Extension Fullness of the Categories of Gelfand-Zeitlin and Whittaker Modules

We prove that the categories of Gelfand-Zeitlin modules of $\mathfrak{g}=\mathfrak{gl}_n$ and Whittaker modules associated with a semi-simple complex finite-dimensional algebra $\mathfrak{g}$ are extension full in the category of all $\mathfrak{g}$-modules. This is used to estimate and in some cases determine the global dimension of blocks of the categories of Gelfand-Zeitlin and Whittaker modules.


Introduction
Homological invariants are useful technical tools in modern representation theory. As classification of all modules of a given (Lie) algebra is a wild problem in almost all non-trivial and interesting cases (see e.g. [Dr, FNP, BKM]), the usual "reasonable" setup for the study of representations of a given (Lie) algebra assumes some fixed subcategory of the category of all modules. Therefore, the problem to compare homological invariants for a given category and some of its subcategories is natural and important.
Motivated by the so-called Alexandru conjecture, see [Fu, Ga], in our previous paper [CM] we compared extensions in the (thick) category O associated with a semi-simple complex finite dimensional Lie algebra g and in the category of all gmodules. As a bonus, we determined the global dimension of the thick category O as well as projective dimensions of its simple objects.
Although category O is, probably, the most studied category of g-modules, there are several other natural and well-studied categories which have rather different flavor. One of them is the category G Z of so-called Gelfand-Zeitlin 1 modules, introduced in [DFO1] for the algebra sl 3 (C), in [DFO2] for the algebra gl n (C) and in [Ma2] for orthogonal Lie algebras. The category G Z can be seen as a generalization of O in the sense that it contains both O and the thick version of O. Study of Gelfand-Zeitlin modules attracted considerable attention, see e.g. [DFO3,Ov1,Ov2,Kh,KM,Ma1,Ma3,MO,MS,FO1,FO2,Ra,FGR] and references therein. As far as we know, simple generic Gelfand-Zeitlin modules give the richest known family of simple gl n (C)-modules, it depends on n(n+1) 2 generic parameters. The first main result of this paper is the following statement proved in Section 3 (we refer to Sections 2 and 3 for more details): Theorem 1. The category G Z is extension full in the category of all gl n -modules. 1 The surname Zeitlin is spelled Ceȋtlin in Russian. It appeared in different transliterations in latin script, in particular, as Cetlin, Zetlin, Tzetlin and Tsetlin. However, it seems that the origin of this surname is the German word "Zeit" which justifies our present version.
The added difficulty of the category G Z in comparison with thick category O in [CM] is that G Z is not a Serre category generated by a well-known category which has enough projective objects (contrary to the relation between category O and thick category O). Therefore to prove Theorem 1 we have to modify and strengthen the abstract results on extension fullness in [CM]. Our arguments also heavily use some properties of Gelfand-Zeitlin modules established by Ovsienko in [Ov2] and by Futorny and Ovsienko in [FO2].
Another big class of g-modules, where now g is an arbitrary semi-simple Lie algebra with a fixed triangular decomposition g = n − ⊕ h ⊕ n + , is the class of so-called Whittaker modules introduced by Kostant in [Ko]. Simple Whittaker modules are simple g-modules on which the algebra U (n + ) acts locally finitely, see also [BM] for a general Whittaker setup. These modules were studied in [MiSo,McD1,McD2,KhMa,Se] in the classical setup. Generalizations of these modules for (infinite dimensional) Lie algebras and some related algebra attracted a lot of attention recently, see [On,OW1,Ch,BO,BCW,BM,GL,OW2] and references therein.
We define the category W of Whittaker modules of finite length and for this category we prove the next statement, which is our second main result (we refer to Section 4 for more details): Theorem 2. The category W is extension full in the category of all g-modules.
Using adjunction, the study of W reduces to the study of locally finite modules over a certain noetherian algebra. An added difficulty compared to the case G Z is that the needed locally finite modules do not decompose into an (infinite) direct sum of finite dimensional modules. It turns out that the indecomposable submodules of restricted Whittaker modules might not be finitely generated. To be able to prove extension fullness we crucially depend on a result of Donkin and Dahlberg from the 80's (see [Do] and [Da]) asserting that essential extensions of locally finite modules over solvable finite dimensional Lie algebras are locally finite.
As a consequence of Theorems 1 and 2, we obtain that the global dimension of the categories G Z and W equals dim g. As both category O and its thick version are full subcategories in G Z or W (thick category O is even a Serre subcategory), combining Theorem 1 and 2 with the results of [CM] gives, in particular, a lower bound on projective dimension of simple highest weight and Verma modules in the category G Z and W.
The paper is organized as follows: in Section 2 we prove some preliminary homological algebra statements, Section 3 deals with the case of Gelfand-Zeitlin modules and Section 4 is devoted to the case of Whittaker modules.

Extension full subcategories
2.1. Extensions in abelian categories. Let A be an abelian category and M, N ∈ A. Recall that, for d ∈ Z ≥0 , the set Ext d The set Ext d A (M, N ) has the natural structure of an abelian group via the Baer sum. If A is k-linear for some field k, then Ext d A (M, N ) has the structure of a k-vector space. We refer to [We,Section 3.4] for further information and details. 2.2. Extension full subcategories. Let A be an abelian category and B a full abelian subcategory of A in the sense that the abelian structure of B is inherited from A. In particular, the natural inclusion functor ι : B → A is exact. Then, for every M, N ∈ B and every d ∈ Z ≥0 , the functor ι induces homomorphisms Let 0 → K → M → N → 0 be a short exact sequence in B. Then, for Q ∈ B, application of Hom B (Q, − ) and Hom A (Q, − ) to this short exact sequence produces the usual long exact sequences in homology for the categories B and A, respectively. Moreover, the homomorphisms ι d Q,− give rise to a homomorphism between these long exact sequences. Similarly for Hom B ( − , Q) and Hom A ( − , Q).
We say that B is extension full in A provided that ι d M,N are bijective for all d ∈ Z ≥0 and for all M, N ∈ B. We refer to [CM,Section 2] for details.
2.3. Checking extension fullness. In this section we formulate and prove three propositions which will be useful for our study of extension fullness later in the paper. The following statement is a modification of [CM,Lemma 4] which also allows for a somewhat stronger formulation. Then B is extension full in A.
Proof. Our proof is similar to that of [CM,Lemma 4]. We prove the statement by induction on d. Since B is assumed to be a Serre subcategory of A, it is clear that ι 0 Q,N and ι 1 Q,N are isomorphisms for all Q, N ∈ B.
To prove the induction step, for Q ∈ B consider a short exact sequence with M ∈ B 0 and K ∈ B, which exists by condition (a). Applying Hom B ( − , N ) and Hom A ( − , N ) gives for each d the following commutative diagram: Let us fix the following notation: for an associative algebra A over a field k denote by A-Mod the category of all A-modules. We also denote by A-mod the full subcategory of A-Mod consisting of all finitely generated modules. We denote by A-flmod the full subcategory of A-Mod consisting of all modules on which the action of A is locally finite. Finally, we denote by A-fmod the full subcategory of A-mod consisting of all finite dimensional modules.
Proposition 5. Consider an associative algebra A, a full abelian subcategory A in A-Mod, and a full abelian subcategory B of A. Assume that these data satisfy the following conditions: Assume that there is a commutative diagram in A with exact rows as follows: For i = 1, 2, . . . , d, let Q i denote the submodule of Z i generated by the images of Y i and X i . Since B is a Serre subcategory of A-Mod, we have that Q i belongs to B. Then diagram (3) restricts to the commutative diagram in which the complex in the second row might be not exact. By assumption, there is a submodule in Z d which surjects onto N and is in B. The sum of that submodule with Q d is also in B since B is a Serre subcategory of A-Mod. We denote this resulting submodule by T d . The kernel K d of the surjection T d ։ N is also in B as B is abelian. By the same reasoning there is a submodule T d−1 of Z d−1 which is in B, contains Q d−1 , maps to T d and surjects onto K d . Proceeding inductively, we construct, for each i = 1, 2, . . . , d − 2, a submodule T i of Z i which is in B, contains Q i , maps to T i+1 and surjects onto the kernel of the map from T i+1 to T i+2 . This gives the commutative diagram in B with exact rows. The above implies that the natural map The construction above also says that for an arbitrary exact sequence with exact rows and such that the second row is in B. This means that the natural map (5) is surjective and hence bijective, completing the proof.
As an immediate corollary from Proposition 5 we obtain: Corollary 6. For an associative algebra A we have that 2.4. Adjunction lemma. The following statement is standard when dealing with categories with enough projective or injective objects. We failed to find it in the literature in the generality we need, so we provide a proof without the use of projective or injective objects.
Proposition 7 (Adjunction lemma). Let A and B be two abelian categories and (F, G) an adjoint pair of exact functors F : A → B and G : B → A. Then for every Proof. Applying F to an exact sequence Denote by K the kernel of the adjunction natural transformation FG → Id B . Then we have the following commutative diagram with exact rows. Here the homomorphism from the third to the last row is given by the natural projection of GF ։ C and the second row is just the corresponding kernel with the morphism from the second to the third row being the canonical injection. In particular, C(N ) := GF(M )/N ′ . The homomorphism from N to N ′ is the natural surjection and, finally, Y ′′ is defined as the pullback and the map from G(Y d−1 ) is given by the universal property of pullbacks. Functoriality of the construction yields a group homomorphism Finally, we claim that Φ and Ψ are inverses of each other. Consider the following commutative diagram: Here the second row is obtained from the first one by applying GF and the homomorphism from the first to the second row is given by adjunction Id A → GF.
The second and the third rows and the homomorphism between them are given by applying the exact functor G to the two middle rows of (6). Note that, by construction, application of G identifies the two last rows of (6), that is G(M ′ ) ∼ = G(M ) and G(X ′ ) ∼ = G(X ′′ ) (this follows from surjectivity of the natural transformation GFG → G given by adjunction). Consequently, from the adjunction identities we have that the composition of the maps in the first column is the identity on G(M ). Finally, the last row and the homomorphism to the third row are given by the definition of Ψ. In particular, from the construction we have that the image of N in the second row (coming from the first row) and in the third row (coming from the last row) coincide. Hence diagram (8) gives rise to a diagram with exact rows, where N ′ is the image of N in GF(N ) and Q ′ is the full preimage of N ′ . Pulling back along the epimorphism N ։ N ′ gives a commutative diagram with exact rows. The last diagram shows that the extensions given by the first and the last rows coincide, which proves that ΨΦ is the identity map.
The claim that ΦΨ is the identity map is proved similarly. This completes the proof.

Gelfand-Zeitlin modules
Notation: For a Lie algebra a we denote by U (a) its universal enveloping algebra and by Z(a) the center of U (a).
3.1. Gelfand-Zeitlin subalgebra of gl n . For k ∈ Z >0 denote by g k the Lie algebra gl k (C). Set U k = U (g k ) and Z k = Z(g k ). We consider the usual chain g 1 ⊂ g 2 ⊂ g 3 ⊂ · · · ⊂ g n of "left upper corner" embeddings of Lie algebras as depicted on the following picture: This gives rise to the chain U 1 ⊂ U 2 ⊂ U 3 ⊂ · · · ⊂ U n of embeddings of associative algebras. The subalgebra Γ of U := U n generated by all centers Z k , where k = 1, 2, . . . , n, is called the Gelfand-Zeitlin subalgebra. We set g := g n .
The algebra Γ is, obviously, commutative. Moreover, the fact that Γ-characters which appear in finite dimensional g-modules form an algebraically dense set in C n(n+1) 2 , see [GZ] or [Zh, Chapter X], implies that Γ is a polynomial algebra in n(n+1) 2 variables (these can be taken to be generators of Z k for k = 1, 2, . . . , n). Furthermore, U is free both as a left and as a right Γ-module, see [Ov1]. Consequently, the usual induction and coinduction functors 3.2. Gelfand-Zeitlin modules. The category G Z of Gelfand-Zeitlin-modules for U is defined as the full subcategory of U -mod which consists of those U -modules M on which the action of Γ is locally finite, that is dim(Γv) < ∞ for all v ∈ M . Basic properties of G Z are collected in the following statement.

Proposition 8. The category G Z is a Serre subcategory of U -Mod and each object in G Z has finite length.
Proof. That G Z is a Serre subcategory of U -Mod follows directly from the definition. If m is a maximal ideal of Γ, then U/U m has finite length by [FO2,Theorem 4.14]. Consequently, Ind U Γ N has finite length for any finite dimensional Γ-module N .
Each M ∈ G Z is finitely generated, in particular, it is generated by a finite dimensional Γ-submodule N . Now, by adjunction, the identity endomorphism of N gives rise to a homomorphism Ind U Γ N ։ M which is surjective as N generates M . Therefore M is a quotient of Ind U Γ N and thus has finite length as well.
Write Specm(Γ) for the set of maximal ideals in Γ. We have the usual decomposition where Γ-fmod m denotes the full subcategory of Γ-fmod consisting of all objects annihilated by some power of m. We have the exact direct summand Ind U Γ,m : Γ-fmod m → G Z of the induction functor, which is left adjoint to the direct summand Res U Γ,m : G Z → Γ-fmod m , of the restriction functor, by [FO2,Corollary 5.3(a)]. The latter can also be defined as first restricting the action of U to the unital subalgebra Γ and then taking the Γ-fmod m -component. Note that modules in G Z are usually infinite dimensional and hence the usual restriction Res U Γ ends up in Γ-lfmod and not in Γ-fmod. The functor Res U Γ,m is left adjoint to the exact direct summand Coind U Γ,m : Γ-fmod m → G Z of the coinduction functor.
3.3. The main result. Our main result in this section is the following: Theorem 9. The category G Z is extension full in U -Mod.
3.4. Proof of Theorem 9. To prove Theorem 9, we would like to apply Proposition 3 for A = U -Mod, B = G Z and B 0 being the full subcategory of B consisting of all U -modules isomorphic to Ind U Γ N for some finite-dimensional Γ-module N . Therefore Theorem 9 follows from the following lemma: Proof. By additivity, we may assume N ∈ Γ-fmod m for some m ∈ Specm(Γ). The image of the functor Ind U Γ,m : Γ-fmod m → A belongs to B. We show that for every d ∈ Z ≥0 , any N ∈ Γ-fmod m and any Q ∈ G Z we have the isomorphisms Since, by construction, these three isomorphisms together with the morphism ι d M,Q will yield a commutative square, we get that ι d M,Q is an isomorphism.
3.5. Estimates for the global dimension.
Proof. It is well-known, see e.g. [We,Corollary 7.7.3], that the trivial g-module C has maximal possible projective dimension in U -Mod, namely dim g. Since we obviously have C ∈ G Z, the claim follows from Theorem 9.
Remark 12. The category G Z decomposes into a direct sum of indecomposable blocks, see [DFO3,Theorem 24]. The proof of Corollary 11 implies that the block containing the trivial g-module has global dimension dim g = n 2 . However, most of the blocks in G Z, namely all the so-called strongly generic blocks in the sense of [MO,Section 3], are equivalent to the category of finite dimension modules over Γ m , the completion of Γ with respect to a maximal ideal m (this equivalence is induced by Ind U Γ,m ), and hence have global dimension n(n+1)

2
, that is the Krull dimension of Γ.

Whittaker modules
4.1. The category of Whittaker modules. Let g be a semi-simple finite dimensional complex Lie algebra with a fixed triangular decomposition g = n − ⊕ h ⊕ n + . Consider the subalgebra R = Z(g)U (n + ) in U (g). Note that R is not commutative unless g is a direct sum of copies of sl 2 . Set U = U (g).
Denote by W the full subcategory of U -mod consisting of all g-modules which are locally finite with respect to the action of R (cf. [McD1,Definition 1.5]). Objects in W will be called Whittaker modules, which is a slight modification of the original notion from [Ko].

4.2.
Simple finite dimensional R-modules. In order to better understand the category W, we start with a classification of simple finite dimensional R-modules. For this we would need the following fact.
Proposition 13. The algebra R is isomorphic to Z(g) ⊗ C U (n + ).
Proof. From the injectivity of the Harish-Chandra homomorphism Z(g) → U (h) we know that the U (h)-components of different elements in Z(g) are different. Hence the PBW Theorem implies that the surjective homomorphism Z(g) ⊗ C U (n + ) ։ R given by multiplication is, in fact, injective, and hence an isomorphism (see also [Ko,Section 3.3

]).
Fix a maximal ideal m in Z(g) and a linear map χ : n + /[n + , n + ] → C and denote by V m,χ the space C endowed with the action of Z(g) via the projection Z(g)/m ∼ = C and with the action of U (n + ) via χ. The following statement shows that this gives a complete and irredundant list of pairwise non-isomorphic simple finite dimensional R-modules.

Proposition 14.
(i) For each m and χ as above, V m,χ is a simple R-module.
(ii) Each simple finite dimensional R-module is isomorphic to V m,χ for some m and χ as above.
Proof. We have that V m,χ is a Z(g) ⊗ C U (n + )-module by construction. Hence claim (i) follows from Proposition 13. Claim (iii) is clear by construction.
To prove claim (ii), we note that R is a finitely generated complex algebra and hence every simple R-module admits a central character by Dixmier's version of Schur's Lemma, see [Di,Proposition 2.6.8]. Therefore from Proposition 13 it follows that simple finite dimensional R-module are exactly simple finite dimensional U (n + )modules with the action of Z(g) given via the projection Z(g)/m ∼ = C for some m.
Since n + is nilpotent, all simple finite dimensional n + have dimension one by Lie's Theorem, see [Di,Corollary 1.3.13]. The claim follows.
4.3. The categories R-fmod and R-lfmod. For a maximal ideal m in Z(g) and a linear map χ : n + /[n + , n + ] → C denote by R-fmod m,χ the full subcategory of R-fmod consisting of all modules for which all simple composition subquotients are isomorphic to V m,χ . Define similarly the subcategory R-lfmod m,χ of R-lfmod. Proof. To prove the claim it is sufficient to check that If m = m ′ , then (12) is clear as Z(g) is central in R. Assume χ = χ ′ and consider some short exact sequence As χ = χ ′ , there is a ∈ n + whose action on M has two different eigenvalues χ(a) = χ ′ (a). Let v and w be the corresponding non-zero eigenvectors in M . Then Cw coincides with the image of V m,χ ′ in M . We claim that Cv is an R-submodule (which would mean that the short exact sequence (13) splits thus completing the proof).
On the other hand, abv = χ(a)bv + [a, b]v. As [a, b] ∈ X 1 , X 1 v ⊂ Cv and v and w are linearly independent, we get [a, b]v = (χ ′ (a) − χ(a))βw = 0, that is β = 0. Therefore bv ∈ Cv and thus X 2 v ⊂ Cv. Proceeding inductively, we get n + v ⊂ Cv and the proof is complete.
Note that the algebra R is isomorphic to the enveloping algebra of the nilpotent Lie algebra n + ⊕ a, where a is an abelian Lie algebra of dimension dim(h), see Proposition 13. In particular R ∼ = U (n) for some solvable Lie algebra n. Therefore we will be able to make use of the following result of [Da, Do] (see also [Fe]), see [Da,Proposition 1].
Lemma 16. Let n be a solvable Lie algebra. Any module V ∈ U (n)-lfmod has an injective hull I V in U (n)-Mod, moreover, I V ∈ U (n)-lfmod.
We note that existence of injective hulls in U (n)-Mod is due, in much bigger generality, to Baer, [Ba], see also [ES]. Lemma 16 implies the following.
Proposition 17. Let n be a solvable Lie algebra. Then U (n)-lfmod has enough injective objects and U (n)-lfmod is extension full in U (n)-Mod.
Proof. As U (n)-lfmod is a Serre subcategory of U (n)-Mod, the injective hulls in U (n)-Mod of Lemma 16 are automatically injective hulls in U (n)-lfmod. This proves that U (n)-lfmod has enough injective objects.
As U (n)-lfmod is a locally noetherian Grothendieck category (see [Kr, Appendix A] or [Ro]), it follows that each injective object in this category is a coproduct of indecomposable injective objects and this decomposition is unique up to isomorphism. Therefore any injective object in U (n)-lfmod is also injective when regarded as a module in U (n)-Mod. The extension fullness thus follows from Proposition 4.
As a consequence, we obtain the following statement.
Corollary 18. The category R-lfmod is extension full in R-Mod.

R versus U (g).
Proposition 19. The algebra U (g) is free both as a left and as a right R-module.
Proof. We prove the statement for the right module structure. For the left module structure the proof is similar. Choose a basis Y 1 , Y 2 , . . . , Y k in n − , a basis H 1 , H 2 , . . . , H l in h and a basis X 1 , X 2 , . . . , X k in n + . Then } is a basis in U (g) by the PBW theorem. For an element v in the above basis we set t v = l i=1 t i . Choose a basis f 1 , f 2 , . . . , f m of U (h) as the free module over the algebra of invariants in U (h) with respect to the dot-action of the Weyl group W of g. This algebra of invariants is exactly the image of Z(g)under the Harish-Chandra isomorphism, moreover, m = |W |. We fix free generators of this algebra of invariants as p 1 , p 2 , · · · , p l . Then we define free generators z 1 , z 2 , . . . , z l of Z(g) as a polynomial algebra, where we impose z i − p i ∈ U (g)n + .
The set B ′ := {z q1 1 z q2 2 · · · z q l l X r1 1 X r2 2 · · · X r k k : q i , r i ∈ Z ≥0 } is a basis of R by Proposition 13. Multiplying Y s1 1 Y s2 2 · · · Y s k k f j on the right with z q1 1 z q2 2 · · · z q l l X r1 1 X r2 2 · · · X r k k yields Y s1 1 Y s2 2 · · · Y s k k f j p q1 1 p q2 2 · · · p q l l X r1 1 X r2 2 · · · X r k k up to a linear combination of elements v in the basis (14) that have strictly lower value of t v than the highest one in the expansion of the product. Therefore we see that B is a basis of U (g) as a free right R-module.
Proposition 20. For every finite dimensional R-module V the induced g-module M (V ) := Ind Proof. It is enough to prove the claim for V = V m,χ , where m is a maximal ideal in Z(g) and χ : n + /[n + , n + ] → C. In this case the statement follows from [McD1,Theorem 2.8].
Corollary 21. Every object in W has finite length.
Proof. Each M ∈ W is generated, as a g-module, by some finite dimensional Rsubmodule V . By adjunction, M is thus a quotient of M (V ) and hence the claim follows from Proposition 20. 4.5. The main result. Our main result in this section is the following: Theorem 22. The category W is extension full in U -Mod. 4.6. Proof of Theorem 22. Before proving Theorem 22, we prove extension fullness of the category W which is defined as the full subcategory of U -Mod consisting of all modules which are locally R-finite. The difference between W and W is that we drop the condition of being finitely generated.
We apply Proposition 3 for A = U -Mod, B = W and B 0 being the full subcategory of B consisting of all U -modules isomorphic to M (V ) for some V ∈ R-flmod.
Lemma 23. Let Q ∈ W and V ∈ R-lfmod. Then ι d M(V ),Q is an isomorphism for every d ∈ Z ≥0 .
Proof. The image of the functor Ind U R : R-lfmod → A belongs to B while the image of the functor Res U R : B → R-Mod belongs to R-lfmod. Therefore, to prove our lemma, it is enough to show that for every d ∈ Z ≥0 , any N ∈ R-lfmod and any Q ∈ W we have the isomorphisms Isomorphisms (15) and (17) follow from Adjunction lemma (Proposition 7) while isomorphism (16) is Proposition 18. The claim follows. Lemma 23 and Proposition 3 thus imply that W is extension full in R-Mod. Hence, to complete the proof of Theorem 22, it remains to note that, by Proposition 5, the category W is extension full in W.

4.7.
Estimates for the global dimension.
Remark 25. The category W has a decomposition where W m,χ is the full subcategory consisting of all modules which restrict to R-lfmod m,χ , see [BM,Theorem 9]. Corollary 11 says that one of these blocks, namely the one corresponding to the trivial central character and trivial χ has global dimension dim g. This particular block contains many simple objects. Most of the blocks contain only one simple object and are expected to have smaller global dimension. Note also that thick category O is a Serre subcategory of W.