Irreducible Generic Gelfand-Tsetlin Modules of $\mathfrak{gl}(n)$

We provide a classification and explicit bases of tableaux of all irreducible generic Gelfand-Tsetlin modules for the Lie algebra $\mathfrak{gl}(n)$.

simplicity we work with gl(n) instead of sl(n). We obtain an explicit construction of all irreducible generic modules providing a Gelfand-Tsetlin type basis.
The organization of the paper is as follows. In Section 3 we introduce some basic definitions and preparatory results on Gelfand-Tsetlin modules. In Section 4 we list the Gelfand-Tsetlin formulas and use them to recall the classical result of Gelfand and Tsetlin for finite-dimensional gl(n)-modules. In Section 5 we introduce the notion of generic Gelfand-Tsetlin module and recall the classification of irreducible generic Gelfand-Tsetlin modules of gl (3). The main theorem in the paper, the classification of irreducible generic Gelfand-Tsetlin gl(n)-modules, is included in Section 6. In the last section we compute the number of irreducible Gelfand-Tsetlin modules in the so-called generic blocks. Acknowledgements. V.F. gratefully acknowledges the hospitality and excellent working conditions at the CRM, University of Montreal, where part of this work was completed. V.F. is supported in part by the CNPq grant (301320/2013-6) and by the Fapesp grant (2010/50347-9). D.G is supported in part by the Fapesp grant (2011/21621-8) and by the NSA grant H98230-13-1-0245. L. E. R. is supported by the Fapesp grant (2012/23450-9).

Notation and conventions
Throughout the paper we fix an integer n ≥ 2. The ground field will be C. For a ∈ Z, we write Z ≥a for the set of all integers m such that m ≥ a. Similarly, we define Z <a , etc. By gl(n) we denote the general linear Lie algebra consisting of all n × n complex matrices, and by {E i,j | 1 ≤ i, j ≤ n} -the standard basis of gl(n) of elementary matrices. We fix the standard Cartan subalgebra h, the standard triangular decomposition and the corresponding basis of simple roots of gl(n). The weights of gl(n) will be written as n-tuples (λ 1 , ..., λ n ).
For a Lie algebra a by U (a) we denote the universal enveloping algebra of a. Throughout the paper U = U (gl(n)). For a commutative ring R, by Specm R we denote the set of maximal ideals of R.
For i > 0 by S i we denote the ith symmetric group. Throughout the paper we set G := S n × · · · × S 1 .

Gelfand-Tsetlin modules
Recall that U = U (gl(n)). Let for m n, gl m be the Lie subalgebra of gl(n) spanned by {E ij | i, j = 1, . . . , m}. We have the following chain It induces the chain U 1 ⊂ U 2 ⊂ . . . ⊂ U n for the universal enveloping algebras U m = U (gl m ), 1 ≤ m ≤ n. Let Z m be the center of U m . The subalgebra of U generated by {Z m | m = 1, . . . , n} will be called the (standard) Gelfand-Tsetlin subalgebra of U and will be denoted by Γ ( [4]).
Definition 3.1. A finitely generated U -module M is called a Gelfand-Tsetlin module (with respect to Γ) if For each m ∈ Specm Γ we have associated a character χ m : Γ → Γ/m ∼ C. In the same way, for each non-zero character χ : Γ → C we have that Ker(χ) is a maximal ideal of Γ. So, we have a natural identification between characters of Γ and elements of Specm Γ. Using characters we can define Gelfand-Tsetlin modules.
Lemma 3.2. Any submodule of a Gelfand-Tsetlin module over gl(n) is a Gelfand-Tsetlin module.
Proof. The proof is standard, but for a sake of completeness, we provide the important details. Let M be a Gelfand-Tsetlin gl(n)-module and N any submodule of M . We will prove that, if {χ 1 , ..., χ k } is a set of distinct Gelfand-Tsetlin char- Without loss of generality we assume that k = 2. Since χ 1 = χ 2 , there exist g ∈ Γ and r ≤ s in Z ≥0 such that χ 1 (g) = χ 2 (g), (g − χ 1 (g)) r (v 1 ) = 0 and (g − χ 2 (g)) s (v 2 ) = 0. Let a := χ 1 (g) and b := χ 2 (g), Then, if We have that y ∈ N on one hand and on the other. As s k (a − b) s−k = 0 for any k, using that (g − a) r−1 y ∈ N , we obtain (g − a) r−1 v 1 ∈ N . Reasoning in the same way, from (g − a) r−i y ∈ N , and One can choose the following generators of Γ: Let Λ be the polynomial algebra in the variables {λ ij | 1 j i n}. The action of the symmetric group S i on {λ ij | 1 j i} induces the action of G = S n ×· · ·×S 1 on Λ. There is a natural embedding ı : Γ−→ Λ given by ı(c mk ) = γ mk (λ) where Hence, Γ can be identified with G−invariant polynomials in Λ.

Remark 3.3.
In what follows, we will identify the set Specm Λ of maximal ideals of Λ with the set C n(n+1) 2 . Then we have a surjective map π : Specm Λ → Specm Γ. Moreover, since Λ is integral over Γ, there are finitely many maximal ideals of Λ that map to a fixed maximal ideal of Γ. The different maximal ideals of Λ are obtained from each other under permutations in the group G.
If π(ℓ) = m for some ℓ ∈ Specm Λ, then we write ℓ = ℓ m and say that ℓ m is lying over m.

Finite dimensional modules of gl(n)
In this section we recall a classical result of Gelfand and Tsetlin which provides an explicit basis for every irreducible finite dimensional gl(n)-module. , by T (L) we will denote the following array with entries · · · ln,n−1 lnn Such an array will be called a Gelfand-Tsetlin tableau of height n. A Gelfand-Tsetlin tableau of height n is called standard if l ki −l k−1,i ∈ Z ≥0 and l k−1,i −l k,i+1 ∈ Z >0 for all 1 ≤ i ≤ k ≤ n − 1.
Note that, for sake of convenience, the second condition above is slightly different from the original condition in [15]. 15]). Let L(λ) be the finite dimensional irreducible module over gl(n) of highest weight λ = (λ 1 , . . . , λ n ). Then there exist a basis of L(λ) consisting of all standard tableaux T (L) = T (l ij ) with fixed top row l nj = λ j − j + 1. Moreover, the action of the generators of gl(n) on L(λ) is given by the Gelfand-Tsetlin formulas: if the new tableau T (L ± δ ki ) is not standard, then the corresponding summand of E k,k+1 (T (L)) or E k+1,k (T (L)) is zero by definition. Furthermore, for s ≤ r, where γ rs are defined in (4).
The formulas above are called Gelfand-Tsetlin formulas for gl(n).

Generic Gelfand-Tsetlin modules of gl(n)
Theorem 4.2 gives an explicit realization of any irreducible finite dimensional gl(n)-module. Using the Gelfand-Tsetlin formulas, Drozd, Futorny and Ovsienko defined the class of infinite-dimensional generic modules for gl(n) in [4].
) has a structure of a gl(n)-module with action of the generators of gl(n) given by the Gelfand-Tsetlin formulas. (ii) The action of the generators of Γ on the basis elements of V (T (L)) is given by (5).
) is a Gelfand-Tsetlin module all of whose Gelfand-Tsetlin multiplicities are 1. and z n1 = . . . = z nn = 0} By a slight abuse of notation we will identify elements in Z such that z n1 = . . . = z nn = 0. This will allow us to write T (L + z) for z ∈ Z n(n−1) 2 .

Remark 5.4.
In what follows, we will apply Lemma 3.2 and use that the elements of Γ separates the tableaux in the submodules of V (T (L)) in the following sense. Let N be a gl(n)-submodule of V (T (L)), g ∈ gl(n), and T (R) be a tableau in N .
Theorem 5.5. If n ∈ Specm Γ is generic, then there exists a unique irreducible Gelfand-Tsetlin module N such that N (n) = 0.
Proof. Let X n = U/U n. We know that X n = U/U n is a Gelfand-Tsetlin module. Furthermore, any irreducible Gelfand-Tsetlin module M with M (n) = 0 is a homomorphic image of X n , and X n (n) maps onto M (n). Since both spaces X n (n) and M (n) have additional structure as modules over certain algebra (see Corollary 5.3, [12]) then the projection X n (n) → M (n) is in fact a homomorphism of modules. Taking into account that dim X n (n) ≤ 1, we conclude that there exist a unique irreducible module N with N (n) = 0.
Definition 5.6. If T (R) is a generic tableau and r ∈ Specm Γ corresponds to R then, the unique module N such that N (r) = 0 is called the irreducible Gelfand-Tsetlin module containing T (R), or simply, the irreducible module containing T (R).
Our goal is to describe explicitly the irreducible Gelfand-Tsetlin module containing T (R) for every generic tableau T (R). Below we recall how this is achieved in the case n = 3 in [26]. One should note that the methods used in [26] involve direct computations based on a case-by-case consideration, while in the present paper we provide an invariant proof. Also, we reformulate the result in [26] in convenient for us terms.

Classification of irreducible generic Gelfand-Tsetlin gl(n)-modules
In this section we prove the main result in the paper, i.e. the generalization of Theorem 5.7 for gl(n). For convenience we introduce and recall some notation.  Proof. In order to prove that W (T (R)) is a submodule, it is enough to prove U ·T (Q) ⊆ W (T (R)) for any T (Q) = T (q ij ) ∈ N (T (R)). We will show g · T (Q) ∈ W (T (R)) for every (standard) generator g of gl(n). Suppose g = E k,k+1 for some 1 ≤ k ≤ n − 1. By the Gelfand-Tsetlin formulas, we have If E k,k+1 (T (Q)) / ∈ W (T (R)), then there exist k and i such that T (Q) ∈ N (T (R)) but T (Q + δ ki ) / ∈ N (T (R)). That implies Ω + (T (R)) ⊆ Ω + (T (Q)) and Ω + (T (R)) Ω + (T (Q + δ ki )).
We consider now each of the two cases separately.
The proof that E k+1,k (T (Q)) ∈ W (T (R)) is analogous to the one for E k,k+1 (T (Q)) ∈ W (T (R)). The case g = E kk is trivial because E kk acts as a multiplication by a scalar on T (Q) and T (Q) ∈ N (T (R)) ⊆ W (T (R)).
Given any tableau T (R), there are three modules containing T (R): V (T (L)), W (T (R)) and U · T (R). We will show that W (T (R)) = U · T (R). For this we need the following lemmas. is such that Ω + (T (L)) ⊆ Ω + (T (L + z)) then, there exist i, j such that z ij = 0 and (6) Ω + (T (L)) ⊆ Ω + (T (L + z ij δ ij )) ⊆ Ω + (T (L + z)) Proof. We will use the following definition in the proof of the lemma.
Lets consider k, l such that z kl = 0. Set for convenience Q := L + z. There exists a maximal chain C in T (Q) of length ℓ, starting in row d such that q kl ∈ C. (6) is obvious for z ij = z kl . Let a and b be the minimum and maximum of {i : z [i] = 0}, respectively. We have , respectively. Now, let a ≤ m ≤ b and consider the following 4 cases. Definition 6.5. Given T (Q) and T (R) in B(T (L)), we say that T (R) (1) T (Q) if there exist g ∈ gl(n) such that T (Q) appears with non-zero coefficient in the decomposition of g · T (R) into a linear combination of tableaux. For any p ≥ 1 we say that T (R) (p) T (Q) if there exist tableaux T (L (1) ),..., T (L (p) ), such that As an immediate consequence of the definition of (p) we have the following.
Proof. By Lemma 5.4 and the definition of the relation (1) , we verify that T (R) (1) T (Q) implies T (Q) ∈ U · T (R). Now, by Lemma 6.6(i), if T (R) (p) T (Q) for some p then T (Q) ∈ U · T (R).
The next theorem provides a convenient basis for the submodule of V (T (L)) generated by a fixed tableau. Recall the definition of N (T (R)) in Notation 6.1(iii)(d).
Theorem 6.8. For any tableau T (R) ∈ B(T (L)), U · T (R) = W (T (R)). In particular, N (T (R)) forms a basis of U · T (R), and the action of gl(n) on U · T (R) is given by the Gelfand-Tsetlin formulas.
Proof. By Proposition 6.2, U ·T (R) ⊆ W (T (R)). To prove that W (T (R)) ⊆ U ·T (R) we will show that T (Q) ∈ U · T (R) for any T (Q) ∈ N (T (R)). By Corollary 6.7, it is enough to prove that T (R) (p) T (Q) for some positive integer p.
Suppose that T (Q) = T (R + z) ∈ N (T (R)) for some z ∈ Z n(n−1) 2 . Let t be the number of non-zero components of z. We will prove that T (R) (p) T (Q) using induction on t.
Let now t = 1 and z ij < 0. Using the same arguments as in the case z ij > 0, we prove that T (R) (−zij) T (Q) using |z ij | applications of E i+1,i . This completes the proof for t = 1.
6.2. Basis for irreducible modules containing a given tableau. By Theorem 6.8, the module generated by a tableau T (R) has basis N (T (R)). For the purpose of the next theorem let us introduce the following equivalence on C n(n+1) 2 : Definition 6.13. We write z ∼ w for z, w ∈ C n(n+1) 2 if and only if one of the two cases hold.
(i) z − w ∈ Z n(n−1) 2 Now we are ready to formulate and prove the main theorem in the paper. Proof. For each tableau T (R), we have an explicit construction of the module containing T (R) (recall Definition 5.6): where the sum is taken over tableaux T (Q) such that T (Q) ∈ U · T (R) and U · T (Q) is a proper submodule of U · T (R). The module M (T (R)) is simple. Indeed, this follows from the fact that for any nonzero tableau T (S) in M (T (R)) we have U · T (S) = U · T (R) and, hence, T (S) generates M (T (R)).
To show that C n(n+1) 2 gen / ∼ parameterizes the set of all irreducible generic Gelfand-Tsetlin modules we use Theorem 5.5 and the fact that ℓ, ℓ ′ ∈ Specm Λ lie over the same m in Specm Γ if and only if ℓ ∈ Gℓ ′ (see Remark 3.3).