HARISH-CHANDRA MODULES FOR YANGIANS

VYACHESLAV FUTORNY, ALEXANDER MOLEV, AND SERGE OVSIENKO

Abstract. We study Harish-Chandra representations of the Yangian Y(𝔤𝔩_{2}) which admit a decomposition with respect to a natural maximal commutative subalgebra Γ and satisfy a polynomial condition. We prove an analogue of Kostant theorem showing that the restricted Yangian Y_{p}(𝔤𝔩_{2}) is a free module over Γ and show that every character of Γ defines a finite number of irreducible Harish-Chandra modules. We study the categories of generic Harish-Chandra modules, describe their simple modules and indecomposable modules in tame blocks.

Mathematics Subject Classification 17B35, 81R10, 17B67

Contents

1. Introduction  1
2. Preliminaries  3
2.1. Harish-Chandra subalgebras  3
2.2. Special PBW algebras  5
3. Freeness of Y_{p}(𝔤𝔩_{2}) over its commutative subalgebra  5
4. Harish-Chandra modules for 𝔤𝔩(2) Yangians  8
4.1. Weight modules  8
5. Properties of Γ as a subalgebra of Y_{p}(𝔤𝔩_{2})  12
6. Category of Harish-Chandra modules over Y_{p}(𝔤𝔩_{2})  15
7. Category of generic Harish-Chandra modules  19
7.1. Category of generic weight modules  20
7.2. Support of irreducible generic weight modules  21
7.3. Indecomposable generic weight modules  22
8. Acknowledgment  25
References  25

1. Introduction

Throughout the paper we fix an algebraically closed field 𝕜 of characteristic 0.

The notion of a Harish-Chandra module with respect to a certain subalgebra is one of the most important in the representation theory of Lie algebras ([Di]). For example, weight modules are Harish-Chandra modules with respect to a Cartan subalgebra. Also the Gelfand-Tsetlin modules ([DFO1]) over the universal enveloping algebra U(𝔤𝔩_{n}) of the general linear Lie algebra 𝔤𝔩_{n} are Harish-Chandra modules with respect to a subalgebra generated by the centers of U(𝔤𝔩_{k}), k = 1, …, n where 𝔤𝔩_{1} ⊂ … ⊂ 𝔤𝔩_{n}. In [DFO2] a general setting has been developed for Harish-Chandra modules over associative algebras. Let U be an associative 𝕜-algebra, U − mod be the category of finitely generated left U-modules and Γ ⊂ U be a subalgebra. Denote by cfs(Γ) a cofinite spectrum of Γ, i.e. the set of maximal two-sided ideals of Γ of finite codimension. A module M ∈ U − mod is called Harish-Chandra module (with respect to Γ) if M = ⊕_{m∈cfs Γ}M(m), where

M(m) = {x ∈ M | there exists k ⩾ 0, such that m^{k}x = 0}.

A key problem in the classification of all irreducible Harish-Chandra modules is to study the liftings from a given m ∈ cfs(Γ) to irreducible Harish-Chandra modules M with M(m) ≠ 0. When such lifting is unique then irreducible Harish-Chandra modules are parametrized by the elements of cfs(Γ). In the case of Gelfand-Tsetlin modules over 𝔤𝔩_{n} it was shown in [Ov] that the number of nonisomorphic irreducible modules defined by a given m ∈ cfs(Γ) is always nonzero and finite.

In this paper we begin a systematic study of Harish-Chandra modules over the Yangians.

The Yangian for 𝔤𝔩_{n} is a unital associative algebra Y(𝔤𝔩_{n}) over 𝕜 with countably many generators t_{ij}^{(1)}, t_{ij}^{(2)}, … where 1 ≤ i, j ≤ n, and the defining relations

(1.1)  (u − v) [t_{ij}(u), t_{kl}(v)] = t_{kj}(u) t_{il}(v) − t_{kj}(v) t_{il}(u),

where

(1.2)  t_{ij}(u) = δ_{ij} + t_{ij}^{(1)}u^{−1} + t_{ij}^{(2)}u^{−2} + ⋯

and u, v are formal variables. This algebra originally appeared in the works on the quantum inverse scattering method; see e.g. Takhtajan–Faddeev [TF], Kulish– Sklyanin [KS]. The term “Yangian” and generalizations of Y(𝔤𝔩_{n}) to an arbitrary simple Lie algebra were introduced by Drinfeld [D1]. He then classified finitedimensional irreducible modules over the Yangians in [D2] using earlier results of Tarasov [T1, T2] for the 𝔰𝔩_{2} case. An explicit construction of all such modules over Y(𝔰𝔩_{2}) is given in those papers by Tarasov and also in the work by Chari and Pressley [CP]. Apart from this case, the structure of a general Yangian representation remains unknown. In the case of Y(𝔤𝔩_{n}) a description of “generic” modules was given in [M1] via Gelfand–Tsetlin bases. A more general class of “tame” representations of Y(𝔤𝔩_{n}) was introduced and explicitly constructed by Nazarov and Tarasov [NT]. An important role in these works is played by the Drinfeld generators [D2]

(1.3)  a_{i}(u), i = 1, …, n,  b_{i}(u), c_{i}(u), i = 1, …, n − 1

of the algebra Y(𝔤𝔩_{n}) which are defined as certain quantum minors of the matrix T (u) = (t_{ij}(u)). The coefficients of the series a_{i}(u), i = 1, …, n form a commutative subalgebra of Y(𝔤𝔩_{n}) which can be regarded as an analogue of a Gelfand-Tsetlin subalgebra of the universal enveloping algebra of 𝔤𝔩_{n} [DFO1] We shall call a representation of Y(𝔤𝔩_{n}) Harish-Chandra if it is a Harish-Chandra module with respect to this subalgebra. In particular, finite-dimensional Harish-Chandra modules are precisely the tame modules of [NT]. Note that Harish-Chandra modules for Y(𝔤𝔩_{n}) are analogs of Gelfand-Tsetlin modules for 𝔤𝔩_{n} [DFO1].

In this paper we are concerned with Harish-Chandra representations of the Yangian Y(𝔤𝔩_{2}). Recall that every irreducible finite-dimensional Y(𝔤𝔩_{2})-module contains a unique vector ξ annihilated by t_{12}(u) and which is an eigenvector for the Drinfeld generators a_{1}(u) and a_{2}(u) defined by

(1.4)  a_{1}(u) = t_{11}(u) t_{22}(u − 1) − t_{21}(u) t_{12}(u − 1),  a_{2}(u) = t_{22}(u);

see [T1, T2] and [CP]. Moreover, there exists an automorphism t_{ij}(u) ↦ c(u) t_{ij}(u) of Y(𝔤𝔩_{2}), where c(u) ∈ 1 + u^{−1} 𝕜[[u^{−1}]], such that the eigenvalues of ξ become polynomials in u^{−1} under the twisted action of the Yangian. This prompts the introduction of the class of Harish-Chandra polynomial modules over Y(𝔤𝔩_{2}), i.e., such Harish-Chandra modules where the operators a_{1}(u) and a_{2}(u) are polynomials. More precisely, by (1.4) it is natural to require that for some positive integer p the polynomials a_{1}(u) and a_{2}(u) have degrees 2p and p, respectively. Note that a_{1}(u) is the quantum determinant of the matrix T (u) [IK], [KS]. Its coefficients are algebraically independent generators of the center of Y(𝔤𝔩_{2}).

We can interpret the definition of Harish-Chandra polynomial modules using the algebra Y_{p}(𝔤𝔩_{2}) called the Yangian of level p; see Cherednik [C1, C2]. It is defined as the quotient of Y(𝔤𝔩_{2}) by the ideal generated by the elements t_{ij}^{(r)} with r ≥ p + 1. A Harish-Chandra polynomial module over Y(𝔤𝔩_{2}) is just a Harish-Chandra module over Y_{p}(𝔤𝔩_{2}) for some positive integer p.

For another interpretation consider the Yangian for 𝔰𝔩_{2} which is the subalgebra Y(𝔰𝔩_{2}) of Y(𝔤𝔩_{2}) generated by the coefficients of the series e(u), f(u) and h(u) [D2] defined by

(1.5)  \begin{aligned}e(u) & = t_{22}(u)^{−1}t_{12}(u), \\ f(u) & = t_{21}(u) t_{22}(u)^{−1}, \\ h(u) & = t_{11}(u) t_{22}(u)^{−1} − t_{21}(u) t_{22}(u)^{−1}t_{12}(u) t_{22}(u)^{−1}. \end{aligned}

Note that the series h(u) can also be given by

(1.6)  h(u) = a_{1}(u) a_{2}(u)^{−1}a_{2}(u − 1)^{−1}

so that the coefficients of h(u) form a commutative subalgebra of Y(𝔰𝔩_{2}). Therefore, the restriction of a Harish-Chandra Y(𝔤𝔩_{2})-module to Y(𝔰𝔩_{2}) admits an eigenbasis for this subalgebra. We also point out that both the above interpretations extend to an arbitrary Yangian Y(𝔤𝔩_{n}).

In this paper we study Harish-Chandra polynomial modules over Y(𝔤𝔩_{2}). We consider the class of modules admitting a central character so that the coefficients of a_{1}(u) act as scalars. This class contains all irreducible Harish-Chandra polynomial modules. We study the properties of the subalgebra Γ of Y(𝔤𝔩_{2}) generated by the coefficients of a_{1}(u) and a_{2}(u). In particular we show that Y(𝔤𝔩_{2}) is free as a left and as a right Γ-module (Theorem 1) which is an analogue of Kostant theorem [K]. Moreover, we show that Γ is a Harish-Chandra subalgebra (Theorem 3) in the sense of [DFO2] and that each character of Γ extends to a finitely many non-isomorphic irreducible Y(𝔤𝔩_{2})-modules (Theorem 4). This gives an equivalence between the category ℍ(Y(𝔤𝔩_{2}), Γ) of Harish-Chandra polynomial modules and the category of finitely generated modules over a certain category 𝒜 whose objects are the maximal ideals of Γ. A full subcategory ℍW (Y(𝔤𝔩_{2}), Γ) consisting of weight polynomial Harish-Chandra modules, when the action of a_{2}(u) is diagonalizable, is equivalent to the category of finitely generated modules over a certain quotient category of 𝒜 (see Section 2.1 for details). An important role in our study is played by certain universal weight polynomial Harish-Chandra modules (Section 3, Theorem 2) such that every irreducible module in ℍW (Y(𝔤𝔩_{2}), Γ) is a quotient of the corresponding universal module. In section 7 we study a full subcategory in ℍW (Y(𝔤𝔩_{2}), Γ) of generic modules, this imposes a certain integrability condition on the eigenvalues of a_{2}(u) while those of a_{1}(u) are arbitrary. In particular, we give a complete description of irreducible modules (Theorem 5) and indecomposable modules in tame blocks of this category (Theorem 6).

2. Preliminaries

2.1. Harish-Chandra subalgebras. In the setting of [DFO2] the subalgebra Γ need not to be commutative. But in this paper we will only deal with the commutative case, hence cfs(Γ) coincides with the set Specm Γ of all maximal ideals in Γ.

When for all m ∈ Specm Γ and all x ∈ M(m) holds mx = 0 such Harish-Chandra module M is called weight (with respect to Γ).

All Harish-Chandra modules (with respect to Γ) form a full abelian subcategory in the category of U − mod which we will denote by ℍ(U, Γ). A full subcategory of ℍ(U, Γ) consisting of weight modules we denote by ℍW (U, Γ). The support of a Harish-Chandra module M is a set Supp M ⊂ Specm Γ consisting of such m that M(m) ≠ 0. For D ⊂ Specm Γ denote by ℍ(U, Γ, D) the full subcategory in ℍ(U, Γ) formed by M such that Supp M ⊂ D. For a given m ∈ Specm Γ let χ_{m} : Γ → Γ/m be a character of Γ. If there exists an irreducible Harish-Chandra module M with M(m) ≠ 0 then we say that χ_{m} extends to M.

The notion of a Harish-Chandra subalgebra ([DFO2]) gives an effective tool for the study of the category ℍ(U, Γ). A commutative subalgebra Γ ⊂ U is called a Harish-Chandra subalgebra in U if for any a ∈ U the Γ−bimodule ΓaΓ is finitely generated as left and as right Γ−module. In this case for a finite-dimensional Γ−module X the module U ⊗_{Γ} X is a Harish-Chandra module.

For a ∈ U let

X_{a} = {(m, n) ∈ Specm Γ × Specm Γ | Γ/n is a subquotient of ΓaΓ/Γam}.

Equivalently, (m, n) ∈ X_{a} if and only if (Γ/n) ⊗_{Γ} ΓaΓ ⊗_{Γ} (Γ/m) ≠ 0. Denote by Δ the minimal equivalence on Specm Γ containing all X_{a}, a ∈ U and by Δ(A, Γ) the set of the Δ−equivalence classes on Specm Γ. Then for any a ∈ U and m ∈ Specm Γ holds

(2.7)  aM(m) ⊂ ∑_{(m,n)∈X_{a}}M(n), ℍ(U, Γ) = ⨁_{D∈Δ(U,Γ)} ℍ(U, Γ, D).

Define a category 𝒜 = 𝒜_{U,Γ} with Ob 𝒜 = Γ and the space of morphisms from m to n being

(2.8) 𝒜(m, n) = lim_{←n,m} U/(n^{n}U + Um^{m}) ( equivalently lim_{←n,m} Γ/n^{n} ⊗_{Γ} U ⊗_{Γ} Γ/m^{m}).

Then we have 𝒜 = ⨁_{D∈Δ(U,Γ)} 𝒜(D), where 𝒜(D) is the restriction of 𝒜 on D.

The category 𝒜 is endowed with the topology of the inverse limit and the category of 𝕜-vector spaces (𝕜 − mod) with the discrete topology. Consider the category 𝒜 − mod_{d} of continuous functors M : 𝒜⟶𝕜 − mod ( discrete modules in [DFO2], 1.5). For any discrete 𝒜−module N define a Harish-Chandra U−module 𝔽(N) = ⊕_{m∈Specm Γ}N(m) and for x ∈ N(m) and a ∈ U define ax = ∑_{n∈Specm Γ} a_{n}x where a_{n} is the image of a in 𝒜(m, n). If f : M⟶N is a morphism in 𝒜 − mod_{d} then define 𝔽(f) = ⊕_{m∈Specm Γ}f(m). Hence we have a functor 𝔽 : 𝒜−mod_{d} ⟶ ℍ(U, Γ).

Proposition 2.1. ([DFO2], Theorem 17) The functor 𝔽 is an equivalence.

We will identify a discrete 𝒜-module N with the corresponding Harish-Chandra module 𝔽(N). Let Γ_{m} = lim_{←m} Γ/m^{m} be the completion of Γ by m ∈ Specm Γ. Then the space 𝒜(m, n) has a structure of Γ_{n} − Γ_{m}-bimodule.

For m ∈ Specm Γ denote by m̂ a completion of m. Consider a two-sided ideal I ⊂ 𝒜 generated by m̂ for all m ∈ Specm Γ and set 𝒜_{W} = 𝒜/I. Then Proposition 2.1 implies the following statement.

Corollary 1. The categories ℍW (U, Γ) and 𝒜_{W} − mod are equivalent.

The subalgebra Γ is called big in m ∈ Specm Γ if 𝒜(m, m) is finitely generated as Γ_{m}−module.

Lemma 2.1. ([DFO2], Corollary 19) If Γ is big in m ∈ Specm Γ then there exist finitely many non-isomorphic irreducible Harish-Chandra U−modules M such that M(m) ≠ 0. For any such module dim M(m) < ∞.

2.2. Special PBW algebras. Let U be an associative algebra over 𝕜 endowed with an increasing filtration {U_{i}}_{i∈ℤ}, U_{−1} = {0}, U_{0} = 𝕜, U_{i}U_{j} ⊂ U_{i+j}. For u ∈ U_{i} \ U_{i−1} set deg u = i. Let \overline{U} = gr U be the associated graded algebra \overline{U} =

⨁_{i=0}^{∞} U_{i}/U_{i−1}. For u ∈ U denote by \overline{u} its image in \overline{U} and for a subset S ⊂ U set \overline{S} =

\{\overline{s} | s ∈ S\} ⊂ \overline{U}. The algebra U is called a special PBW algebra if any element of U can be written uniquely as a linear combination of ordered monomials in some fixed generators of U and if \overline{U} is a polynomial algebra. Such algebras were introduced in [FO].

Let Λ = 𝕜[X_{1}, …, X_{n}] be a polynomial algebra. For g_{1}, …, g_{t} ∈ Λ denote by V(g_{1}, …, g_{t}) a set of all zeroes of the ideal generated by the elements g_{1}, …, g_{t}. A sequence g_{1}, …, g_{t} ∈ Λ is called regular (in Λ) if the class of g_{i} in Λ/(g_{1}, …, g_{i−1}) is non-invertible and is not a zero divisor for any i = 1, …, t.

Next proposition contains the basic properties of regular sequences which can be easily checked or can be found in [BH].

Proposition 2.2. (1) The sequence X_{1}, …, X_{r}, G_{1}, … G_{t} with G_{1}, …, G_{t} ∈ Λ is regular in Λ if and only if the sequence g_{1}, …, g_{t} is regular in 𝕜[X_{r+1}, …, X_{n}], where g_{i}(X_{r+1}, …, X_{n}) = G_{i}(0, …, 0, X_{r+1}, …, X_{n}).

(2) A sequence g_{1}, … g_{t} is regular in Λ if and only if the variety V (g_{1}, …, g_{t}) is equidimensional of dimension n − t.

(3) A sequence g_{1}g′_{1}, g_{2}, …, g_{t} is regular if and only if the sequences g_{1}, g_{2}, …, g_{t} and g′_{1}, g_{2}, …, g_{t} are regular.

The following analogue of Kostant theorem ([K]) is valid for special PBW algebras.

Proposition 2.3. ([FO]) Let U be a special PBW algebra and let g_{1}, …, g_{t} ∈ U be mutually commuting elements such that \overline{g}_{1}, …, \overline{g}_{t} is a regular sequence in \overline{U}, Γ = 𝕜[g_{1}, …, g_{t}]. Then U is a free left (right) Γ−module. Moreover Γ is a direct summand of U.

3. Freeness of Y_{p}(𝔤𝔩_{2}) over its commutative subalgebra

Let p be a positive integer. The level p Yangian Y_{p}(𝔤𝔩_{2}) for the Lie algebra 𝔤𝔩_{2} [C2] can be defined as the algebra over 𝕜 with generators t_{ij}^{(1)}, …, t_{ij}^{(p)}, i, j = 1, 2, subject to the relations

(3.9)  [T_{ij}(u), T_{kl}(v)] = \frac{1}{u − v} (T_{kj}(u)T_{il}(v) − T_{kj}(v)T_{il}(u)),

where u, v are formal variables and

(3.10)  T_{ij}(u) = δ_{ij} u^{p} + ∑_{k=1}^{p} t_{ij}^{(k)} u^{p−k} ∈ Y_{p}(𝔤𝔩_{2})[u].

Explicitly, (3.9) reads

(3.11)  [t_{ij}^{(r)}, t_{kl}^{(s)}] = ∑_{a=1}^{min(r,s)} (t_{kj}^{(a−1)}t_{il}^{(r+s−a)} − t_{kj}^{(r+s−a)}t_{il}^{(a−1)}),

where t_{ij}^{(0)} = δ_{ij} and t_{ij}^{(r)} = 0 for r ≥ p + 1. Note that the level 1 Yangian Y_{1}(𝔤𝔩_{2}) coincides with the universal enveloping algebra U(𝔤𝔩_{2}). Set deg t_{ij}^{(k)} = k for i, j, k = 1, …, p. This defines a natural filtration on the Yangian Y_{p}(𝔤𝔩_{2}). The corresponding graded algebra will be denoted by \overline{Y}_{p}(𝔤𝔩_{2}). We have the following analog of the Poincaré–Birkhoff–Witt theorem for the algebra Y_{p}(𝔤𝔩_{2}).

Proposition 3.1. ([C2]; see also [M2]) Given an arbitrary linear ordering on the set of the generators t_{ij}^{(k)}, any element of the algebra Y_{p}(𝔤𝔩_{2}) is uniquely written as a linear combination of ordered monomials in these generators. Moreover, the algebra \overline{Y}_{p}(𝔤𝔩_{2}) is a polynomial algebra in generators \overline{t}_{ij}^{(k)}.

Proposition 3.1 implies that Y_{p}(𝔤𝔩_{2}) is a special PBW algebra. Denote by D(u) the quantum determinant

(3.12)  \begin{aligned}D(u) & = T_{11}(u)T_{22}(u − 1) − T_{21}(u)T_{12}(u − 1) \\ & = T_{11}(u − 1)T_{22}(u) − T_{12}(u − 1)T_{21}(u). \end{aligned}

It was shown in [C1, C2] (see also [M2] for a different proof) that the coefficients of the polynomial D(u) are algebraically independent generators of the center of the algebra Y_{p}(𝔤𝔩_{2}).

Denote by Γ the subalgebra of Y_{p}(𝔤𝔩_{2}) generated by the coefficients of D(u) and t_{22}^{(k)}, k = 1, …, p. This algebra is obviously commutative. We will show later (Corollary 3) that Γ is a Harish-Chandra subalgebra in Y_{p}(𝔤𝔩_{2}).

Lemma 3.1. The sequence \overline{t}_{22}^{(1)}, …, \overline{t}_{22}^{(p)}, \overline{d}_{1}, …, \overline{d}_{2p} of the images of the generators of Γ is regular in \overline{Y}_{p}(𝔤𝔩_{2}).

Proof. Denote t_{i} = \overline{t}_{11}^{(i)} + \overline{t}_{22}^{(i)}, i = 1, …, p, Δ_{i,j} = \overline{t}_{11}^{(i)}\overline{t}_{22}^{(j)} − \overline{t}_{21}^{(i)}\overline{t}_{12}^{(j)}, i, j = 1, …, p, i ≠ j. It follows from 3.12 that

\overline{D}(u) = u^{2p} + ∑_{i=1}^{2p} \overline{d}_{i}u^{2p−i},

where \overline{d}_{i} = t_{i} + ∑_{j=1}^{i−1} Δ_{j,i−j} for i = 1, …, p and \overline{d}_{i} = ∑_{j=i−p}^{p} Δ_{i,i−j} for i = p + 1, …, 2p. Hence we need to show that the sequence

\overline{t}_{22}^{(1)}, …, \overline{t}_{22}^{(p)}, t_{1}, t_{2} + Δ_{11}, …, t_{p} + ∑_{i=1}^{p−1} Δ_{i,p−i}, ∑_{i=1}^{p} Δ_{i,p+1−i}, …, Δ_{pp}

is regular. We will denote by ∇_{i} the result of the substitution \overline{t}_{22}^{(1)} = … = \overline{t}_{22}^{(p)} = 0 in \overline{d}_{i}, i = 1, …, 2p. By Proposition 2.2, (1) we only need to show the regularity of the sequence

∇_{1}, …, ∇_{2p}.

Consider the following triangular automorphism ϕ of \overline{Y}_{p}(𝔤𝔩_{2})/I: \overline{t}_{11}^{(i)} ↦ \overline{t}_{11}^{(i)} + ∑_{j=1}^{i−1} Δ_{i,i−j}, \overline{t}_{21}^{(i)} → \overline{t}_{21}^{(i)}, \overline{t}_{12}^{(i)} → \overline{t}_{12}^{(i)}, i = 1, …, p, where I is an ideal generated

by \overline{t}_{22}^{(1)}, …, \overline{t}_{22}^{(p)}. Applying ϕ^{−1} to the sequence ∇_{1}, …, ∇_{2p} we see that it is enough to show the regularity of the sequence

\overline{t}_{11}^{(1)}, …, \overline{t}_{11}^{(p)}, ∇_{p+1}, …, ∇_{2p}.

Again by Proposition 2.2, (1) this is equivalent to the regularity of the sequence ∇_{p+1}, …, ∇_{2p}. For each pair i, j, i, j = 1, …, p, i+j ≥ p+1 consider the following elements of 𝕜[\overline{t}_{12}^{(i)}, \overline{t}_{21}^{(i)}| i, j = p + 1, …, 2p] arranged in the table s_{ij} below

( \overline{t}_{21}^{(i)}\overline{t}_{12}^{(j)} ; \overline{t}_{21}^{(i−1)}\overline{t}_{12}^{(j)} + \overline{t}_{21}^{(i)}\overline{t}_{12}^{(j−1)} ; \overline{t}_{21}^{(i−2)}\overline{t}_{12}^{(j)} + \overline{t}_{21}^{(i−1)}\overline{t}_{12}^{(j−1)} + \overline{t}_{21}^{(i)}\overline{t}_{12}^{(j−2)} ; ⋮ ; \overline{t}_{21}^{(p+1−j)}\overline{t}_{12}^{(j)} + \overline{t}_{21}^{(p−j)}\overline{t}_{12}^{(j+1)} + . . . . . . + \overline{t}_{21}^{(i+1)}\overline{t}_{12}^{(p−i)} + \overline{t}_{21}^{(i)}\overline{t}_{12}^{(p+1−i)} )

Note that when i = j = p the rows of the table are exactly the elements ∇_{i}, i = p + 1, …, 2p. We will show by induction on i + j that the rows of this table form a regular sequence. Let i + j = p + 1. Then s_{ij} consists of the unique element \overline{t}_{21}^{(i)}\overline{t}_{12}^{(j)} and the corresponding variety is obviously equidimensional. Hence the statement follows from Proposition 2.2, (2). Applying Proposition 2.2, (3) to the table above we obtain the following two tables s′_{ij} and s′′_{ij}

( \overline{t}_{21}^{(i)} ; \overline{t}_{21}^{(i−1)}\overline{t}_{12}^{(j)} + \overline{t}_{21}^{(i)}\overline{t}_{12}^{(j−1)} ; ⋮ ; \overline{t}_{21}^{(p+1−j)}\overline{t}_{12}^{(j)} + … + \overline{t}_{21}^{(i)}\overline{t}_{12}^{(p+1−i)} ); ( \overline{t}_{12}^{(j)} ; \overline{t}_{21}^{(i−1)}\overline{t}_{12}^{(j)} + \overline{t}_{21}^{(i)}\overline{t}_{12}^{(j−1)} ; ⋮ ; \overline{t}_{21}^{(p+1−j)}\overline{t}_{12}^{(j)} + … + \overline{t}_{21}^{(i)}\overline{t}_{12}^{(p+1−i)} )

Next we apply Proposition 2.2, (1) substituting \overline{t}_{21}^{(i)} = 0 in s′_{ij} and \overline{t}_{12}^{(i)} = 0 in s′′_{ij}. It is easy to see that after the substitution we obtain the tables s_{i−1j} and s_{ij−1}. Applying the induction to these sequences we conclude their regularity which implies the regularity of the sequence s_{ij} for all i, j = 1, …, p, i + j ≥ p + 1 by Proposition 2.2, (3). In particular, the sequence s_{pp} is regular which completes the proof. □

We immediately obtain the following

Corollary 2. The generators t_{22}^{(1)}, …, t_{22}^{(p)}, d_{1}, …, d_{2p} of Γ are algebraically independent.

We will denote by K(Γ) the field of fractions of Γ.

Combining Lemma 3.1 with Proposition 2.3 we obtain the following

Theorem 1. (1) Y_{p}(𝔤𝔩_{2}) is free as a left (right) module over Γ. Moreover Γ

is a direct summand of Y_{p}(𝔤𝔩_{2}).

(2) For any m ∈ Specm Γ the character χ_{m} extends to an irreducible Y_{p}(𝔤𝔩_{2})- module.

For a subset P ⊂ Y_{p}(𝔤𝔩_{2}) denote by 𝔻(P) the set of all x ∈ Y_{p}(𝔤𝔩_{2}) such that there exists z ∈ Γ, z ≠ 0 for which zx ∈ P.

Corollary 3. Let P ⊂ Y_{p}(𝔤𝔩_{2}) be a finitely generated left Γ−module then 𝔻(P) is a finitely generated left Γ−module.

Proof. Since Γ is a domain then 𝔻(P) is a Γ-submodule in Y_{p}(𝔤𝔩_{2}). Using the fact that Y_{p}(𝔤𝔩_{2}) is a free left Γ−module we conclude that Y_{p}(𝔤𝔩_{2}) ≃ F_{P} ⊕ F where F_{P} and F are free left Γ−modules, F_{P} has a finite rank and P ⊂ F_{P}. Then 𝔻(P) ⊂ F_{P} and hence it is finitely generated as a module over a noetherian ring. □

4. Harish-Chandra modules for 𝔤𝔩(2) Yangians

Let L be a polynomial algebra in variables b_{1}, …, b_{p}, g_{1}, … g_{2p}. Define a 𝕜-mo-nomorphism ı : Γ → L such that ı(t_{22}^{(k)}) = σ_{k,p}(b_{1}, …, b_{p}), ı(d_{i}) = σ_{i,2p}(g_{1}, …, g_{2p}) where σ_{i,j} is the i-th elementary symmetric polynomial in j variables. We will identify the elements of Γ with their images in L and treat them as polynomials in variables b_{1}, …, b_{p}, g_{1}, … g_{2p} invariant under the action of the group S_{p} × S_{2p} . Set L = Specm L. We will identify L with 𝕜^{3p}. If β = (β_{1}, …, β_{p}), γ = (γ_{1}, …, γ_{2p}) and ℓ = (β_{1}, …, β_{p}, γ_{1}, …, γ_{2p}) then we will write ℓ = (β, γ). The map ı induces an epimorphism ı^{∗} : L → Specm Γ. If ℓ ∈ L and m = ı^{∗}(ℓ) then D(ℓ) will denote the equivalence class of m in Δ(Y_{p}(𝔤𝔩_{2}), Γ).

Let L_{0} ⊂ L, L_{0} ≃ ℤ^{p}, be a lattice generated by δ_{i} ∈ 𝕜^{3p}, i = 1, …, p, where δ_{i} = (δ_{i}^{1}, …, δ_{i}^{3p}), δ_{i}^{j} = δ_{ij}, j = 1, …, 3p. Then L_{0} acts on L by shifting δ_{i}(ℓ) := ℓ + δ_{i}. Also the group S_{p} × S_{2p} acts on L by permutations. Thus the semidirect product 𝕎 of the groups S_{p} × S_{2p} and L_{0} acts on L and L. Denote by S a multiplicative set in L generated by the elements b_{i} − b_{j} − m for all i ≠ j and all m ∈ ℤ and by 𝕃 the localization of L by S. Note that S is invariant under the action of 𝕎 and hence 𝕎 acts on 𝕃.

Let L_{1} = Specm 𝕃 ⊂ L, i.e. L_{1} consists of generic 3p-tuples ℓ = (β, γ) such that β_{i} − β_{j} ∉ ℤ for all i ≠ j. If ℓ ∈ L_{1} then the modules from the category ℍ(Y_{p}(𝔤𝔩_{2}), Γ, D(ℓ)) are called generic Harish-Chandra modules.

Fix ℓ = (β, γ) ∈ L. Let I_{ℓ} be the left ideal of Y_{p}(𝔤𝔩_{2}) generated by the coefficients of the polynomials T_{22}(u)−β(u) and D(u)−γ(u). Define the corresponding quotient module over Y_{p}(𝔤𝔩_{2}) by

M(ℓ) = Y_{p}(𝔤𝔩_{2})/I_{ℓ}.

It follows from Theorem 1 that I_{ℓ} is a proper ideal of Y_{p}(𝔤𝔩_{2}) and so M(ℓ) is a non-trivial module. Therefore, the image of 1 in M(ℓ) is nonzero. We shall denote it by ξ. The next proposition shows the universality of the module M(ℓ).

Proposition 4.1. Let ℓ = (β, γ) ∈ L and let V be a weight Y_{p}(𝔤𝔩_{2})-module with a central character γ generated by a nonzero η ∈ V_{β}. Then V is a homomorphic image of M(ℓ).

Proof. Indeed, there is a homomorphism f : M(ℓ) → V which maps ξ to η. Since η generates V the statement follows. □

4.1. Weight modules. For ℓ = (β, γ) ∈ L the category ℍW (Y_{p}(𝔤𝔩_{2}), Γ, D(ℓ)) consists of finitely generated weight modules V with central character γ and with Supp V ⊂ D(ℓ). For simplicity we will denote it by R_{ℓ}. If ℓ ∈ L_{1} then the modules from R_{ℓ} will be called generic weight modules.

Let ℓ = (β, γ) ∈ L, β = (β_{1}, …, β_{p}), γ = (γ_{1}, …, γ_{2p}), β(u) = (u + β_{1}) … (u + β_{p}), γ(u) = (u + γ_{1}) … (u + γ_{2p}).

A Y_{p}(𝔤𝔩_{2})-module V is an object of R_{ℓ} if V is a direct sum of its weight subspaces:

(4.13) V = ⨁_{ℓ∈L} V_{ℓ}, where V_{ℓ} = {η ∈ V | T_{22}(u)η = β(u)η, D(u)η = γ(u)η}.

If V ∈ R_{ℓ} then we shall simply write V_{β} instead of V_{ℓ} and identify Supp V with the set of all β such that the subspace V_{β} is nonzero.

Lemma 4.1. (compare with (2.7)) Let V be a generic weight Y_{p}(𝔤𝔩_{2})-module and let β = (β_{1}, …, β_{p}) ∈ Supp V. Then

(4.14)  T_{21}(u)V_{β} ⊆ ∑_{i=1}^{p} V_{β+δ_{i}}  and  T_{12}(u)V_{β} ⊆ ∑_{i=1}^{p} V_{β−δ_{i}}

where β ± δ_{i} = (β_{1}, …, β_{i} ± 1, …, β_{p}).

Proof. First we show that T_{21}(−β_{i})V_{β} ⊆ V_{β+δ_{i}} for all i = 1, …, p. Since

T_{22}(u − 1)T_{21}(u) = T_{21}(u − 1)T_{22}(u)

we have

T_{22}(−β_{i} − 1)T_{21}(−β_{i}) η = T_{21}(−β_{i} − 1)T_{22}(−β_{i}) η = 0

for all η ∈ V_{β}. Also,

\begin{aligned}T_{22}(−β_{j})T_{21}(−β_{i}) η & = (β_{i} − β_{j})^{−1}(T_{21}(−β_{i})T_{22}(−β_{j}) − T_{21}(−β_{j})T_{22}(−β_{i})) η \\ & + T_{21}(−β_{i})T_{22}(−β_{j}) η = 0 \end{aligned}

since T_{22}(−β_{k}) η = 0 for all k = 1, …, p. Using the fact that β_{i} − β_{j} ∉ ℤ we conclude that T_{21}(−β_{i})V_{β} ⊆ V_{β+δ_{i}} for all i = 1, …, p. Since T_{21}(u) is a polynomial of degree p − 1 in u and β_{i} ≠ β_{j} if i ≠ j, we have that T_{21}(u)V_{β} ⊆ ∑_{i=1}^{p} V_{β+δ_{i}}. The case of T_{12}(u) is treated analogously using the identity T_{22}(u)T_{12}(u − 1) = T_{12}(u)T_{22}(u − 1). □

Corollary 4. If V is indecomposable generic weight module over Y_{p}(𝔤𝔩_{2}) and β ∈ Supp V then Supp V ⊆ β + ℤ^{p}.

Lemma 4.2. If V is a generic weight Y_{p}(𝔤𝔩_{2})-module with central character γ(u) then for any β = (β_{1}, …, β_{p}) ∈ Supp V and any η ∈ V_{β} we have

T_{12}(−β_{r})T_{21}(−β_{s}) η = T_{21}(−β_{s})T_{12}(−β_{r}) η,

if s ≠ r, and

\begin{aligned}T_{12}(−β_{i} − 1)T_{21}(−β_{i}) η & = −γ(−β_{i}) η, \\ T_{21}(−β_{i} + 1)T_{12}(−β_{i}) η & = −γ(−β_{i} + 1) η. \end{aligned}

Proof. The first equality follows from the defining relations (1.1). The others follow from (3.12). □

Corollary 5. Let V be a generic weight Y_{p}(𝔤𝔩_{2})-module with a central character γ and let β = (β_{1}, …, β_{p}) ∈ Supp V.

(i) If γ(−β_{i}) ≠ 0 then Ker T_{21}(−β_{i}) ∩ V_{β} = 0.

(ii) If γ(−β_{i} + 1) ≠ 0 then Ker T_{12}(−β_{i}) ∩ V_{β} = 0.

(iii) If V is indecomposable and γ(−β_{i} + k) ≠ 0 for all k ∈ ℤ then

Ker T_{21}(−ψ_{i}) ∩ V_{ψ} = Ker T_{12}(−ψ_{i}) ∩ V_{ψ} = 0

for all ψ = (ψ_{1}, …, ψ_{p}) ∈ Supp V.

Given (k) = (k_{1}, …, k_{p}) ∈ ℤ^{p} define the corresponding vector of the module M(ℓ) by

\begin{matrix}ξ^{(k)} = & ∏_{i, k_{i}>0} T_{21}(−β_{i} − k_{i} + 1) ⋯ T_{21}(−β_{i} − 1) T_{21}(−β_{i}) \\ × & ∏_{i, k_{i}<0} T_{12}(−β_{i} − k_{i} − 1) ⋯ T_{12}(−β_{i} + 1) T_{12}(−β_{i}) ξ. \end{matrix}

Theorem 2. The vectors ξ^{(k)}, (k) ∈ ℤ^{p} form a basis of M(ℓ). Moreover, we have the formulas

(4.15)  T_{22}(u) ξ^{(k)} = ∏_{i=1}^{p}(u + β_{i} + k_{i}) ξ^{(k)},

(4.16)

\begin{aligned}T_{21}(u) ξ^{(k)} & = ∑_{i=1}^{p} A_{i}(k) \frac{(u + β_{1} + k_{1}) ⋯ ∧_{i} ⋯ (u + β_{p} + k_{p})}{(β_{1} − β_{i} + k_{1} − k_{i}) ⋯ ∧_{i} ⋯ (β_{p} − β_{i} + k_{p} − k_{i})} ξ^{(k+δ_{i})}, \\ T_{12}(u) ξ^{(k)} & = ∑_{i=1}^{p} B_{i}(k) \frac{(u + β_{1} + k_{1}) ⋯ ∧_{i} ⋯ (u + β_{p} + k_{p})}{(β_{1} − β_{i} + k_{1} − k_{i}) ⋯ ∧_{i} ⋯ (β_{p} − β_{i} + k_{p} − k_{i})} ξ^{(k−δ_{i})}, \end{aligned}

where

A_{i}(k) = \begin{cases}1 & if k_{i} ≥ 0 \\ −γ(−β_{i} − k_{i}) & if k_{i} < 0 \end{cases}

and

B_{i}(k) = \begin{cases}−γ(−β_{i} − k_{i} + 1) & if k_{i} > 0 \\ 1 & if k_{i} ≤ 0. \end{cases}

The action of T_{11}(u) is found from the relation

(4.17)  (T_{11}(u)T_{22}(u − 1) − T_{21}(u)T_{12}(u − 1)) ξ^{(k)} = γ(u) ξ^{(k)}.

Proof. We start by proving the formulas for the action of the generators of Y_{p}(𝔤𝔩_{2}). Relation (4.15) follows by induction from the defining relations (1.1). By Lemma 4.2 we have: if k_{i} > 0 then

(4.18)  \begin{aligned}T_{21}(−β_{i} − k_{i}) ξ^{(k)} & = ξ^{(k+δ_{i})}, \\ T_{12}(−β_{i} − k_{i}) ξ^{(k)} & = −γ(−β_{i} − k_{i} + 1) ξ^{(k−δ_{i})}; \end{aligned}

if k_{i} < 0 then

(4.19)  \begin{aligned}T_{12}(−β_{i} − k_{i}) ξ^{(k)} & = ξ^{(k−δ_{i})}, \\ T_{21}(−β_{i} − k_{i}) ξ^{(k)} & = −γ(−β_{i} − k_{i}) ξ^{(k+δ_{i})}; \end{aligned}

and if k_{i} = 0 then

(4.20)  \begin{aligned}T_{12}(−β_{i}) ξ^{(k)} & = ξ^{(k−δ_{i})}, \\ T_{21}(−β_{i}) ξ^{(k)} & = ξ^{(k+δ_{i})}. \end{aligned}

Applying the Lagrange interpolation formula we obtain the remaining formulas.

To show that the vectors ξ^{(k)} form a basis of M(ℓ), denote by T_{β} the subspace of Y_{p}(𝔤𝔩_{2}) spanned by the elements

\begin{matrix}τ^{(k)} = & ∏_{i, k_{i}>0} T_{21}(−β_{i} − k_{i} + 1) ⋯ T_{21}(−β_{i} − 1) T_{21}(−β_{i}) \\ × & ∏_{i, k_{i}<0} T_{12}(−β_{i} − k_{i} − 1) ⋯ T_{12}(−β_{i} + 1) T_{12}(−β_{i}), \end{matrix}

where (k) runs over ℤ^{p}. It suffices to prove the vector space decomposition

(4.21)  Y_{p}(𝔤𝔩_{2}) = T_{ℓ} ⊕ I_{ℓ}.

Due to the formulas proved above, Y_{p}(𝔤𝔩_{2}) = T_{ℓ} + I_{ℓ}. We now need to show that the vectors τ^{(k)} are linearly independent modulo the left ideal I_{ℓ}. By (4.15) and the genericity assumption, the elements τ^{(k)} mod I_{ℓ} are eigenvectors for T_{22}(u) with distinct eigenvalues. So the claim will follow if we demonstrate that each τ^{(k)} is nonzero modulo I_{ℓ}. Suppose first that γ is generic: γ(−β_{i} − k) ≠ 0 for all k ∈ ℤ and all i. Then we deduce from (4.18)–(4.20) that τ^{(k)} ≠ 0 mod I_{ℓ} since 1 ≠ 0 mod I_{ℓ} which gives (4.21) for generic γ.

Let now γ be arbitrary. Suppose that a nonzero element τ belongs to the intersection T_{ℓ} ∩ I_{ℓ}. Then

(4.22)  τ = ∑_{i=1}^{p} a_{i} (t_{22}^{(i)} − β^{(i)}) + ∑_{i=1}^{2p} b_{i} (D^{(i)} − γ^{(i)}),

where D^{(i)}, β^{(i)} and γ^{(i)} are the coefficients of the polynomials D(u), β(u) and γ(u), respectively, while a_{i}, b_{i} ∈ Y_{p}(𝔤𝔩_{2}). Let γ̃ be generic. Then we can rewrite (4.22) as

(4.23)  τ = ∑_{i=1}^{p} a_{i} (t_{22}^{(i)} − β^{(i)}) + ∑_{i=1}^{2p} b_{i} (D^{(i)} − γ^{(i)}) + ∑_{i=1}^{2p} b_{i} (γ^{(i)} − γ^{(i)}).

Consider the unique decompositions of the elements b_{j} in accordance with (4.21) where γ(u) is taken to be γ̃(u):

(4.24)  b_{j} = τ_{j} + ∑_{i=1}^{p} a_{ij} (t_{22}^{(i)} − β^{(i)}) + ∑_{i=1}^{2p} b_{ij} (D^{(i)} − γ^{(i)})

for some a_{ij}, b_{ij} ∈ Y_{p}(𝔤𝔩_{2}). Using the decomposition (4.21) for generic γ̃(u) we must have

(4.25)  τ = ∑_{j=1}^{2p} τ_{j} (γ^{(j)} − γ^{(j)}).

for all such γ̃(u). This means that the T_{ℓ}-component of each element b_{j} (γ^{(j)} − γ^{(j)}) is independent of γ̃(u). However, due to the formulas (4.15)–(4.17), this is only possible if all b_{j} are zero. Finally, the elements a_{i} must be zero too by the decomposition (4.22) with generic γ. So, (4.21) holds for all γ(u). □

Remark 1. Given two monic polynomials α(u) and β(u) of degree p define the corresponding Verma module V (α(u), β(u)) as the quotient of Y_{p}(𝔤𝔩_{2}) by the left ideal generated by the coefficients of the polynomials T_{11}(u)−α(u), T_{22}(u)−β(u) and T_{12}(u); cf. [T1, T2]. Then the same argument as above shows that V (α(u), β(u)) has a basis {ξ^{(k)}} parameterized by p-tuples of nonnegative integers (k) = (k_{1}, …, k_{p}). The formulas of Theorem 2 hold for the basis vectors ξ^{(k)}, where γ(u) should be taken to be α(u) β(u − 1) which defines the central character γ of V (α(u), β(u)). In fact, V (α(u), β(u)) is isomorphic to the quotient of the corresponding universal module M(ℓ), ℓ = (β, γ) by the submodule spanned by the vectors {ξ^{(k)}} such that (k) contains at least one negative component k_{i}.

Corollary 6. Let ℓ = (β, γ) ∈ L_{1}.

(1) The module M(ℓ) is a generic weight Y_{p}(𝔤𝔩_{2})-module with central character γ, Supp M(ℓ) = ℤ^{p} and all weight spaces are 1-dimensional.

(2) The module M(ℓ) has a unique maximal submodule and hence a unique irreducible quotient.

(3) The equivalence class D(ℓ) coincides with the set ℓ + L_{0}.

Proof. Statement (1) follows immediately from Theorem 2. By Proposition 4.1 the sum of all proper submodules is again a proper submodule. Thus M(ℓ) has a unique maximal submodule which implies (2). The statement (3) follows immediately from (1). □

We will denote the unique irreducible quotient of M(ℓ) by L(ℓ). It follows from Corollary 6 that all weight spaces of L(ℓ) are 1-dimensional. Using Proposition 4.1 we can now describe all irreducible generic weight Y_{p}(𝔤𝔩_{2})-modules.

Corollary 7. Let ℓ = (β, γ) ∈ L_{1}.

(1) There exists an irreducible generic weight Y_{p}(𝔤𝔩_{2})-module L(ℓ) with L(ℓ)_{β} ≠ 0 and with central character γ. Moreover, dim L(ℓ)_{ψ} = 1 for all ψ ∈ Supp L(ℓ).

(2) Any irreducible weight module over Y_{p}(𝔤𝔩_{2}) with central character γ generated by a nonzero vector of weight β is isomorphic to L(ℓ).

5. Properties of Γ as a subalgebra of Y_{p}(𝔤𝔩_{2})

In this section we adapt the results from [DFO2] and [Ov] for the Yangians. In particular, we show that Γ is a Harish-Chandra subalgebra.

For any ℓ_{0} ∈ L_{1} the module M(ℓ_{0}) has a basis ξ^{(k)}, (k) ∈ ℤ^{p} with the action of generators of Y(𝔤𝔩_{2}) defined by formulas (4.15)–(4.17). Then we can relabel the basis elements of M(ℓ_{0}) by ξ_{ℓ}, ℓ ∈ ℓ_{0} +L_{0}. It follows from Theorem 2 that for every x ∈ Y_{p}(𝔤𝔩_{2}) there exists a finite subset L_{x} ⊂ L_{0} consisting of elements δ such that

(5.26)  ξ_{ℓ} = ∑_{δ∈L_{x}} θ(x, ℓ, δ)ξ_{ℓ+δ}

where θ(x, ℓ, δ) = θ(x, b, δ)(ℓ), θ(x, b, δ) ∈ 𝕃, b = (b_{1}, …, b_{p}, g_{1}, …, g_{2p}). Clearly, the set L_{x} is S_{p} × S_{2p}-invariant. Note that for a given x this formula does not depend on ℓ_{0}.

Let M_{L_{0}}(𝕃) be the ring of locally finite (with the finite number of non-zero elements in each row and each column) matrices over 𝕃 with the entries indexed by the elements of L_{0}. Any ℓ ∈ ℒ_{1} defines the evaluation homomorphism χ_{ℓ} : 𝕃⟶ 𝕜, which induces the homomorphism of matrix algebras M_{ℒ_{0}} (ℓ) : M_{ℒ_{0}} (𝕃) ⟶ M_{ℒ_{0}} (𝕜). For ℓ, ℓ′ ∈ L_{0} denote by e_{ℓ ℓ′} the corresponding matrix unit in M_{L_{0}}(𝕃). The group 𝕎 acts on M_{L_{0}}(𝕃) as follows: (w^{−1} · X)_{ℓ,ℓ′} = w^{−1} · X_{w(ℓ)w(ℓ′)} for all w ∈ 𝕎, X = (X_{ℓ ℓ′})_{ℓ,ℓ′∈L_{0}}, ℓ, ℓ′ ∈ L_{0}. Note that this action induces an action of S_{p} × S_{2p} on the free 𝕃-module X_{0} = ∑_{δ∈L_{0}} 𝕃e_{δ,\overline{0}} where \overline{0} is a zero element in L_{0}.

Define a map

G : Y_{p}(𝔤𝔩_{2}) → M_{L_{0}}(𝕃)

such that for any x ∈ Y_{p}(𝔤𝔩_{2}) and any ℓ ∈ L_{0}, G(x)_{ℓ ℓ′} = θ(x, b + ℓ, δ) if ℓ′ − ℓ = δ and 0 otherwise.

Lemma 5.1. (1) G is a representation of Y_{p}(𝔤𝔩_{2}).

(2) G(x) is 𝕎-invariant for any x ∈ Y_{p}(𝔤𝔩_{2}). In particular, G(x)_{\overline{0} \overline{0}} ∈ K(Γ).

(3) If x = x(b_{1}, …, b_{p}, g_{1}, …, g_{2p}) ∈ Γ then G(x)_{ℓℓ} = x(b_{1} + l_{1}, …, b_{p} +

l_{p}, g_{1}, …, g_{2p}) where ℓ = (l_{1}, …, l_{p}, 0, …, 0) ∈ L_{0}.

(4) G(Γ) consists of 𝕎-invariant diagonal matrices X such that X_{\overline{0} \overline{0}} ∈ Γ. In

particular, X_{\overline{0} \overline{0}} ∈ Γ determines X.

Proof. Let T be a free (non-commutative) algebra with generators t_{ij}^{(k)}, i, j = 1, 2, k = 1, …, p, π : T → Y_{p}(𝔤𝔩_{2}), t_{ij}^{(k)} ⟼ t_{ij}^{(k)}, be a canonical projection. Define a homomorphism g : T → M_{L_{0}}(𝕃) by g(t_{ij}^{(k)}) = G(t_{ij}^{(k)}) for all suitable i, j, k. To prove (1) it is enough to show that g(Ker π) = 0. Let f ∈ Ker π and suppose that g(f)_{ℓ′ℓ′′} ∈ 𝕃 is nonzero for some ℓ′, ℓ′′ ∈ L_{0}. Then M_{ℒ_{0}} (ℓ)(g(f)) = 0 and thus g(f)_{ℓ′ℓ′′}(ℓ) = 0 for any ℓ ∈ L_{1}. Since L_{1} is dense in Specm L we conclude that g(f) = 0 implying (1).

The image of G is 𝕎-invariant since it holds for the generators of Y_{p}(𝔤𝔩_{2}) (4.15)– (4.17). For any σ ∈ S_{p} × S_{2p}, (σ^{−1} · G)(x)_{\overline{0} \overline{0}} = σ^{−1}(G(x)_{σ(\overline{0}) σ(\overline{0})})= σ^{−1}(G(x)_{\overline{0} \overline{0}}). Hence G(x)_{\overline{0} \overline{0}} is S_{p} × S_{2p}-invariant proving (2). The statement (3) follows from (2) if we apply a shift by ℓ ∈ L_{0} to an arbitrary x ∈ Y_{p}(𝔤𝔩_{2}). The statement (4) follows immediately from (2) and (3). □

The composition r_{ℓ} of G and M_{ℒ_{0}} (ℓ) defines a representation G_{ℓ} of Y_{p}(𝔤𝔩_{2}). It is easy to see that the corresponding Y_{p}(𝔤𝔩_{2})−module coincides with the module M(ℓ) from Theorem 2.

Proposition 5.1. The representation G : Y_{p}(𝔤𝔩_{2})⟶ M_{L_{0}}(𝕃) is faithful.

Proof. It is clear that Ker G ⊂ ∩_{ℓ∈L_{1}} Ker r_{ℓ}. Hence it is enough to prove that

⋂_{ℓ∈L_{1}} Ker r_{ℓ} = 0.

Let ℓ = (β, γ). Then Ker r_{ℓ} = Ann M(ℓ) dy definition. Since M(ℓ) = Y_{p}(𝔤𝔩_{2})/I_{ℓ} we have that Ker r_{ℓ} ⊂ I_{ℓ}. Therefore, it is enough to show that ∩_{ℓ∈L_{1}}I_{ℓ} = 0. By Theorem 1, (1) the Yangian Y_{p}(𝔤𝔩_{2}) is free as a right module over Γ. Let x_{i}, i ∈ I be a basis of Y_{p}(𝔤𝔩_{2}) over Γ. If x = ∑_{i∈I} x_{i}z_{i} for some z_{i} ∈ Γ then x ∈ I_{ℓ} if and only if z_{i}(ℓ) = 0 for all i ∈ I. Since L_{1} is dense in L in Zariski topology it follows immediately that if x ∈ ∩_{ℓ∈L_{1}}I_{ℓ} then z_{i} = 0 for all i ∈ I and thus x = 0. This completes the proof. □

Immediately from the proof of the theorem above and the density of L_{1} in L we obtain the following analogue of the Harish-Chandra Theorem for Lie algebras [Di].

Corollary 8. Let x ∈ Y_{p}(𝔤𝔩_{2}) be such that xM(ℓ) = 0 for any ℓ ∈ L_{1}. Then x = 0.

Corollary 9. (1) Γ is a maximal commutative subalgebra in Y_{p}(𝔤𝔩_{2}). (2) If for x ∈ Y_{p}(𝔤𝔩_{2}) the matrix G(x) is diagonal then x ∈ Γ.

Proof. Consider an element x ∈ Y_{p}(𝔤𝔩_{2}) which commutes with every z ∈ Γ. Suppose there exist ℓ_{1}, ℓ_{2} ∈ ℒ_{0}, ℓ_{1} ≠ ℓ_{2} such that G(x)_{ℓ_{1}ℓ_{2}} ≠ 0. There exists z ∈ Γ such that z(ℓ_{1}) ≠ z(ℓ_{2}) and thus G(z)_{ℓ_{1}ℓ_{1}} ≠ G(z)_{ℓ_{2}ℓ_{2}} by Lemma 5.1, (3). Then we have G(xz)_{ℓℓ′} = G(x)_{ℓℓ′}G(z)_{ℓ′}_{ℓ′} = G(zx)_{ℓℓ′} = G(z)_{ℓℓ}G(x)_{ℓℓ′} and therefore G(x) is diagonal. To conclude the maximality of Γ it is enough to prove the statement (2).

By Lemma 5.1, (2), G(x)_{\overline{0} \overline{0}} = \frac{f}{g} ∈ 𝕃 where f, g ∈ Γ are relatively prime. Suppose

that g ∉ 𝕜. By Lemma 5.1, (4) we have that G(x)G(g) = G(f) and xg = f by Proposition 5.1. It implies that x ∈ Γ by Theorem 1, (1). This completes the proof. □

Corollary 10. Let p : M_{L_{0}}(L)⟶ X_{0} be the projection. Then the composition r : Y_{p}(𝔤𝔩_{2}) ⟶^{G} M_{L_{0}}(𝕃) ⟶^{p} X_{0} is a monomorphism of Y_{p}(𝔤𝔩_{2})−modules. The map p commutes with the action of S_{p} × S_{2p} and in particular, r(Y_{p}(𝔤𝔩_{2})) is S_{p} × S_{2p}−invariant.

Proof. Note that for any x ∈ Y_{p}(𝔤𝔩_{2}) the matrix G(x) ∈ M_{L_{0}}(𝕃) is determined completely by its column p(G(x)). Thus r(x) = 0 implies G(x) = 0 and x = 0 by faithfulness of G. Hence r is a monomorphism. Other statements follow immediately from the definitions and Lemma 5.1, (2). □

As in [DFO2], we identify the (Γ − Γ)-bimodule structure on Y_{p}(𝔤𝔩_{2}) with the corresponding Γ ⊗_{𝕜} Γ−module structure. Let b = (b_{1}, …, b_{p}, g_{1}, …, g_{2p}). For any z ∈ Γ and any S ⊂ L introduce the following polynomial

F_{S,z} = ∏_{δ∈S}(z ⊗ 1 − 1 ⊗ z(b + δ)) = ∑_{i=0}^{|S|} z^{i} ⊗ a_{i}, a_{i} ∈ 𝕃.

Proposition 5.2. ([DFO2], Lemma 25). Let S be a finite S_{p}×S_{2p}-invariant subset

in L and z be any element of Γ, F_{S,z} = ∑_{i=0}^{|S|} z^{i} ⊗ a_{i}, a_{i} ∈ 𝕃.

(1) a_{i} ∈ Γ, i = 0, …, |S|.
(2) For any x ∈ Y_{p}(𝔤𝔩_{2}) such that L_{x} ⊂ S holds ∑_{i=0}^{q} z^{i}xa_{i} = 0.

Proof. If S is S_{p} × S_{2p}-invariant then the coefficients of the polynomial F_{S,z} are S_{p} ×S_{2p}-invariant and hence belong to Γ which proves (1). It is enough to check the statement (2) for S = L_{x} since F_{S,z} = F_{S\L_{x},z}F_{L_{x},z}. Denote q = |S|. Let ℓ ∈ L_{1} and let ξ_{ℓ} be a basis element of M(ℓ). Then

∑_{i=0}^{q} z^{i}xa_{i}(ξ_{ℓ}) = ∑_{i=0}^{q} z^{i}xa_{i}(ℓ)(ξ_{ℓ}) =

∑_{i=0}^{q} z^{i}a_{i}(ℓ) ∑_{δ∈L_{x}} θ(x, ℓ, δ)ξ_{ℓ+δ} =

∑_{δ∈L_{x}} θ(x, ℓ, δ) ∑_{i=0}^{q} a_{i}(ℓ)(z^{i}ξ_{ℓ+δ}) =

∑_{δ∈L_{x}} θ(x, ℓ, δ) ∑_{i=0}^{q} a_{i}(ℓ)z(ℓ + δ)^{i}ξ_{ℓ+δ} = ∑_{δ∈L_{x}} θ(x, ℓ, δ)F_{L_{x},z}(z(ℓ + δ), ℓ)ξ_{ℓ+δ} = 0

since F_{L_{x},z}(z(ℓ + δ), ℓ) = 0 for every δ ∈ L_{x}. Applying Corollary 8 we obtain the statement of the proposition. □

The main result of this section is the following

Theorem 3. Γ is a Harish-Chandra subalgebra of Y_{p}(𝔤𝔩_{2}).

Proof. Following [DFO2], Proposition 8, it is enough to show that a Γ-bimodule Γ t_{ij}^{(k)} Γ is finitely generated both as left and as right module for every possible choice of indices i, j, k. It is obvious for i = j = 2 since t_{22}^{(k)} ∈ Γ. We prove it for i = 2, j = 1. Since d_{i} is central for every i = 1, …, 2p we have d_{i}t_{21}^{(k)} = t_{21}^{(k)}d_{i}. From formulas (4.16) follows that L_{t_{21}^{(k)}} = {δ_{i}|i = 1, …, p}. Then

F_{L_{t{(k) \atop 21}} ,t_{22}^{(i)}} = z^{p} ⊗ 1 + ∑_{l=0}^{p−1} z^{l} ⊗ a_{l}, a_{l} ∈ Γ

and

(5.27)  (t_{22}^{(i)})^{p}t_{21}^{(k)} + ∑_{l=0}^{p−1}(t_{22}^{(i)})^{l}t_{21}^{(k)}a_{l} = 0

by Proposition 5.2, (2). Hence the elements (∏_{i=1}^{p}(t_{22}^{(i)})^{k_{i}})t_{21}^{(k)}, 0 ≤ k_{i} < p form the generators of Γt_{21}^{(k)}Γ as a right Γ-module.

Applying a suitable automorphism we conclude that Γt_{21}^{(k)}Γ is finitely generated as a left Γ-module.

The cases i = 1, j = 2 and i = j = 1 can be treated analogously since L_{t_{12}^{(k)}} =

{−δ_{i}|i = 1, …, p} and L_{t_{11}^{(k)}} = {δ_{i} − δ_{j}|i, j = 1, …, p}. Hence Γt_{ij}^{(k)}Γ is finitely generated as a right and as a left Γ-module. □

6. Category of Harish-Chandra modules over Y_{p}(𝔤𝔩_{2})

Since Γ is a Harish-Chandra subalgebra of Y_{p}(𝔤𝔩_{2}) we can apply all the statements from Section 2.1. Denote 𝒜 = 𝒜_{Y_{p}(𝔤𝔩_{2}),Γ}. Then by Proposition 1, the categories 𝒜 − mod_{d} and ℍ(Y_{p}(𝔤𝔩_{2}), Γ) are equivalent. Also the full subcategory ℍW (Y_{p}(𝔤𝔩_{2}), Γ) consisting of weight modules is equivalent to the module category 𝒜_{W} −mod. If ℓ ∈ L then the category R_{ℓ} is equivalent to the block 𝒜_{W} (D(ℓ))−mod of the category 𝒜_{W} − mod.

We will show that each character of Γ extends to a finite number of irreducible Harish-Chandra modules over Y_{p}(𝔤𝔩_{2}). This is an analogue of the corresponding result in the case of a Lie algebra 𝔤𝔩_{n} which was conjectured in [DFO1] and proved in [Ov]. In this section we use the techniques of [DFO2] and [Ov].

Lemma 6.1. For any x ∈ Y_{p}(𝔤𝔩(2)), f ∈ Γ ⊗ Γ, ℓ, ℓ′ ∈ L_{0} holds

G(f · x)_{ℓℓ′} = f(b + ℓ, b + ℓ′)G(x)_{ℓℓ′}.

Proof. Let f = ∑_{i} z_{i} ⊗ z′_{i} ∈ Γ ⊗ Γ. Then G(f · x) = ∑_{i} G(z_{i})G(x)G(z′_{i}) and hence

G(f · x)_{ℓℓ′} = ∑_{i} G(z_{i})_{ℓ ℓ}G(x)_{ℓ ℓ′}G(z′_{i})_{ℓ′ ℓ′} = G(x)_{ℓ ℓ′} ∑_{i} G(z_{i})_{ℓ ℓ}G(z′_{i})_{ℓ′ ℓ′} =

G(x)_{ℓ ℓ′} ∑_{i} z_{i}(b + ℓ)z′_{i}(b + ℓ′) = G(x)_{ℓ ℓ′}f(b + ℓ, b + ℓ′).

□

Lemma 6.2. ([DFO2], Lemma 25). Let z ∈ Γ, S ⊂ L be a S_{p} × S_{2p}-invariant set and x ∈ Y_{p}(𝔤𝔩_{2}) be such that G(x)_{ℓ ℓ′} = 0 for all ℓ, ℓ′, ℓ − ℓ′ ∉ S then F · x = 0.

Proof. Let F in the form F = ∑_{i} z^{i} ⊗ a_{i} where a_{i} ∈ L. If ℓ − ℓ′ ∈ S then G(F ·x)_{ℓ′ℓ} = F (b+ℓ, b+ℓ′)G(x)_{ℓ ℓ′} by Lemma 6.1. Then h = z ⊗1−1⊗z(b+ℓ−ℓ′) divides F, h(b + ℓ, b + ℓ′) = 0, F (b + ℓ, b + ℓ′) = 0 and F · x = 0. □

Let S ⊂ ℒ_{0} be a finite S_{p} × S_{2p}−invariant set. Define Y^{S}= {x ∈ Y_{p}(𝔤𝔩_{2}) | ℒ_{x} ⊂ S}. Clearly Y^{S} is a Γ−subbimodule in Y_{p}(𝔤𝔩_{2}). We have the following characterization of the bimodule Y^{S}.

Lemma 6.3. Let x ∈ Y_{p}(𝔤𝔩_{2}). Then

(1) x ∈ Y^{S} if and only if whenever G(x)_{ℓ,ℓ′} ≠ 0, for some ℓ, ℓ′ ∈ ℒ_{0}, implies that ℓ − ℓ′ ∈ S.

(2) y = F_{ℒ_{x}\S,z} · x ∈ Y^{S} for any z ∈ Γ.

(3) Y^{S} is a finitely generated left (right) Γ−module and Y^{S} = 𝔻(Y^{S}).

(4) Y^{{0}} = Γ.

Proof. The statement (1) follows from definitions. Let F = F_{ℒ_{x}\S,z} . To prove (2) calculate the matrix element G(y)_{ℓℓ′} provided ℓ − ℓ′ ∉ S. If ℓ − ℓ′ ∉ L_{x} then G(x)_{ℓℓ′} = 0 and hence G(y)_{ℓℓ′} = 0. Suppose that ℓ − ℓ′ ∈ ℒ_{x} \ S then by Lemma 6.1, G(y)_{ℓℓ′} = G(F · x)_{ℓℓ′} = F (b + ℓ, b + ℓ′)G(x)_{ℓℓ′}. But

F (b + ℓ, b + ℓ′) = ∏_{δ∈ℒ_{x}\S} (z(b + ℓ) − z(b + ℓ′ + δ))

which is equal to zero. This proves (2).

Let x ∈ 𝔻(Y^{S}) and z ∈ Γ is such that z ≠ 0 and zx ∈ Y^{S}. Since G(zx)_{ℓℓ′} = z(b + ℓ)G(x)_{ℓℓ′} then G(zx)_{ℓℓ′} = 0 if and only if G(x)_{ℓℓ′} = 0 implying that x ∈ Y^{S}. Hence Y^{S} = 𝔻(Y^{S}).

Consider r(Y^{S}) as a Γ−submodule of X_{0} where r : Y_{p}(𝔤𝔩_{2})⟶ X_{0} is defined in Corollary 10. Then r(Y^{S}) belongs to a free 𝕃−submodule of X_{0} of finite rank ∑_{ℓ∈S} 𝕃e_{\overline{0}ℓ}. Hence 𝕃 · r(Y^{S}) is finitely generated L-module. Without loss of generality we can assume that it is generated by the elements r(x_{1}), …, r(x_{s}) ∈ r(Y^{S}),

i.e. 𝕃 · r(Y^{S}) = ∑_{i=1}^{s}𝕃 · r(x_{i}). Since 𝔻(Y^{S}) = Y^{S} we have that 𝔻(∑_{i=1}^{s}Γx_{i}) ⊂

Y^{S}. Fix x ∈ Y^{S}. Then r(x) = ∑_{i=1}^{s}t_{i}r(x_{i}), t_{i} ∈ 𝕃. Note that for any y ∈ Y^{S}

and any σ ∈ S_{p} × S_{2p}, σ · r(y) = r(y). Hence p!(2p)!r(x) = ∑_{σ∈S_{p}×S_{2p}} σ · r(x) =

∑_{σ∈S_{p}×S_{2p}}∑_{i=1}^{s}(σ · t_{i})σ · r(x_{i}) which can be rewritten as follows

r(x) = \frac{1}{p!(2p!)}∑_{i=1}^{s}u_{i}r(x_{i}),

where u_{i} = ∑_{σ∈S_{p}×S_{2p}}σ · t_{i}. Since each u_{i} is S_{p} × S_{2p}-invariant then it belongs to

the field of fractions K(Γ) for all i = 1, …, s. Multiplying both parts of the last

equality by the common denominator of u_{i} we obtain that x ∈ 𝔻(∑_{i=1}^{s}Γx_{i}) and thus

𝔻(∑_{i=1}^{s}Γx_{i}) = Y^{S}. Applying Corollary 3 we conclude that Y^{S} is finitely generated

over Γ. This proves (3). By the definition of Y^{S}, x ∈ Y^{{0}} if and only if G(x) is diagonal. Hence x ∈ Γ by Corollary 9, (2). □

Let m, n ∈ Specm Γ, ℓ_{m}, ℓ_{n} ∈ L are such that ı^{∗}(ℓ_{m}) = m and ı^{∗}(ℓ_{n}) = n. Denote

S(m, n) = {σ_{1}ℓ_{n} − σ_{2}ℓ_{m} | σ_{1}, σ_{2} ∈ S_{p} × S_{2p}} ∩ ℒ_{0}.

Consider the following subset in L

ℒ_{2} = {ℓ ∈ ℒ |ℓ_{i} − ℓ_{j} ∉ ℤ \ {0}, i, j = 1, …, p}

and set Ω = ı^{∗}(ℒ_{2}).

Proposition 6.1. (1) For all m, n ∈ Specm Γ and all m, n ≥ 0 holds

Y_{p}(𝔤𝔩_{2}) = Y^{S} + n^{n}Y_{p}(𝔤𝔩_{2}) + Y_{p}(𝔤𝔩_{2})m^{m},

where S = S(m, n).

(2) For all m, n ∈ Specm Γ a system of generators of Y^{S} as a left Γ-module (as a right Γ-module) generates 𝒜(m, n) as a left Γ_{n}−module (as a right Γ_{m}−module), i.e. 𝒜(m, n) is finitely generated as a left Γ_{n} and as a right Γ_{m}−module. In particular, the algebra Γ is big in every n ∈ Ob 𝒜.

(3) If S(m, n) = ∅ then 𝒜(m, n) = 0 (cf. [DFO2], Corollary 27).

(4) If S(m, n) = {0} then 𝒜(m, n) is generated as a left Γ_{n} and as a right Γ_{m}-module by the image of 1 in 𝒜(m, n).

(5) If S(m, m) = {0} then m ∈ Ω, 𝒜(m, m) is a quotient algebra of Γ and χ_{m} extends uniquely to an irreducible Y_{p}(𝔤𝔩_{2})-module.

(6) If ℓ_{m} ∈ L_{1} then 𝒜(m, m) = Γ_{m}.

(7) Let ℓ ∈ L_{1}, m = ı^{∗}(ℓ) and n = ı^{∗}(ℓ + δ_{i}), i ∈ {1, …, p}. Then 𝒜(m, n) is a free of rank 1 right Γ_{m}- ( left Γ_{n}-) module.

Proof. (1) It is enough to show that for any x ∈ Y_{p}(𝔤𝔩_{2}) and any k ⩾ 1 there exists x_{k} ∈ Y^{S} such that

(6.28)  x ∈ x_{k} + ∑_{i=0}^{k}n^{k−i}xm^{i}.

The statement will follow if we choose k = m + n + 1. We will use induction on k. If L_{x} ⊂ S then x ∈ Y^{S} and there is nothing to prove. Note that by the definition of the set S for any ℓ ∈ ℒ_{x} \ S the S_{p} × S_{2p}-orbits of ℓ_{n} and ℓ_{m} + ℓ are disjoint. Hence there exists z ∈ Γ such that z(ℓ_{n}) ≠ z(ℓ_{m} + ℓ) for any ℓ ∈ ℒ_{x} \ S. Let F = F_{ℒ_{x}\S,z} . Then F (ℓ_{m}, ℓ_{n}) = ∏_{ℓ∈ℒ_{x}\S} (z(ℓ_{n})− z(ℓ_{m} + ℓ)) ≠ 0 since every factor F is non-zero. We can assume that F (ℓ_{m}, ℓ_{n}) = 1. Hence we obtain that F = 1 + u where u ∈ n ⊗ Γ+ Γ ⊗ m. It follows from Lemma 6.3, (2) that x_{1} = F · x belongs to Y^{S}. Hence we have x_{1} = (1 + u) · x ∈ x + nxΓ + Γxm and x ∈ x_{1} + nxΓ+ Γxm. This proves the base of induction. Assume that 6.28 holds for some k ≥ 1. Then

x ∈ x_{k} + ∑_{i=0}^{k}n^{k−i}(x_{k} + ∑_{j=0}^{k}n^{k−j}xm^{j})m^{i} ⊂ x_{k} + ∑_{i=0}^{k}n^{k−i}x_{k}m^{i} + ∑_{i=0}^{k+1}n^{k+1−i}xm^{i}.

Since Y^{S} is a Γ-bimodule we conclude that x_{k} + ∑_{i=0}^{k}n^{k−i}x_{k}m^{i} ⊂ Y^{S} which implies

the statement (1). In particular,

x_{k+1} − x_{k} ∈ ∑_{i=0}^{k} n^{k−i}Y^{S}m^{i}.

(2) We prove the statement for the case of left module, the case of the right module can be treated analogously. By (1) the image \overline{x} of every x ∈ Y^{S} in 𝒜(n, m) is the limit of the sequence (\overline{x}_{k})_{k⩾1}, x_{k} ∈ Y^{S}. Let y_{1}, …, y_{m} be a finite system of generators of Y^{S} as a left Γ−module. Then for every N > 1 there exists a maximal d_{N} such that

y_{i}m^{N} ⊂ ∑_{j=1}^{m} n^{d_{N}} y_{j}

for all i = 1, …, m. Note that by the proof of (1), x_{k+1} − x_{k} ∈ ∑_{i=0}^{k}n^{k−i}Y^{S}m^{i} ⊂

n^{R_{k}}Y^{S} where R_{k} = min{[k/2], d_{[k/2]}}. Since Y^{S} is a finitely generated right Γmodule and Γ is noetherian then the intersection ∩_{k≥1}Y^{S}m^{k} = 0. It follows that d_{N} → ∞ while N → ∞. Since

\overline{x} = \overline{x}_{1} + ∑_{k=1}^{∞} \overline{(x_{k+1} − x_{k})}

we have \overline{x} ∈ ∑_{k=1}^{∞} \overline{n^{R_{k}}Y^{S}} ⊂ ∑_{l=1}^{m} Γ_{n}\overline{y_{l}}. Note that the first sum is well defined since

R_{k} → ∞ when k → ∞. We conclude that 𝒜(n, m) is finitely generated as left Γ_{n}−module. This completes the proof of (2).

(3) If S = ∅, then Y^{S} = 0 and the statement follows from (1) and the definition of the category 𝒜 (2.8).

(4) By the definition of Y^{S} for every x ∈ Y^{{ 0}} the matrix G(x) is diagonal. Following Corollary 9, (2) it means x ∈ Γ, in particular A(m, n) is generated (both as a left and as a right module) by the image of 1 ∈ Γ.

(5) By (4), Y^{0} = Γ, i.e. A(m, n) is 1-generated as a left Γ_{m}-module. Then the 𝕜-algebra homomorphism î_{m} : Γ_{m}⟶A(m, m), z ⟼ z · 1_{m}, where 1_{m} is a unit morphism, is an epimorphism which shows that A(m, m) is a quotient algebra of Γ_{m}. The uniqueness of extension follows from the uniqueness of the simple A(m, m)-module and [DFO2], Theorem 18.

(6) Let ℓ = ℓ_{m}. Since ℓ ∈ L_{1} then for any k > 0 there exists a canonical projection π_{k} : 𝕃⟶ 𝕃/ℓ^{k}𝕃. It induces a homomorphism of the matrix algebras π_{k} : M_{L_{0}}(𝕃) ⟶ M_{L_{0}}(𝕃/ℓ^{k}) and defines a Harish-Chandra module by the following composition

Y_{p}(𝔤𝔩_{2})⟶^{G}M_{L_{0}}(𝕃) ⟶^{π_{k}} M_{L_{0}}(𝕃/ℓ^{k}).

For any x ∈ Γ there exists k > 0, such that x ∉ (ℓ)^{k}. Then π_{k}G(x)_{\overline{0},\overline{0}} = x+(ℓ)^{k} ≠ 0 that completes the proof.

(7) The proof is analogous to the proof of (6). Let z ∈ Γ, z ≠ 0. Suppose 𝒜(m, n)z = 0. Then by the construction of the equivalence 𝔽 : 𝒜 − mod_{d} ⟶ ℍ(U, Γ) for any Harish-Chandra module M and any x ∈ Y_{p}(𝔤𝔩_{2}) the linear operator xz on M induces a zero map between M(m) and M(n). It is enough to construct a Harish-Chandra module where this is failed. For k ≥ 1 consider a natural map π_{k} : 𝕃 → 𝕃/(ℓ)^{k} and a composition π_{k} · G : Y_{p}(𝔤𝔩_{2}) → M_{L_{0}}(𝕃/(ℓ)^{k}). It defines a Harish-Chandra module structure on a free 𝕃/(ℓ)^{k}-module X̄ = ∑_{δ∈L_{0}} 𝕃/(ℓ)^{k}e_{δ,\overline{0}}. Consider x ∈ Y_{p}(𝔤𝔩_{2}) such that G(x)_{δ_{i}\overline{0}} ≠ 0. Then G(xz)_{δ_{i}\overline{0}} = G(x)_{δ_{i}\overline{0}}G(z)_{\overline{0}\overline{0}} = G(x)_{δ_{i}\overline{0}}z ≠ 0. Choose k such that G(xz)_{δ_{i}\overline{0}} ∉ (ℓ)^{k}. Hence (π_{k} · G)(xz)_{δ_{i},\overline{0}} ≠ 0 and the linear operator xz induces a non-zero map between X̄(m) = 𝕃/(ℓ)^{k} and X̄(n) = 𝕃/(ℓ + δ_{i})^{k}. The obtained contradiction shows that 𝒜(m, n)z ≠ 0. The case z𝒜(m, n) = 0 is treated analogously. □

Now we are in the position to state the main result of this section which follows immediately from Lemma 2.1 and Proposition 6.1, (2).

Theorem 4. Let m ∈ Specm Γ. Then the left ideal Y_{p}(𝔤𝔩(2))m is contained in finitely many maximal left ideals of Y_{p}(𝔤𝔩(2)). In particular, m extends to a finitely many (up to an isomorphism) irreducible Y_{p}(𝔤𝔩(2))−modules and for each such module M, dim M(n) < ∞ for all n ∈ Specm Γ.

7. Category of generic Harish-Chandra modules

Lemma 7.1. Let ℓ ∈ L_{1}, ℓ = (β, γ), m = ı^{∗}(ℓ) ∈ Specm Γ, n = ı^{∗}(ℓ + δ_{i}), i ∈ {1, …, p}. If β_{i} ∉ {γ_{1}, …, γ_{2p}} then the objects of 𝒜 represented by m and n are isomorphic.

Proof. Choose z_{1}, z_{2} ∈ Γ such that z_{1}(ℓ+δ_{j}) = δ_{ij}, z_{2}(ℓ+δ_{i}−δ_{j}) = δ_{ij}, j = 1, …, p. Denote z = z_{2}t_{12}^{(1)}z_{1}t_{21}^{(1)}. Then G(z) is diagonal by Lemma 6.1 and hence z ∈ Γ by Corollary 9, (2). We will show that the image of z in Γ_{m} is invertible. Clearly, this is equivalent to the fact that z(m) ≠ 0. Note that z(m) = z(ℓ). Thus applying formulas (4.15)–(4.17) we have z(m) = γ(−β_{i}) ≠ 0 since ℓ ∈ L_{1}. Denote by T_{1} (respectively T_{2}) the generator of Γ̂-bimodule 𝒜(m, n) (respectively 𝒜(n, m)) (Proposition 6.1, (7)). Then z_{2}t_{12}^{(1)} = z_{m}T_{2}, z_{1}t_{21}^{(1)} = T_{1}z′_{m} for some z_{m}, z′_{m} ∈ Γ_{m} and z = z_{m}T_{2}T_{1}z′_{m}. Since z(m) ≠ 0 it follows that z′_{m}(m) ≠ 0, z_{m}(m) ≠ 0 and hence T_{2}T_{1} = z_{m}^{−1}z(z′_{m})^{−1} is invertible in Γ_{m}. The similar argument shows that T_{1}T_{2} is invertible in Γ_{n}. Therefore the objects m and n are isomorphic. □

Corollary 11. Let ℓ ∈ L_{1}, ℓ = (β, γ), β_{i} −γ_{j} ∉ ℤ and m = ı^{∗}(ℓ) ∈ Specm Γ. Then the category ℍ(Y_{p}(𝔤𝔩_{2}), Γ, D(ℓ)) is hereditary. Moreover,

dim Ext_{ℍ(Y_{p}(𝔤𝔩_{2}),Γ,D(ℓ))}^{1}(L(ℓ), L(ℓ)) = 3p.

Proof. By Lemma 7.1 and our assumptions all objects of the category 𝒜(D(ℓ)) are isomorphic and hence the category 𝒜(D(ℓ)) − mod _{d} is equivalent to the category of finite-dimensional modules over Γ_{m}. Applying Proposition 2.1 we conclude that the category ℍ(Y_{p}(𝔤𝔩_{2}), Γ, D(ℓ)) is hereditary. Since Γ_{m} is an algebra of power series in 3p variables the statement about dim Ext^{1} follows. □

7.1. Category of generic weight modules. Fix ℓ ∈ L_{1}, m = ı^{∗}(ℓ), n = ı^{∗}(ℓ + δ_{i}) ∈ Specm Γ, i ∈ {1, …, p}. Then 𝒜_{W} (m, m) ≃ Γ_{m}/Γ_{m}m ≃ 𝕜 by Proposition 6.1, (6) and dim 𝒜_{W} (m, n) = 1 by Proposition 6.1, (7). We will give a direct construction of the category 𝒜_{W} (D(ℓ)).

Suppose ℓ = (β, γ), β = (β_{1}, …, β_{p}) ∈ 𝕜^{p}, γ = (γ_{1}, …, γ_{2p}) ∈ 𝕜^{2p} and

(7.29)  γ(u) = ∏_{i=1}^{2p}(u + γ_{i}).

Since ℓ ∈ L_{1} then β_{i} − β_{j} ∉ ℤ for all i, j = 1, …, p, i ≠ j. Consider the following category K_{ℓ}: Ob (K_{ℓ}) = ℤ^{p} and the morphisms are generated by

(7.30)  f_{i}(k) : (k) ↦ (k + δ_{i})  and  e_{i}(k) : (k) ↦ (k − δ_{i}),

where i = 1, …, p and (k) = (k_{1}, …, k_{p}) ∈ ℤ^{p} with the following relations:

\begin{aligned}f_{j}(k + δ_{i}) f_{i}(k) & = f_{i}(k + δ_{j}) f_{j}(k), \\ e_{j}(k − δ_{i}) e_{i}(k) & = e_{i}(k − δ_{j}) e_{j}(k), \\ e_{i}(k + δ_{j}) f_{j}(k) & = f_{j}(k − δ_{i}) e_{i}(k) for i ≠ j, \\ e_{i}(k + δ_{i}) f_{i}(k) & = −γ(−β_{i} − k_{i}) 1_{(k)}, \\ f_{i}(k − δ_{i}) e_{i}(k) & = −γ(−β_{i} − k_{i} + 1) 1_{(k)}. \end{aligned}

It follows immediately from Lemmas 4.1 and 4.2 that any module in the category R_{ℓ} can be naturally viewed as a module over the category K_{ℓ} which defines a functor F : R_{ℓ} → K_{ℓ}−mod. Consider the cyclic subalgebra C_{ℓ}(a) = Hom_{K_{ℓ}}(a, a) for any a ∈ ℤ^{p}. Clearly, C_{ℓ}(a) ≃ 𝕜 for any a ∈ ℤ^{p} due to the defining relations of K_{ℓ}. For any a = (k_{1}, …, k_{p}) ∈ ℤ^{p} we can construct a universal module M(ℓ, a) ∈ K_{ℓ}−mod. Consider 𝕜 as a C_{ℓ}(a)-module with

\begin{aligned}e_{i}(k + δ_{i}) f_{i}(k) 1 & = −γ(−β_{i} − k_{i}), \\ f_{i}(k − δ_{i}) e_{i}(k) 1 & = −γ(−β_{i} − k_{i} + 1). \end{aligned}

Let A_{ℓ,a} be an algebra of paths in K_{ℓ} originating in a. Now construct a ℤ^{p}-graded K_{ℓ}-module

M(ℓ, a) = A_{ℓ,a} ⊗_{C_{ℓ}(a)} 𝕜.

Clearly, all graded components of M(ℓ, a) are 1-dimensional and M(ℓ, a)_{a} = 1_{a} ⊗𝕜. A module M(ℓ, a) contains a unique maximal ℤ^{p}-graded submodule which intersects M(ℓ, a)_{a} trivially and hence has a unique irreducible quotient L(ℓ, a) with L(ℓ, a)_{a} ≃ 𝕜 and dim L(ℓ, a)_{b} ≤ 1 for all b ∈ ℤ^{p}. If V is another irreducible K_{ℓ}- module with V_{a} ≠ 0 then there exists a non-trivial C_{ℓ}(a)-homomorphism from 𝕜 to V_{a} which can be extended to an epimorphism from M(ℓ, a) to V. Since V is irreducible we conclude that V ≃ L(ℓ, a).

Obviously, we can view M(ℓ) as a module over the category K_{ℓ} with a natural action of the morphisms of K_{ℓ} and F (M(ℓ)) = M(ℓ, β). Thus a K_{ℓ}-module M(ℓ, β) can be extended to a Y_{p}(𝔤𝔩_{2})-module M(ℓ). Moreover, the functor F preserves the submodule structure of M(ℓ). In particular, F (L(ℓ)) = L(ℓ, β).

Proposition 7.1. If ℓ ∈ L_{1} then the categories K_{ℓ}−mod and R_{ℓ} are equivalent.

Proof. Let ℓ = (β, γ). We already have a functor F : R_{ℓ} → K_{ℓ}−mod. Suppose that V ∈ K_{ℓ}−mod. We want to show that V can be extended to a Y_{p}(𝔤𝔩_{2})-module. Fix v ∈ V_{(k)} \ {0}. Let W ⊆ V be a submodule generated by v. Then W_{(k)} = 𝕜v and there is an epimorphism from M(ℓ, a) to W, where a = (k_{1}, …, k_{p}), which maps 1_{a} ⊗ 1 to v. Since F (M(ℓ′)) = M(ℓ, a), where ℓ′ = (β + a, γ), then W can be extended to a corresponding quotient of M(ℓ′). Since v was an arbitrary element of V we conclude that V can be extended to a Y_{p}(𝔤𝔩_{2})-module and will denote that module by G(V). Clearly, G defines a functor from K_{ℓ}−mod to R_{ℓ} (action on morphisms is obvious). One can easily see that the functors F and G define an equivalence between the categories K_{ℓ}−mod and R_{ℓ}. □

7.2. Support of irreducible generic weight modules. To complete the classification of irreducible modules we have to know when two irreducible modules L(ℓ) and L(ℓ′) are isomorphic. For that we need to describe the support Supp L(ℓ).

We shall say that the weight subspaces M(ℓ)_{ψ} and M(ℓ)_{ψ+δ_{i}} are strongly isomorphic if γ(−ψ_{i}) ≠ 0 where ψ = (ψ_{1}, …, ψ_{p}). This implies

f_{i}(ψ_{1}, …, ψ_{p}) M(ℓ)_{ψ} ≠ 0 and e_{i}(ψ_{1}, …, ψ_{i} + 1, …, ψ_{p}) M(ℓ)_{ψ+δ_{i}} ≠ 0.

The statement below follows immediately from the relations in K_{ℓ} (cf. also Corollary 5).

Lemma 7.2. If M(ℓ)_{ψ} and M(ℓ)_{ψ+δ_{i}} are strongly isomorphic, then M(ℓ)_{ψ±δ_{j}} and M(ℓ)_{ψ+δ_{i}±δ_{j}} are strongly isomorphic for all i, j = 1, …, p, i ≠ j. Moreover, if

f_{i}(ψ_{1}, …, ψ_{p}) M(ℓ)_{ψ} = 0  or  e_{i}(ψ_{1}, …, ψ_{p}) M(ℓ)_{ψ} = 0

then

\begin{aligned}f_{i}(ψ_{1}, …, ψ_{j} ± 1, …, ψ_{p}) M(ℓ)_{ψ±δ_{j}} & = 0 or \\ e_{i}(ψ_{1}, …, ψ_{j} + 1, …, ψ_{p}) M(ℓ)_{ψ±δ_{j}} & = 0, \end{aligned}

respectively, for all j ≠ i.

Let a_{i}, a′_{i} ∈ ℤ ∪ {±∞}, a_{i} ≤ a′_{i}, i ∈ {1, …, p}. Denote

P (a_{1}, …, a_{p}, a′_{1}, …, a′_{p}) = {(x_{1}, …, x_{p}) ∈ ℤ^{p} | a_{i} ≤ x_{i} ≤ a′_{i}, i = 1, …, p},

a parallelepiped in ℤ^{p}. Note that some faces of the parallelepiped can be infinite in some directions. In particular, in the case a_{i} = −∞, a′_{i} = ∞ for all i, the parallelepiped coincides with ℤ^{p}.

Theorem 5. For any irreducible weight module L(ℓ) over Y_{p}(𝔤𝔩_{2}) there exist elements a_{i}, b_{i} ∈ ℤ ∪ {±∞}, a_{i} ≤ a′_{i}, i ∈ {1, …, p} such that

Supp L(ℓ) = P (a_{1}, …, a_{p}, a′_{1}, …, a′_{p}).

Proof. Let ℓ = (β, γ) ∈ L_{1}. Fix i ∈ {1, …, p}. If γ(−β_{i} + k) ≠ 0 for all k ∈ ℤ then (k_{1}, …, k_{i} + m, …, k_{p}) ∈ Supp L(ℓ) as soon as (k_{1}, …, k_{p}) ∈ Supp L(ℓ). This follows immediately from Lemma 7.2. In this case we set a_{i} = −∞ and a′_{i} = ∞. Let now γ(−β_{i} + k) = 0 for some k ∈ ℤ. Let m ≥ 0 be the smallest integer (if exists) such that γ(−β_{i} − m) = 0 and let n ≤ 0 be the largest integer (if exists) such that γ(−β_{i} − n + 1) = 0. It follows from Lemma 7.2 that

Supp L(ℓ) ∩ {β + kδ_{i} | k ∈ ℤ} = {β + nδ_{i}, …, β, …, β + mδ_{i}}.

If β + sδ_{j} ∈ Supp L(ℓ), j ≠ i then

Supp L(ℓ)∩{β +sδ_{j} +kδ_{i} | k ∈ ℤ} = {β +sδ_{j} +nδ_{i}, …, β +sδ_{j}, …, β +sδ_{j} +mδ_{i}}.

In this case we set a_{i} = β_{i} + n and a′_{i} = β_{i} + m. The statement of the theorem now follows. □

7.3. Indecomposable generic weight modules. Fix ℓ = (β, γ) ∈ L_{1}. A full subcategory S ⊆ K_{ℓ} is called a skeleton of K_{ℓ} provided the objects of S are pairwise non-isomorphic and any object of K_{ℓ} is isomorphic to some object of S. In this case the categories of K_{ℓ}−mod and S−mod are equivalent.

For each i ∈ {1, …, p} consider a set I_{i} = {k ∈ ℤ | γ(−β_{i} − k) = 0}. Define a category S_{ℓ} as a 𝕜-category with the set of objects

S_{0} = {0, …, |I_{1}|} × … × {0, …, |I_{p}|}

and with morphisms generated by

\begin{matrix}r_{(i_{1},...,i_{p})}^{k} : (i_{1}, …, i_{p}) & ↦ (i_{1}, …, i_{k} + 1, …, i_{p}), \\ s_{(j_{1},...,j_{p})}^{k} : (j_{1}, …, j_{p}) & ↦ (j_{1}, …, j_{k} − 1, …, j_{p}), \end{matrix}

where k ∈ {1, …, p} is such that I_{k} ≠ ∅, i_{k} < |I_{k}|, j_{k} > 0, subject to the relations:

s_{(i_{1},...,i_{k}+1,...,i_{p})}^{k}r_{(i_{1},...,i_{p})}^{k} = r_{(i_{1},...,i_{p})}^{k}s_{(i_{1},...,i_{k}+1,...,i_{p})}^{k} = 0

and

x_{(a_{1},...,a_{p})}^{k}y_{(e_{1},...,e_{p})}^{r} = y_{(c_{1},...,c_{p})}^{r}x_{(e_{1},...,e_{p})}^{k}

for all k ≠ r and all possible x, y ∈ {r, s}, a_{i}, e_{i}, c_{i}, 1 ≤ i ≤ p for which this equality makes sense.

It follows from the construction that S_{ℓ} is the skeleton of the category K_{ℓ}. Note that the corresponding algebra is finite-dimensional. In particular, S_{ℓ} is semisimple when I_{k} = ∅ for all 1 ≤ k ≤ p, i.e. when γ(−β_{k} + r) ≠ 0 for all k ∈ ℤ and all i = 1, …, p. Hence it is enough to describe all indecomposable modules over S_{ℓ}.

Fix a ∈ S_{0} and define a simple S_{ℓ}-module S_{a} such that S_{a}(b) = δ_{a,b}𝕜 for all b ∈ S_{0} and all morphisms are trivial. Since S_{ℓ} defines a finite-dimensional algebra we have the following

Proposition 7.2. Any simple module over S_{ℓ} is isomorphic to S_{a} for some a ∈ S_{0}.

This is another confirmation of the fact that all weight spaces in any irreducible generic weight Y_{p}(𝔤𝔩_{2})-module are 1-dimensional. But this need not to be the case for indecomposable modules. We restrict ourselves to a full subcategory R_{ℓ}^{f} ⊆ R_{ℓ} which consists of weight modules V with dim V_{ψ} < ∞ for all ψ ∈ Supp V. We will establish the representation type of the category R_{ℓ}^{f} (finite, tame or wild). For necessary definitions we refer to [Dr].

To establish the representation type of the category R_{ℓ}^{f} it is enough to consider the category S_{ℓ}−mod^{f}, of modules over the category S_{ℓ} with finite-dimensional weight spaces. Denote X_{ℓ} = {k ∈ {1, …, p} | I_{k} ≠ ∅}.

7.3.1. Indecomposable modules in the case |X_{ℓ}| = 1. In this section we describe all indecomposable modules over S_{ℓ} in the case |X_{ℓ}| = 1. Let X_{ℓ} = {i} and let |I_{i}| = r > 0. In this case the category S_{ℓ} has the following quiver A with relations:

[figure: a_{1}; b_{1}; 1; 2; a_{r}; b_{r}; r + 1; ∘; a_{i} b_{i} = b_{i} a_{i} = 0; r; ∘; ∘; ∘ …]

We denote by S_{i}, i ∈ {1, …, r+1}, the simple module corresponding to the point i. These modules correspond to all irreducible modules in R_{ℓ}^{f} by Proposition 7.2. Now describe remaining indecomposable modules for a quiver above. Fix integers 1 ≤ k_{1} < k_{2} ≤ r + 1 and a function ξ_{k_{1},k_{2}} : {k_{1}, k_{1} + 1, …, k_{2}} → {0, 1}. Define a module M = M(k_{1}, k_{2}, ξ_{k_{1},k_{2}}) as follows: M(i) = 𝕜e_{i}, k_{1} ≤ i ≤ k_{2}, M(j) = 0 otherwise, a_{i}e_{i} = e_{i+1}, b_{i}e_{i+1} = 0 if ξ_{k_{1},k_{2}}(i) = 1 and a_{i}e_{i} = 0, b_{i}e_{i+1} = e_{i} if ξ_{k_{1},k_{2}}(i) = 0 for all 1 ≤ i < k_{2}.

The proof of the following proposition is standard; see e.g. [GR].

Proposition 7.3. The modules S_{i}, 1 ≤ i ≤ r + 1 and M(k_{1}, k_{2}, ξ_{k_{1},k_{2}}) with 1 ≤ k_{1} < k_{2} ≤ r + 1 and

ξ_{k_{1},k_{2}} : {k_{1}, k_{1} + 1, …, k_{2}} → {0, 1},

exhaust all non-isomorphic indecomposable modules for A.

7.3.2. Indecomposable modules in the case |X|_{ℓ} = 2. In this section we describe the indecomposable modules for S_{ℓ} when |X|_{ℓ} = 2 and |I_{k}| = 1 for each k ∈ X_{ℓ}. Then S_{ℓ} is isomorphic to the following category B considered in [BB].

[figure: B :; a_{2}; b_{2}; b_{0}; a_{0}; a_{1}; b_{1}; b_{3}; a_{3}; 1; 2; 0; 3; a_{i}b_{i} = b_{i}a_{i} = 0,  i = 0, …, 3,; a_{i}a_{j} = b_{l}b_{m} for any i, j, l, m ∈ {0, 1, 2, 3},; where possible.; ∘; ∘; ∘; ∘]

By Proposition 7.2 this category has four non-isomorphic simple modules S_{i}, 0 ≤ i ≤ 3, with a support in a chosen point i. The indecomposable modules were described in [BB]. For the sake of completeness we repeat here this classification.

We will treat the objects of B as elements of ℤ/4ℤ. Consider the following three families of non-simple indecomposable modules.

Finite family. Fix an 0 ≤ i ≤ 3 and define the B-module M_{i} such that M_{i}(j) = 𝕜e_{j} for each j = 0, …, 3 and a_{i}e_{i} = e_{i+1}, a_{i+1}e_{i+1} = e_{i+2}, b_{i−1}e_{i} = e_{i−2}, b_{i−2}e_{i−1} = e_{i−2} and u_{j}e_{k} = 0 for all other cases of u ∈ {a, b} and j, k = 0, …, 3. Obviously, M_{i} is indecomposable module for any i.

Infinite discrete families. Let n ∈ ℕ, n > 1, and j ∈ ℤ_{4}. Define a B-module M_{n,j,1} (resp., M_{n,j,2}) as follows. Consider n elements e_{1}, …, e_{n}. A 𝕜-basis of the vector space M_{n,j,1}(l) (resp., M_{n,j,2}(l)) is the set of e_{k} such that j + k − 1 ≡ l(mod 4). The elements a_{l} and b_{l−1} act as follows:

a_{l}e_{k} = \{ \begin{matrix}e_{k+1}, & if l is even (resp., odd), k < n and j + k − 1 ≡ l(mod 4); \\ 0, & otherwise. \end{matrix}

b_{l−1}e_{k} = \{ \begin{matrix}e_{k−1}, & if l is even (resp., odd), k > 1 and j + k − 1 ≡ l(mod 4); \\ 0, & otherwise. \end{matrix}

All modules M_{n,j,1} and M_{n,j,2}, n > 1, 0 ≤ j ≤ 3 are non-isomorphic indecomposable B-modules.

Infinite continuous families. For each λ ∈ 𝕜, λ ≠ 0, and d ∈ ℤ, d > 0 define the B-modules M_{d,λ,1} and M_{d,λ,2} as follows. Set

\begin{aligned}M_{d,λ,1}(i) & = 𝕜^{d}, \\ M_{d,λ,1}(a_{0}) & = M_{d,λ,1}(a_{2}) = M_{d,λ,1}(b_{1}) = I_{d}, \\ M_{d,λ,1}(b_{0}) & = M_{d,λ,1}(b_{2}) = M_{d,λ,1}(a_{1}) = M_{d,λ,1}(a_{3}) = 0, \\ M_{d,λ,1}(b_{3}) & = J_{d,λ} \end{aligned}

and

\begin{aligned}M_{d,λ,2}(i) & = 𝕜^{d}, \\ M_{d,λ,2}(b_{0}) & = M_{d,λ,2}(b_{2}) = M_{d,λ,2}(a_{1}) = I_{d}, \\ M_{d,λ,2}(a_{0}) & = M_{d,λ,2}(a_{2}) = M_{d,λ,2}(b_{1}) = M_{d,λ,2}(b_{3}) = 0, \\ M_{d,λ,2}(a_{3}) & = J_{d,λ}, \end{aligned}

where J_{d,λ} is the Jordan cell of dimension d with the eigenvalue λ.

All modules M_{d,λ,k}, k = 1, 2 are indecomposable and corresponding indecomposable modules in R_{ℓ}^{f} have all weight spaces of dimension d.

Proposition 7.4. ([BB], Proposition 3.3.1). The modules S_{i}, M_{i}, M_{n,i,1}, M_{n,i,2}, M_{d,λ,1}, M_{d,λ,2} where 0 ≤ i ≤ 3, d is a positive integer, λ ∈ 𝕜, λ ≠ 0, and n ≥ 2 is an integer, constitute an exhaustive list of pairwise non-isomorphic indecomposable B-modules.

The following theorem which describes the representation type of R_{ℓ}^{f}.

Theorem 6. (i) If |X_{ℓ}| = 0 then R_{ℓ}^{f} is a semisimple category with a unique

indecomposable (=irreducible) module; (ii) If |X_{ℓ}| = 1 then R_{ℓ}^{f} has finite representation type; (iii) If |X_{ℓ}| = 2 then R_{ℓ}^{f} has tame representation type if and only if |I_{k}| = 1 for

all k ∈ X. Otherwise, R_{ℓ}^{f} has wild representation type;

(iv) If |X_{ℓ}| > 2 then R_{ℓ}^{f} has wild representation type.

Proof. In the case when |X_{ℓ}| = 1 all indecomposable modules for S_{ℓ} are described in Proposition 7.3. Hence R_{ℓ}^{f} has finite representation type. If |X_{ℓ}| = 2 and |I_{k}| = 1 for each k ∈ X then all indecomposable modules for S_{ℓ} are described in Proposition 7.4. It follows from the definition that R_{ℓ}^{f} has tame representation type in this case. If |I_{k}| > 1 for at least one k then it is easy to construct a family of indecomposable modules that depends on two continuous parameters. Hence, in this case R_{ℓ}^{f} has wild representation type. Suppose now that |X_{ℓ}| > 2. Then S_{ℓ} contains a full subcategory of wild representation type considered in [BB], Theorem 1. We immediately conclude that R_{ℓ}^{f} has wild representation type. This completes the proof. □

Corollary 12. (1) If |X_{ℓ}| = 0 then the category R_{ℓ} is a semisimple category

with a unique indecomposable module.

(2) If |X_{ℓ}| = 1 then R_{ℓ} has finite representation type with indecomposable modules as in Proposition 7.3.

Proof. Since cases |X_{ℓ}| ≤ 1 correspond to finite representation type then the corresponding categories do not admit infinite-dimensional indecomposable modules by [A] and hence every indecomposable module belongs to R_{ℓ}^{f}. □

8. Acknowledgment

The first author is a Regular Associate of the ICTP and is supported by the CNPq grant (Processo 300679/97-1) The second and the third authors are grateful to FAPESP for the financial support (Processos 2001/13973-0 and 2002/01866-7) and to the University of São Paulo for the hospitality during their visits.

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Universidade de São Paulo, Caixa Postal 66281- CEP 05315-970, São Paulo, Brazil

E-mail address: futorny@ime.usp.br

School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

E-mail address: alexm@maths.usyd.edu.au

Faculty of Mechanics and Mathematics, Kiev Taras Shevchenko University, Vladimirskaya 64, 00133, Kiev, Ukraine

E-mail address: ovsienko@sita.kiev.ua