≤ k, 1 ≤ i ≤ i − 1;

4. l_{i−1,j} − l_{i,j+1} ∈ ℤ_{+}, 2 ≤ i ≤ k, 1 ≤ i ≤ i − 1.

Let N([t]) be the ℂ-vectorspace with C([t]) as a basis.

Theorem 7. Gelfand-Zetlin formulae define on N([t]) the structure of an U(𝔤l(n, ℂ))- module, which is weight and has finite length. N([t]) is simple if and only if t_{i,j} − t_{i+1,s} ∉ ℤ for all i = k, …, n − 1 and all j, s. With respect to the subalgebra 𝔤l(k, ℂ), embedded with respect to the upper left corner, N([t]) is a direct sum of simple finite-dimensional modules.

Proof. First we show how to adjust arguments from Lemma 1 to this case. We write u[l] = ∑_{[t]∈I(u,[l])} f(u, [l], [t])[l + t] for some [l] ∈ C([t]) and some relation u = 0 in U(𝔤). Now we apply u to all tableaux [s] in all simple finite-dimensional U(𝔤)-modules, which satisfy s_{i,j} = l_{i,j}, 1 ≤ j ≤ i ≤ k. It is easy to see that this set of tableaux is big enough to conclude that all f(u, [l], [t]) = 0 as in Lemma 1. This gives us the U(𝔤l(n, ℂ))-module structure on N([t]) and it is obvious that N([t]) is weight. The finite length arguments are the same as in Theorem 5. Moreover, these arguments also immediately imply the simplicity criterion for N([t]).

Consider the subalgebra 𝔞_{k} = 𝔤l(k, ℂ) of 𝔤l(n, ℂ), embedded with respect to the upper left corner and fix [l] ∈ C([t]). Denote by D([l]) the set of all [s] ∈ C([t]) satisfying s_{i,j} = l_{i,j}, k ≤ i ≤ n, 1 ≤ j ≤ i. By Gelfand-Zetlin formulae and definition of N([t]), the linear span T_{1} of all [s] ∈ D([l]) and the linear span T_{2} of all other tableaux are closed under the 𝔞_{k}-action, i.e. N([t]) = T_{1} ⊕T_{2} as 𝔞_{k}-module. Moreover, by our restrictions on elements in C([t]), the set D([t]) is finite, and hence T_{1} is finite-dimensional. This completes the proof.

The easiest case is k = n − 1, in which it is easy to see that N([t]) is a direct sum of non-isomorphic simple finite-dimensional modules. Again, by out general philosophy of quantum deformation of Gelfand-Zetlin formulae we immediately get the following:

Theorem 8. Assume that 2(t_{i,j} − t_{i,s}) ∉ 1(q) + 2ℤ, k < i < n, 1 ≤ j ≤ i, 1 ≤ s ≤ i + 1. Then quantum Gelfand-Zetlin formulae define on N([t]) the structure of an U_{q}(𝔤l(n, ℂ))- module, which is weight and has finite length. We will denote this module by N_{q}([t]). N_{q}([t]) is simple if and only if 2(t_{i,j} − t_{i+1,s}) ∉ 1(q) + 2ℤ for all i = k, …, n − 1 and all j, s. With respect to the canonical subalgebra U_{q}(𝔤l(k, ℂ)), the module N_{q}([t]) is a direct sum of simple finite-dimensional modules.

Proof. Is the quantum version of that of Theorem 7.

We also remark that in the case k = n − 1 the additional condition of Theorem 8 disappears and hence in this case all N([t]) admit a quantum deformation.

As we have mentioned in the proof of Theorem 5, all simple subquotients of all modules M(m, x), M([t]) and N([t]) considered above correspond to particular subsets in B(m, x), B([t]) and C([t]) respectively. Hence analogous quantum construction for them give us quantum deformation of all these modules, in particular, of all simple 𝔤l(k, ℂ) − 𝔤l(n, ℂ) Harish-Chandra modules, which can be realized as simple subquotients of N([t]) constructed above.

7 Quantum deformation of simple dense modules in the general case

In this section we retain all notation from Sections 1 and 2. Let M be an irreducible coherent family and λ ∈ 𝔥^{∗} be such that M(λ) is simple (hence dense) and has a realization via Gelfand-Zetlin construction. As we have already mentioned in Section 3, most of λ satisfy this condition. We fix one of them. Let μ ∈ 𝔥^{∗} be such that the module M(μ) = ⊕_{ν∈μ+Q}M_{ν} is simple. Then μ − λ = ∑_{i=1}^{n−1} a_{i}α_{i} for some uniquely defined a_{i} ∈ ℂ. For k ∈ {1, …, n − 1} and x ∈ ℂ denote by F_{k,x} the composition of the following functors: U_{α_{k}} ⊗_{U(𝔤)} _{−}, followed by θ_{x}-twist, followed by restriction to U(𝔤). It is a direct corollary from the definitions of θ_{x} and coherent families (see also [M, Proposition 4.8]) that all simple subquotients of F_{k,x}(M) belong to M. The following two lemmas give us a direct way to represent M(μ) in terms of M(λ) and F_{k,x}.

Lemma 6. Let ν ∈ 𝔥^{∗} and x ∈ ℂ. Then the modules M(ν + xα_{k}) and F_{k,x}(M(ν)) have the same composition subquotients with the same multiplicity.

Proof. By polynomiality properties of F_{k,x} and coherent families, F_{k,x} “preserves” M, i.e. sends any module from M to a module, all simple subquotients of which are simple subquotients of M. Hence it is enough to show that the characters of M(ν + xα_{k}) and F_{k,x}(M(ν)) coincide. But the first step in F_{k,x} is the induction to U_{α_{k}}, which means that for any weight τ of M(ν) it is enough to find some N ∈ ℕ such that X_{−α_{k}} acts injectively on ⊕_{i>N}M(ν)_{τ−iα_{k}}. Now we recall that ⊕_{i∈ℤ}M(ν)_{τ−iα_{k}} is an admissible dense 𝔰l(2, ℂ)-module. Hence the kernel of X_{−α_{k}} is finite-dimensional on it. This completes the proof.

Lemma 7. M = F_{n−1,a_{n−1}} ∘ ⋯ ∘ F_{1,a_{1}}(M(λ)) ≃ M(μ).

Proof. If M ≠ 0 then it has finite length by [Fe, Section 4], all its simple subquotients belong to M and supp(M) ⊂ λ + ∑_{i=1}^{n} a_{i}α_{i} + Q = μ + Q. Hence M ≃ M(μ) as M(μ) is the only simple subquotient of M, whose support intersects μ + Q and the dimension of weight subspaces of M is less or equal than that of M(μ). So, it is enough to prove that M ≠ 0. But the last follows directly from Lemma 6.

Consider now U_{q}(𝔤). By [Jo, 4.4.12], the set F = {F_{k}^{l}|l ∈ ℕ} is an Ore set in U_{q}(𝔤) and we denote by U_{q}^{(k)} the localization of U_{q}(𝔤) with respect to F. Now we need certain analogs of θ_{x} for U_{q}^{(k)}.

Lemma 8. Let u ∈ U_{q}^{(k)}. Then the vectorspace T (u) ⊂ U_{q}^{(k)}, spanned by all F_{k}^{l}uF_{k}^{−l}, l ∈ ℤ_{+}, is finite-dimensional.

Proof. Clearly, we can assume that u is a monomial in U_{q}^{(k)}, say u = Z_{1} … Z_{m}, where each Z_{j} equals one of F_{i}, E_{i}, K_{i}^{±1}, K_{i}^{±1}F_{k}^{−1}, F_{k}^{−1}, or F_{k}F_{i}F_{k}^{−1}. We will say that another monomial, u′ = Z′_{1} … Z′_{m}, is obtained from u by admissible substitution if for all j = 1, …, m the pair (Z_{j}, Z′_{j}) belongs to the following list:

Z_{j}: K_{i}^{±1} K_{k}^{±1}K_{k+1}^{∓1}F_{k}^{−1}  E_{k}  E_{i}, i ≠ k  F_{i}  F_{k}F_{i}F_{k}^{−1} F_{k}^{−1}
Z′_{j}: K_{i}^{±1} K_{k}^{±1}K_{k+1}^{∓1}F_{k}^{−1}  E_{k}  E_{i}  F_{i}  F_{i}  F_{k}^{−1}
K_{k}K_{k+1}^{−1}F_{k}^{−1}  F_{k}F_{i}F_{k}^{−1} F_{k}F_{i}F_{k}^{−1}
K_{k}^{−1}K_{k+1}F_{k}^{−1}

It is obvious that, for fixed u, the set S(u) of monomials, obtained from u by admissible substitutions, is finite. Now we claim that if u′ is obtained from u by admissible substitution, then F_{k}u′F_{k}^{−1} is a sum with some complex coefficients of monomials from S(u). We write F_{k}u′F_{k}^{−1} = F_{k}Z′_{1}F_{k}^{−1}F_{k}Z′_{2}F_{k}^{−1} … F_{k}Z′_{m}F_{k}^{−1} and thus reduce the problem to the case m = 1, in which it follows from the defining relations for U_{q}(𝔤), see Subsection 2.1. Indeed, from these relations we have: F_{k}K_{i}F_{k}^{−1} = q^{δ_{i,k}−δ_{i,k+1}}K_{i}; F_{k}K_{i}^{−1}F_{k}^{−1} = q^{−δ_{i,k}+δ_{i,k+1}}K_{i}^{−1}; F_{k}E_{k}F_{k}^{−1} = E_{k}− (q − q^{−1})^{−1}K_{k}K_{k+1}^{−1}F_{k}^{−1}+ (q − q^{−1})^{−1}K_{k}^{−1}K_{k+1}F_{k}^{−1}; F_{k}E_{i}F_{k}^{−1} = E_{i}, i ≠ k; F_{k}F_{k}^{−1}F_{k}^{−1} = F_{k}^{−1}; F_{k}F_{i}F_{k}^{−1} = F_{i}, |i − j| > 1. Finally, from the quantum Serre’s relations we get F_{k}^{2}F_{k±1}F_{k}^{−2} = −F_{k±1}+ (q + q^{−1})F_{k}F_{k±1}F_{k}^{−1}. This completes the proof.

Lemma 9. There exists a unique family, ϑ_{x}^{q}, of automorphisms of U_{q}^{(k)} satisfying the following two conditions:

1. ϑ_{x}^{q}(u) = F_{α}^{x}uF_{α}^{−x}, x ∈ ℤ;

2. the map ℂ ∋ x ↦ ϑ_{x}^{q}(u) is polynomial in q^{±x/2} for any u ∈ U_{q}^{(k)}.

Proof. Let u ∈ U_{q}^{(k)} and u_{i}, i ∈ I, be the list of all monomials in u. Set S(u) = ∪_{i∈I}S(u_{i}). Then S(u) is a finite union of finite sets defined in Lemma 8. So, the vector subspace T ⊂ U_{q}^{(k)}, spanned by S(u) is finite-dimensional and contains all F_{α}^{k}uF_{α}^{−k}, k ∈ ℕ. From defining relations in U_{q}(𝔤) and construction of S(u_{i}) in Lemma 8 it follows immediately that the map u ↦ F_{α}^{k}uF_{α}^{−k}, k ∈ ℕ, is polynomial in q^{±k/2} on T, hence admits a unique extension for all k ∈ ℂ. This completes the proof.

Corollary 2. The automorphisms ϑ_{x}^{q} tend to θ_{x} under q → 1.

Proof. Under q → 1 the relations of U_{q}(𝔤) tend to the relations of U(𝔤), which are polynomial in x. Hence, by construction of ϑ_{x}^{q}, its limit will be a family, t_{x}, of automorphisms of U(𝔤), such that t_{x}(u) = X_{−α}^{x}uX_{−α}^{−x}, x ∈ ℤ, and the map u ↦ t_{x}(u) is polynomial in x. But then t_{x} = θ_{x} as θ_{x} is the unique family with this property by [M, Lemma 4.3].

For k ∈ {1, …, n − 1} and x ∈ ℂ we denote by F_{k,x}^{q} the composition of the following three functors: U_{q}^{(k)} ⊗_{U_{q}(𝔤)} _{−}, followed by ϑ_{x}^{q}-twist, followed by restriction to U_{q}(𝔤). Now we can state our main result.

Theorem 9. 1. Let M_{q} be a true quantum deformation of a weight module, M, k ∈ {1, …, n − 1} and x ∈ ℂ. If the dimensions of the weight spaces of M are uniformly bounded, then F_{k,x}^{q}(M) is a true quantum deformation of F_{k,x}(M).

2. Let λ, μ and {a_{i}} be as in the first paragraph of this section. Denote by M_{q}(λ) the true quantum deformation of M(λ), constructed in Section 5. Then the module M_{q}(μ) = F_{n−1,a_{n−1}}^{q} ∘ ⋯ ∘ F_{1,a_{1}}^{q}(M_{q}(λ)) is a true quantum deformation of M(μ).

Proof. The second statement follows in an obvious way from the first one and Lemma 7. So we will prove only the first statement.

Because of the definitions of F_{k,x}^{q} and F_{k,x}, which are based on a single root, α_{k} ∈ π, the first statement naturally reduces to the case 𝔤 = 𝔰l(2, ℂ). The module M still have uniformly bounded dimensions of the weight spaces and hence is a module from the BGGcategory O, [Jo, 4.1.4]. The list of indecomposable modules in O for 𝔰l(2, ℂ) is very short and contains simple finite-dimensional modules, Verma module, dual Verma modules and the so-called big projective modules, which can be obtained from a Verma module by tensoring with a finite-dimensional modules. The main point is that all Verma modules possess a true quantum deformation in a natural way, [Jo, Chapter 4], [Ja, Chapter 5]. And hence, tensoring with finite-dimensional modules over U_{q}(𝔰l(2, ℂ)) one obtains a deformation of the big projective modules as well. All finite-dimensional modules are, certainly, killed by both F_{k,x}^{q} and F_{k,x}, and the part of a dual Verma module, not killed but these functors, is a Verma module. Hence over 𝔰l(2, ℂ) the question reduces to calculations with Verma modules and big projective modules, which are filtered by Verma modules and this Verma filtration is unique and has length two.

First we note that if M is a Verma modules, the necessary statement follows immediately from Section 5, since all Verma modules over 𝔰l(2, ℂ) can be realized via Gelfand-Zetlin construction. This is enough to prove the second statement for 𝔰l(2, ℂ), but not enough to prove the first one in the general case, as the restriction of M to 𝔰l(2, ℂ) can also have big projective modules as direct summands. So, to complete the proof we have to prove our statement for big projective modules.

Let 0 → M_{1} → N → M_{2} → 0 be an exact sequence, where M_{i}, i = 1, 2, are Verma modules and N is a big projective modules. Then there is a corresponding sequence, 0 → M_{1}^{q} → N^{q} → M_{2}^{q} → 0, for quantum deformations. As X_{−α_{k}} (resp. F_{k}) acts injectively on all modules in the sequence, the definition of F_{k,x} (resp. F_{k,x}^{q}) implies that the sequence 0 → F_{k,x}(M_{1}) → F_{k,x}(N) → F_{k,x}(M_{2}) → 0 (resp. 0 → F_{k,x}^{q}(M_{1}^{q}) → F_{k,x}^{q}(N^{q}) → F_{k,x}^{q}(M_{2}^{q}) → 0) is also exact. Hence the corresponding weight spaces of F_{k,x}(N) and F_{k,x}^{q}(N^{q}) have the same dimension (in fact, it is either 2 or 0). Now, as F_{k,x}^{q}(N^{q}) is a weight module with finite-dimensional weight spaces, we can apply Corollary 2 and get that the action of U_{q}^{(k)} on F_{k,x}^{q}(N^{q}) tends to the action of U_{α} on F_{k,x}(N) and hence the action of U_{q}(𝔰l(2, ℂ)) on F_{k,x}^{q}(N^{q}) tends to the action of U(𝔰l(2, ℂ)) on F_{k,x}(N), when q → 1. So, F_{k,x}^{q}(N^{q}) is a true quantum deformation of F_{k,x}(N) and the proof is complete.

We remark that the first statement of Theorem 9 can be generalized to weight modules with finite-dimensional weight spaces with respect to the Gelfand-Zetlin subalgebra.

Acknowledgments

The research was done during the visit of the author to Max-Planck-Institut für Mathematik in Bonn, Germany. The financial support, hospitality, accommodation and perfect working conditions provided by the Institut are gratefully acknowledged. I also thank L.Turowska for helpful discussions.

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Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111, Bonn, Germany, e-mail:

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