ON THE EXISTENCE OF (𝔤, 𝔨)-MODULES OF FINITE TYPE

Ivan Penkov ^{1}, Vera Serganova ^{2} and Gregg Zuckerman ^{3}

Abstract. Let 𝔤 be a reductive Lie algebra over an algebraically closed field of characteristic zero, and 𝔨 be a subalgebra reductive in 𝔤. We prove that 𝔤 admits an irreducible (𝔤, 𝔨)-module M which has finite 𝔨-multiplicities and which is not a (𝔤, 𝔨′)-module for any proper inclusion of reductive subalgebras 𝔨 ⊂ 𝔨′ ⊂ 𝔤, if and only if 𝔨 contains its centralizer in 𝔤. The main point of the proof is a geometric construction of (𝔤, 𝔨)-modules which is analogous to cohomological induction. For 𝔤 = 𝔤𝔩 (n) we show that, whenever 𝔨 contains its centralizer, there is an irreducible (𝔤, 𝔨)-module M of finite type over 𝔨 such that 𝔨 coincides with the subalgebra of all g ∈ 𝔤 which act locally finitely on M. Finally, for a root subalgebra 𝔨 ⊂ 𝔤𝔩 (n), we describe all possibilities for the subalgebra 𝔩 ⊃ 𝔨 of all elements acting locally finitely on some M.

Math. Subject Classification 2000: Primary 17B10 , Secondary 22E46 .

To Israel Moiseevich Gelfand on his 90^{th} birthday

1. Introduction

Let 𝔤 be a reductive Lie algebra over an algebraically closed field of characteristic zero and 𝔨 ⊂ 𝔤 be a subalgebra reductive in 𝔤. In his program talk [G], I. Gelfand has introduced the notion of a (𝔤, 𝔨)-module with finite 𝔨-multiplicities. The present paper focuses on a new notion relevant to Gelfand’s program: we call 𝔨 primal if 𝔤 admits an irreducible (𝔤, 𝔨)-module with finite 𝔨-multiplicities which is not a (𝔤, 𝔨′)-module for any proper inclusion of reductive subalgebras 𝔨 ⊂ 𝔨′ ⊂ 𝔤. Our central result is that 𝔨 is primal if and only if 𝔨 contains it centralizer in 𝔤, or equivalently, if and only if 𝔨 is a direct sum of a semisimple subalgebra 𝔨′ in 𝔤 and a Cartan subalgebra of the centralizer C(𝔨′) in 𝔤. This provides a complete description of all primal subalgebras, as the semisimple subalgebras of a reductive Lie algebra have been classified by E. Dynkin, [D].

2 ON THE EXISTENCE OF (𝔊, 𝔎)-MODULES OF FINITE TYPE

Here is a brief account of our motivation. It is common wisdom that classifying all irreducible representations of a reductive Lie algebra 𝔤 is not a well-posed problem. In contrast with that, classifying irreducible representations with natural finiteness properties has remained a core problem in representation theory since the work of E. Cartan and H. Weyl. A landmark success has been the celebrated classification of irreducible Harish-Chandra modules (see [V], Ch. 6 and [KV], Ch. 11). The case of (G_{2}, 𝔰𝔩(3))-modules has been considered by P. Kekäläinen in [Ke], and by G. Savin in [S]. In 1998 O. Mathieu (following up on work of S. Fernando and others) obtained a very important different classification: of irreducible weight modules with finite-dimensional weight spaces, [M].

1Work supported in part by an NSF GIG grant, by the Max-Planck-Institute for Mathematics in Bonn, and by the MSRI.

2Work supported in part by an NSF grant and the Max-Planck-Institute for Mathematics in Bonn.

3Work supported in part by an NSF grant and the MSRI.

In [PS] it was noticed that both classifications, of irreducible Harish-Chandra modules and of irreducible weight modules, are particular cases of the problem of classifying irreducible 𝔤-modules which have finite type over their Fernando-Kac subalgebra. The Fernando-Kac subalgebra 𝔤 [M] associated to an irreducible 𝔤-module M is by definition the set of all elements in 𝔤 which act locally finitely on M. The fact that 𝔤 [M] is a Lie subalgebra in 𝔤 was discovered independently by S. Fernando, [F], and V. Kac, [K]. Furthermore, M is of finite type over a given subalgebra 𝔩 ⊂ 𝔤 [M] if the multiplicity of an arbitrary fixed irreducible 𝔩-module in any (varying) finite-dimensional 𝔩-submodule of M is bounded. The subalgebra 𝔩 is called a Fernando-Kac subalgebra of finite type if 𝔤 admits an irreducible 𝔤-module M with 𝔤 [M] = 𝔩 which is of finite type over 𝔩. The problem of classifying all, not necessarily reductive, Fernando-Kac subalgebras of finite type is of fundamental importance for the structure theory of 𝔤-modules. In this article we classify the reductive parts of Fernando-Kac subalgebras of finite type, as a subalgebra is primal if and only if it is a reductive part of a Fernando-Kac subalgebra of finite type.

A short outline of the paper is as follows. In Section 3 we establish some necessary, (but in general not sufficient) conditions for a subalgebra 𝔩 ⊂ 𝔤 to be a Fernando-Kac subalgebra of finite type. We show in particular that a Fernando-Kac subalgebra of finite type 𝔩 is algebraic and admits a natural decomposition 𝔩 = 𝔩_{red}⊃^{+} 𝔫_{𝔩}, where 𝔩_{red} is a reductive in 𝔤 subalgebra which contains its centralizer, and 𝔫_{𝔩} is a nilpotent ideal in 𝔩. We also characterize completely all solvable Fernando-Kac subalgebras of finite type in 𝔤. In Section 4 we fix an arbitrary algebraic subalgebra 𝔨, reductive in 𝔤, and construct irreducible (𝔤, 𝔨)-modules M of finite type over 𝔨. The construction of M is a D-module version of cohomological induction: M equals the global sections of a D^{μ}- module supported on the preimage in G/B of K · P ⊂ G/P for a suitable parabolic subgroup P ⊂ G. Here G is a connected algebraic group with Lie algebra 𝔤 and K is a subgroup with Lie algebra 𝔨. We show then, that if 𝔨 contains its centralizer in 𝔤, 𝔤 [M]_{red} = 𝔨 for some M. Therefore, 𝔨 is primal if and only if it contains its centralizer. Furthermore, as a corollary we obtain that any semisimple subalgebra of 𝔤 is the derived subalgebra of a primal subalgebra, and that any subalgebra which is not a proper subalgebra of a maximal root subalgebra is a Fernando-Kac subalgebra of finite type. In Section 5 we consider in more detail the case 𝔤 = 𝔤𝔩 (n). We prove that here any primal subalgebra 𝔨 is itself a reductive Fernando-Kac subalgebra of finite type, and also give an explicit description of all Fernando-Kac subalgebras of finite type which contain a Cartan subalgebra.

ON THE EXISTENCE OF (𝔤, 𝔨)-MODULES OF FINITE TYPE 3

In conclusion, for an arbitrary reductive Lie algebra 𝔤, we give a complete description of all primal subalgebras 𝔨 ⊂ 𝔤, and for each primal subalgebra 𝔨 we construct certain “series” of irreducible (𝔤, 𝔨)-modules of finite type over 𝔨. A direct comparison with known results in the case of a symmetric pair (𝔤, 𝔨), shows that the (𝔤, 𝔨)-modules obtained by our construction are only a part of all irreducible (𝔤, 𝔨)-modules. Consequently the problem of classifying all irreducible (𝔤, 𝔨)-modules of finite type over an arbitrary primal subalgebra 𝔨 ⊂ 𝔤 is still open.

2. General preliminaries

The ground field F is algebraically closed of characteristic zero. If X is a topological space and F is a sheaf of abelian groups on X, then Γ (F) denotes the global sections of F on X. If f: X → Y is a continuous map of topological spaces, f^{−1} denotes the topological inverse image functor from sheaves on Y to sheaves on X. If X is an algebraic variety, O_{X} stands for the structure sheaf of X, and if f: X → Y is a morphism of algebraic varieties, f^{∗} (respectively f_{∗}) denotes the inverse image (resp. direct image) functor of O-modules. A multiset is defined as a map from a set Y into ℤ_{+} ∪∞, where ℤ_{+} := {0, 1, 2, 3, ...}, or, more informally, as a set whose elements have finite or infinite multiplicities.

Throughout this paper 𝔤 is a fixed reductive Lie algebra, and G stands for a connected algebraic group with Lie algebra 𝔤. Denote by C (𝔩) (respectively N (𝔩)) the centralizer (respectively normalizer) of a subalgebra 𝔩 ⊂ 𝔤. Furthermore, U (𝔩) stands for the universal enveloping algebra of 𝔩, Z (𝔩) stands for the center of 𝔩, 𝔯_{𝔩} stands for the solvable radical of 𝔩, and 𝔫_{𝔩} stands for the maximal ideal in 𝔩 which acts nilpotently on 𝔤. The sign ⊂^{+} denotes the semi-direct sum of Lie algebras, and 𝔩_{ss} is a Levi component of 𝔩. If 𝔩 is reductive, then 𝔩_{ss} simply equals the derived subalgebra [𝔩, 𝔩]. For a Borel subalgebra 𝔟 ⊂ 𝔤 which contains a Cartan subalgebra 𝔥, ρ_{𝔟} denotes as usual the half-sum of the roots of 𝔟. In what follows a root subalgebra 𝔩 ⊂ 𝔤 means a subalgebra containing a Cartan subalgebra of 𝔤.

By definition a 𝔤-module M is a (𝔤, 𝔩)-module if 𝔩 ⊂ 𝔤 [M]. M is a strict (𝔤, 𝔩)- module if 𝔩 = 𝔤 [M]. We also need the following definition from [PS]: M is an isotropic (𝔤, 𝔩)-module if for each 0 ≠ m ∈ M the set of elements g ∈ 𝔤 acting finitely on m coincides with 𝔩. An irreducible strict (𝔤, 𝔩)-module is automatically isotropic.

The following statement is a reformulation of Lemma 1 in [PS]. Lemma 2.1. Let 𝔥 be a Cartan subalgebra in 𝔤, 𝔩 ⊃ 𝔥 be a solvable subalgebra and M be an isotropic strict (𝔤, 𝔩) of finite type over 𝔥. Then there exists a parabolic subalgebra 𝔮 ⊂ 𝔤 with 𝔤 = 𝔩 + 𝔮, 𝔮 ∩ 𝔩 = 𝔥, and such that the semisimple part of 𝔮 is a direct sum of simple Lie algebras of types A and C.

3. Necessary conditions for 𝔩 to be of a Fernando-Kac subalgebra of finite type

Theorem 3.1. Let 𝔩 ⊂ 𝔤 be a Fernando-Kac subalgebra of finite type.

(1) N (𝔩) = 𝔩; hence 𝔩 is an algebraic subalgebra of 𝔤.

(2) There is a decomposition 𝔩 = 𝔫_{𝔩}⊂^{+} 𝔩_{red}, unique up to an inner automorphism of 𝔩, where 𝔩_{red} is a (maximal) subalgebra of 𝔩 reductive in 𝔤.

(3) Any irreducible (𝔤, 𝔩)-module M of finite type over 𝔩 has finite type over 𝔩_{red} and 𝔩_{red} acts semi-simply on M.

(4) C (𝔩_{red}) = Z (𝔩_{red}), and Z (𝔩_{red}) is a Cartan subalgebra of C (𝔩_{ss}).

(5) 𝔩 ∩ C (𝔩_{ss}) is a solvable Fernando-Kac subalgebra of finite type of C (𝔩_{ss}).

Proof. Let M be an irreducible strict (𝔤, 𝔩)-module and M_{0} ⊂ M be an irreducible finite-dimensional 𝔩-submodule. To prove 1, assume that N (𝔩) ≠ 𝔩. Then one can choose x ∈ N (𝔩) \𝔩 such that [x, 𝔩_{ss}] = 0 for a fixed Levi decomposition 𝔩 = 𝔩_{ss}⊃^{+} 𝔯_{𝔩}. Since x ∉ 𝔩, x acts freely on any non-zero vector in M. Set

M_{n} := M_{0} + x · M_{0} + x^{2} · M_{0} + ... + x^{n} · M_{0}

A simple calculation, using [x, 𝔩_{ss}] = 0 and [x, 𝔯_{𝔩}] ⊂ 𝔯_{𝔩}, shows that M_{n} is 𝔩-invariant and M_{n}/M_{n−1} is isomorphic to M_{0} as an 𝔩-module. Therefore the multiplicity of M_{0} in M is infinite. Contradiction. To show the algebraicity of 𝔩, consider the normalizer J of 𝔩 in G. The Lie subalgebra of 𝔤 corresponding to J is N(𝔩). Hence, N(𝔩) = 𝔩 is an algebraic subalgebra of 𝔤.

Claim 2 follows from 1 via some well known statements. For instance, Corollary 1 in [B], §5 implies that a self-normalizing subalgebra 𝔩 is splittable, i.e. for y ∈ 𝔩 the semisimple and nilpotent parts of y are contained in 𝔩. Proposition 7 in [B] , §5 claims that any splittable subalgebra has a decomposition as required in 2.

To prove 3 note first that M is a quotient of the induced module U (𝔤)⊗_{U(𝔩)}M_{0}. As the adjoint action of 𝔩_{red} on U (𝔤) is semisimple, 𝔩_{red} acts semi-simply on U (𝔤)⊗_{U(𝔩)}M_{0}, and therefore also on M. Now note that there exists ν ∈ 𝔫_{𝔩}^{∗} such that

x · m = ν (x) m

for any m ∈ M_{0} and x ∈ 𝔫_{𝔩}. Since the adjoint action of 𝔫_{𝔩} on U (𝔤) is locally nilpotent, we obtain that, for any x ∈ 𝔫_{𝔩}, x−ν (x) acts locally nilpotently on U (𝔤)⊗_{U(𝔩)}M_{0}, and hence on M. Therefore 𝔫_{𝔩} acts via the character ν on any irreducible 𝔩-subquotient of M, and consequently two irreducible 𝔩-subquotients of M are isomorphic if and only if they are isomorphic as 𝔩_{red}-modules. This implies that M has also finite type over 𝔩_{red}, and 3 is proved.

4. By 2 any irreducible strict (𝔤, 𝔩)-module M has an 𝔩_{red}-decomposition

M = ⊕_{i}M′_{i}

for finite-dimensional isotypic components M′_{i}. Clearly each M′_{i} is C (𝔩_{red})-invariant, and, as it is finite-dimensional, C (𝔩_{red}) ⊂ 𝔤 [M] = 𝔩. Note that C (𝔩_{red}) ∩ 𝔩 is solvable. Consequently, since C (𝔩_{red}) = C (𝔩_{ss}) ∩ C (Z (𝔩_{red})) ⊂ 𝔩, the centralizer of Z (𝔩_{red}) in C (𝔩_{ss}) is solvable. On the other hand, as C (𝔩_{ss}) is reductive and Z (𝔩_{red}) is reductive in C (𝔩_{ss}), the centralizer of Z (𝔩_{red}) in C (𝔩_{ss}) is reductive. Therefore Z (𝔩_{red}) coincides with its centralizer in C (𝔩_{ss}). This implies that C (𝔩_{red}) = C (𝔩_{ss}) ∩ C (Z (𝔩_{red})) = Z (𝔩_{red}), and that Z (𝔩_{red}) is a Cartan subalgebra of C (𝔩_{ss}).

To show 5 decompose M as

M = ⊕_{i} (M_{i} ⊗ V_{i}) ,

where M_{i} are pairwise non-isomorphic irreducible 𝔩_{ss}-modules, and V_{i} are C (𝔩_{ss})- modules. Then each V_{i} is a strict isotropic (C (𝔩_{ss}) , 𝔩 ∩ C (𝔩_{ss}))-module of finite type over 𝔩∩C (𝔩_{ss}) . Furthermore 𝔩∩C (𝔩_{ss}) is solvable, and 5 follows from Lemma 2.1. □

The conditions in Theorem 3.1 are not sufficient for 𝔩 to be a Fernando-Kac subalgebra of finite type: see the Example in subsection 5.3. In general, the problem of a complete characterization of a Fernando-Kac subalgebra of finite type is open. However, for a solvable 𝔩 we have the answer. Proposition 3.2. A solvable subalgebra 𝔩 ⊂ 𝔤 is a Fernando-Kac subalgebra of finite type if and only if 𝔩 = 𝔥⊃^{+} 𝔫_{𝔩}, where 𝔥 is a Cartan subalgebra of 𝔤 and 𝔫_{𝔩} is the nilradical of a parabolic subalgebra of 𝔤 whose simple components are all of types A and C.

Proof. Here 𝔩_{ss} = 0, C (𝔩_{ss}) = 𝔤, and Theorem 3.1 4 implies that 𝔥 := 𝔩_{red} is a Cartan subalgebra of 𝔤. The claim of the Corollary follows now immediately from [PS], Sect. 3 where a criterion for 𝔩 to be a Fernando-Kac subalgebra of finite type is established under the assumption that 𝔩 ⊃ 𝔥. □

Note that Theorem 3.1 3, applied to a solvable 𝔩, yields that any strict irreducible (𝔤, 𝔩)-module of finite type over 𝔩 is a weight module with finite-dimensional weight spaces. Such modules are classified by O. Mathieu in [M]. More precisely, any irreducible weight module M with finite-dimensional weight spaces is the unique irreducible quotient of an induced module U (𝔤) ⊗_{U(𝔭)} M^{𝔫_{𝔭}}, where 𝔭 is a parabolic subalgebra and M^{𝔫_{𝔭}} is the 𝔭-submodule of 𝔫_{𝔭}-invariants in M. The Fernando-Kac subalgebra 𝔤 [M] of M equals (𝔤 [M] ∩ 𝔭_{red}) ⊃^{+} 𝔫_{𝔭}, and it is solvable if and only if 𝔤 [M] ∩ 𝔭_{red} is a Cartan subalgebra of 𝔤 (in general 𝔤 [M] ∩ 𝔭_{red} is the semi-direct sum of a Cartan subalgebra and an ideal in 𝔭_{ss}).

4. A construction of irreducible (𝔤, 𝔨)-modules of finite type

4.1. A geometric set up. Let 𝔨 ⊂ 𝔤 be an algebraic subalgebra, reductive in 𝔤 and such that 𝔨_{ss} is proper in 𝔤_{ss}. Denote by K the subgroup of G with Lie algebra 𝔨, and let K_{ss} be the subgroup corresponding to 𝔨_{ss}. By H_{K} we denote a fixed Cartan subgroup of K, with Lie algebra 𝔥_{𝔨}. Fix an element h ∈ 𝔥_{𝔨} such that C(F h) ⊂ C(𝔥_{𝔨}∩ 𝔨_{ss}) and for which the operator ad_{h} : 𝔤 → 𝔤 has rational eigenvalues. The element h defines the parabolic subalgebra

(4.1)  𝔭 := ⊕_{γ≥0}𝔤_{h}^{γ},

where 𝔤_{h}^{γ} is the γ-eigenspace of ad_{h} : 𝔤 → 𝔤. Clearly 𝔟_{𝔨} := 𝔭 ∩ 𝔨 is a Borel subalgebra of 𝔨 containing 𝔥_{𝔨}. Let P be the subgroup of G corresponding to 𝔭 and B ⊂ P be a Borel subgroup of G such that B_{K} = B ∩ K has Lie algebra 𝔟_{𝔨}. Set X := G/B, Y := G/P and let π: X → Y be the natural projection. Denote by S the K-orbit of the closed point in Y corresponding to P, and put V := π^{−1} (S). Lemma 4.1. V ≅ S × Z, where Z := P/B.

Proof. V is a relative flag variety over S wtih fiber Z = P/B ≅ P_{ss}/(P_{ss} ∩ B). Moreover, V = K_{ss} ×_{K_{ss}∩P} Z. By our assumption on K and P, K_{ss} ∩ P_{ss} is the identity element. Hence the action of K_{ss} ∩ P on Z is trivial, and the bundle V → S is trivial. □

4.2. D-module preliminaries. For any μ ∈ 𝔥^{∗} let D^{μ} denote the twisted sheaf of differential operators on X defined in [BB]. A D^{μ}-module is by convention a sheaf F of D^{μ}-modules on X which is quasicoherent as a sheaf of O_{X}-modules. The support of F is the closure of the subvariety of all closed points for which the sheaf-theoretic fiber of F is non-zero. A weight μ ∈ 𝔥^{∗} defines the character θ^{μ} of the center of U (𝔤) via the Harish-Chandra map (see [B], §6).

When the ground field F is arbitrary, by a dominant weight we mean an element μ ∈ 𝔥^{∗} whose value on all B-positive co-roots is a nonnegative rational number. For F = ℂ it suffices that the value has nonnegative real part. The Beilinson-Bernstein localization theorem claims that, for a regular dominant μ, the functor of global sections

Γ: D^{μ}-mod → U (𝔤) / (ker θ^{μ}) -mod

is an equivalence between the category of D^{μ}-modules and the category of U (𝔤) / (ker θ^{μ})- modules, where (ker θ^{μ}) stands for the two-sided ideal in U (𝔤) generated by the kernel of the central character θ^{μ}. The inverse equivalence (usually referred to as localization) is given by the functor

R ↦ D^{μ} ⊗_{Γ(D^{μ})} R,

where the U (𝔤) / (ker θ^{μ})-module R is endowed with a Γ(D^{μ})-module structure via the natural isomorphism U (𝔤) / (ker θ^{μ}) → Γ(D^{μ}), see [BB].

Let i : W → X define a non-singular locally closed subvariety of X; we denote by D_{W}^{μ} the sheaf of right i^{∗}D^{μ}-module endomorphisms of the inverse image sheaf i^{∗}D^{μ} which are left O_{W} -module differential operators. Furthermore, the inverse image functor i^{∗} of O-modules yields a functor

i^{⋆}: D^{μ}-mod → D_{W}^{μ} -mod.

If W is a closed subvariety we will also consider the direct image functor

i_{⋆} : D_{W}^{μ} -mod → D^{μ}-mod,
F ↦ D_{←W}^{μ} ⊗_{D_{W}^{μ}} F,

where D_{←W}^{μ} := i^{⋆} (D^{μ} ⊗_{O_{X}} Ω_{X}^{∗})⊗_{O_{W}} Ω_{W} and Ω stands for volume forms. Kashiwara’s theorem claims that i_{⋆} is an equivalence between the category of D_{W}^{μ} -modules and the category of D^{μ}-modules supported in W. It will also be important for us that the sheaf i^{−1}i_{⋆}F has a natural O_{W} -module filtration with successive quotients

(4.7)  Λ^{max} (N_{W |X}) ⊗_{O_{W}} S^{i} (N_{W |X}) ⊗_{O_{W}} F,

where i ∈ ℤ_{+}, N_{W |X} denotes the normal bundle of W in X, S^{i} stands for i-th symmetric power and Λ^{max} stands for maximal exterior power.

In [PS] the following lemma is proven. Lemma 4.2. If Q is the support of a D^{μ}-module F, then 𝔤 [Γ (F)] ⊂ Stab_{𝔤} Q, where Stab_{𝔤} Q is the Lie algebra of the subgroup of G which stabilizes Q.

4.3. The construction. Let L be an irreducible (𝔭, 𝔥_{𝔨})-module of finite type over 𝔥_{𝔨} with trivial action of 𝔫_{𝔭} + (Z(𝔭_{red}) ∩ 𝔨_{ss}) and with 𝔭_{red}-central character θ_{𝔭_{red}}^{ν} for some P_{ss} ∩ B-dominant weight ν ∈ 𝔥^{∗}. Consider Z = P/B as a (non-singular) closed subvariety of X = G/B. Set L := D_{Z}^{η} ⊗_{Γ(D_{Z}^{η})} L, where η = ν + ρ_{𝔟∩𝔭_{red}} − ρ_{𝔟}. Let, furthermore, O_{S} (ζ) be the invertible K_{ss}-sheaf of local sections on S of the line bundle K ×_{K∩P} (F_{w(ζ)}), where w is the longest element in the Weyl group of 𝔨_{ss}, ζ is a 𝔨_{ss}-integral weight in 𝔥^{∗} and F_{ξ} stands for the one-dimensional 𝔥-module of weight ξ. Then F := O_{S} (ζ) ⊠ L is a D_{V}^{μ} -module for μ = ζ + η, and M = i_{⋆}F is a D^{μ}-module. Finally, set M = Γ (M). Theorem 4.3. Assume that ζ is dominant and μ is regular and dominant. Then

(1) M is an infinite-dimensional irreducible 𝔤-module;

(2) 𝔤 [M] = 𝔨_{ss}⊃^{+} 𝔪_{L}, where 𝔪_{L} is the maximal 𝔨_{ss}-invariant subspace in 𝔭 [L]; moreover 𝔤 [M] is the unique maximal subalgebra in 𝔭 [L] + 𝔨 which contains 𝔨;

(3) M is a (𝔤, 𝔨)-module of finite type over 𝔨.

Proof. D_{Z}^{η} is a sheaf of twisted differential operators on the flag variety Z. By the Beilinson-Bernstein theorem applied to Z, L is an irreducible D_{Z}^{η}-module. Furthermore, F is an irreducible D_{V}^{μ} -module. Since V is a non-singular closed subvariety, M is an irreducible D^{μ}-module by Kashiwara’s theorem. Finally, by the Beilinson-Bernstein theorem applied to X, M = Γ (M) is an irreducible 𝔤-module. 1 is proven.

To prove 2 consider the subalgebra Stab_{𝔤} Q, where Q is the support of the D^{μ}- module M. Note that Q ⊂ V and that V = π^{−1}(π(Q)). Hence, Stab_{𝔤} Q is a subalgebra of 𝔰𝔱 := Stab_{𝔤} V. One can check easily that

(4.8)  𝔰𝔱 = 𝔨_{ss}⊃^{+} 𝔪,

where 𝔪 is the maximal 𝔨_{ss}-invariant subspace in 𝔭. Thus 𝔰𝔱 is a maximal subalgebra in 𝔨 + 𝔭 containing 𝔨. By Lemma 4.2, 𝔤 [M] ⊂ Stab_{𝔤} Q ⊂ 𝔰𝔱 and therefore 𝔤 [M] = 𝔰𝔱 [M].

Recall now that by (4.7) i^{−1}M = i^{−1}i_{⋆}F (considered as an 𝔰𝔱-sheaf) has a natural 𝔰𝔱-sheaf filtration with successive quotients

Λ^{max} (N_{V |X}) ⊗_{O_{V}} S^{i} (N_{V |X}) ⊗_{O_{V}} F.

In particular, M_{0} := Λ^{max} (N_{V |X}) ⊗_{O_{V}} F is a subsheaf of i^{−1}M. As N_{V |X} ≅ N_{S|Y} ⊠

O_{Z}, Λ^{max} (N_{V |X}) ≅ O_{S} (τ) ⊠ O_{Z}, where τ = −w (∑_{α∈Δ(𝔫_{𝔭})} α) − 2ρ_{𝔟∩𝔨_{ss}}. Therefore M_{0} ≅ O_{S} (τ + ζ) ⊠ L and

M_{0} := Γ (M_{0}) ≅ Γ (π_{∗}M_{0}) ≅ Γ (O_{S} (τ + ζ)) ⊗ L.

Both weights τ and ζ are dominant. Hence τ + ζ is 𝔨_{ss}-dominant, M_{0} ≠ 0, and by the irreducibility of M,

(4.11)  𝔤 [M] = 𝔰𝔱 [M] = 𝔰𝔱 [M_{0}] .

To calculate 𝔰𝔱 [M_{0}] we use that Γ (M_{0}) ≅ Γ (π_{∗}M_{0}). Observe that π_{∗}M_{0} is the sheaf of sections of the induced vector bundle K_{ss} ×_{K_{ss}∩P} (F_{w(ζ+τ)} ⊗ L). The latter is a K_{ss}-sheaf, hence 𝔨_{ss} ⊂ 𝔰𝔱 [M_{0}]. By (4.8) and (4.11), 𝔤 [M] = 𝔨_{ss}⊃^{+} 𝔪_{L}, where 𝔪_{L} = 𝔤 [M] ∩ 𝔪. To calculate 𝔪_{L}, let’s write down the action of 𝔪 on Γ (π_{∗}M_{0}). An element of Γ (π_{∗}M_{0}) is a function ϕ : K_{ss} → F_{w(ζ+τ)} ⊗ L satisfying the condition ϕ(ab) = b^{−1}ϕ(a) for all a ∈ K_{ss}, b ∈ K_{s}s ∩ P. For x ∈ 𝔪 and a ∈ K_{ss} we have

(4.12)  (L_{x}ϕ) (a) = Ad_{a}^{−1} (x) (ϕ (a)) ,

where L_{x}ϕ stands for the action of x on ϕ. This formula immediately implies that

𝔪_{L} ⊂ {x ∈ 𝔪 | Ad_{K_{ss}} (x) ⊂ 𝔪 [F_{w(ζ+τ)} ⊗ L] = 𝔪 [L]} .

To see that 𝔪_{L} is equal to the right hand side, let U be a unipotent subgroup of K_{ss} complementary to K_{ss} ∩ P. U acts simply transitively on an open dense subset of S. Consider a U-invariant function f : K_{ss} → L. For any a ∈ U we have f(a) = f(1). Let x be in 𝔪[L] and assume x is Ad U-invariant. Then by (4.12), x acts locally finitely on f, and, by the irreducibility of M, x acts locally finitely on M. Finally, any y obtained from x by the action of K_{ss} also acts locally finitely on M. Hence

(4.14)  𝔪_{L} = {x ∈ 𝔪 | Ad_{K_{ss}} (x) ⊂ 𝔪 [F_{w(ζ+τ)} ⊗ L] = 𝔪 [L]} .

In other words, 𝔪_{L} is the maximal 𝔭_{ss}-invariant subspace in 𝔪[L], or equivalently in 𝔭[L]. Consequently 𝔨_{ss}⊃^{+} 𝔪_{L} is the maximal subalgebra in 𝔨 + 𝔭 [L] containing 𝔨, and 2 is proven.

It remains to prove 3. Let j : S → Y be the natural embedding. Observe that the isomorphism N_{V |X} ≅ N_{S|Y} ⊠ O_{Z} yields an isomorphism of 𝔨-sheaves

j^{−1}j_{⋆}O_{S} (ζ) ⊠ L ≅ i^{−1}i_{⋆}(O_{S} (ζ) ⊠ L) ≅ i^{−1}M.

Therefore we have an isomorphism of 𝔨-modules

Γ (M) ≅ Γ (π_{∗}M) ≅ Γ (j_{⋆}O_{S} (ζ)) ⊗ L,

where the action of 𝔨_{ss} on L is trivial and the action of Z (𝔨) is induced by the embedding Z (𝔨) ⊂ 𝔭_{red}. By (4.7) j^{−1}j_{⋆}O_{S} (ζ) has a filtration by 𝔨-sheaves with successive quotients

S^{i} (N_{S|Y} ) ⊗_{O_{S}} O_{S} (ζ + τ) .

Consequently, M has a 𝔨-module filtration whose associated graded 𝔨-module is a submodule of

Γ (S^{·} (N_{S|Y} ) ⊗_{O_{S}} O_{S} (ζ + τ)) ⊗ L.

The sheaf S^{·} (N_{S|Y} ) ⊗_{O_{S}} O_{S} (ζ + τ) is locally free on S, and has a filtration with invertible successive quotients O_{S} (κ), where κ runs over the multiset Θ of weights in 𝔥_{𝔨}^{∗}

Θ = {ζ + τ + ∑_{n_{α}∈ℕ}n_{α}α | n_{α} ∈ ℤ_{+}}.

Here we take the summation over all weights α of the 𝔥_{𝔨}-module 𝔫_{𝔭}/ (𝔫_{𝔭} ∩ 𝔨). Thus the multiplicity of the irreducible 𝔨-module with the highest weight κ in M is majorized by the multiplicity of κ in Θ + Θ_{L}, where Θ_{L} is the multiset of 𝔥_{𝔨}-weights of L. Our goal is to show that the multiset Θ + Θ_{L} has finite multiplicities. For any multiset C ⊂ 𝔥_{𝔨}^{∗} and t ∈ F, set C^{t} := {κ ∈ C | κ (h) = t}. Then Θ_{L} = Θ_{L}^{t_{0}} for some t_{0} ∈ F, and (Θ + Θ_{L})^{t} = Θ^{t−t_{0}} +Θ_{L}. As L has finite type over 𝔥_{𝔨}, Θ_{L} has finite multiplicities. Furthermore, Θ^{t−t_{0}} is a finite multiset as α (h) are all positive. Therefore (Θ + Θ_{L})^{t} has finite multiplicities, and thus Θ + Θ_{L} also has finite multiplicities. Theorem 4.3 is proven. □

The construction in Theorem 4.3 does not provide all irreducible (𝔤, 𝔨)-modules of finite type over 𝔨. Consider for instance the case when 𝔨 is symmetric, i.e. 𝔨 is stable under an involution of 𝔤. Here irreducible (𝔤, 𝔨)-modules of finite type over 𝔨 are nothing but irreducible Harish-Chandra modules. The Beilinson-Bernstein classification of irreducible Harish-Chandra modules implies that the supports of their corresponding localizations (the latter are D^{μ}-modules on X = G/B) run over the closures of all K-orbits in X. In particular, there are infinite-dimensional irreducible Harish-Chandra modules whose localizations are supported on the closure X of the open orbit of K in G/B. These latter modules do not appear among the modules constructed in Theorem 4.3, as all D^{μ}-modules M considered above are supported on a closed proper subvariety of X.

4.4. Description of primal subalgebras. Theorem 4.4. Let 𝔨 be a reductive in 𝔤 subalgebra with C (𝔨) = Z (𝔨). Then 𝔨 is primal, i.e. there exists a Fernando-Kac subalgebra of finite type 𝔩 ⊂ 𝔤 such that 𝔩_{red} = 𝔨. In addition, 𝔩 can be chosen so that 𝔫_{𝔩} is the nilradical of a Borel subalgebra of C(𝔨_{ss}).

Proof. The assumption C (𝔨) = Z (𝔨) implies that Z (𝔨) is a Cartan subalgebra of C (𝔨_{ss}). Let h′ be a semisimple element in 𝔨_{ss} such that C (F h′) = C (𝔥_{𝔨} ∩ 𝔨_{ss}) and ad_{h′} : 𝔤 → 𝔤 has rational eigenvalues γ′_{i}. Let furthermore h′′ ∈ Z (𝔨) be a regular element in C (𝔨_{ss}) for which ad_{h′′} : 𝔤 → 𝔤 has rational eigenvalues γ′′_{j}, and such that

(4.20)  |γ′′_{j}| < min_{γ′_{i}≠0} |γ′_{i}|

for all j. Denote by 𝔭 the parabolic subalgebra of 𝔤 defined by the element h := h′+h′′ and let L be a 1-dimensional 𝔭-module. Theorem 4.3 applies to the triple (𝔨, 𝔭, L) (as 𝔨 is automatically algebraic) and hence yields an irreducible (𝔤, 𝔨)-module M of finite type over 𝔨. Put 𝔩 := 𝔤 [M]. Then 𝔩 = 𝔨_{ss}⊃^{+} 𝔪, where as in the proof of Theorem 4.3 𝔪 is the maximal 𝔨_{ss}-invariant subspace in 𝔭. Let κ be the 𝔭 ∩ 𝔨_{ss}-lowest weight of an irreducible 𝔨_{ss}-submodule in 𝔪. We have κ (h′) = γ′_{i} ≤ 0 for some i. On the other hand, as 𝔪 ⊂ 𝔭, κ (h′ + h′′) ≥ 0. Condition (4.20) gives κ (h′) = 0, i.e. 𝔪 = C (𝔨_{ss})∩𝔭. As h′′ is regular in C (𝔨_{ss}), 𝔪 is a Borel subalgebra in C (𝔨_{ss}), hence 𝔪 is solvable and 𝔪 = Z (𝔨_{ss}) + [𝔪, 𝔪]. Therefore 𝔩_{red} = 𝔨, 𝔫_{𝔩} = [𝔪, 𝔪] and [𝔫_{𝔩}, 𝔨_{ss}] = 0. □

Corollary 4.5. A reductive in 𝔤 subalgebra 𝔨 is primal if and only if C (𝔨) = Z (𝔨).

Proof. The statement follows directly from Theorem 4.4 and 3.1 4. □

Corollary 4.5, together with the remark that 𝔨 is primal if and only if 𝔨 = 𝔩_{red} for a Fernando-Kac subalgebra of finite type, reduces the problem of classifying all Fernando-Kac subalgebras of finite type to the problem of describing all nilpotent subalgebras 𝔫 such that 𝔨⊃^{+} 𝔫 is a Fernando-Kac subalgebra of finite type, 𝔨 being a fixed primal subalgebra of 𝔤. The latter problem is open. In the next section we solve this problem in the case when 𝔤 = 𝔤𝔩(n) and 𝔨 is a root subalgebra, and show also that every primal subalgebra of 𝔤𝔩(n) is a Fernando-Kac subalgebra of finite type.

For simple Lie algebras not of type A, it is not true that any primal subalgebra is itself a Fernando-Kac subalgebra of finite type. Indeed, Proposition 3.2 implies that a Cartan subalgebra of a simple Lie algebra 𝔤 (which is always primal) is a Fernando-Kac subalgebra of finite type if and only if 𝔤 is of type A or C. (This was proved first by S. Fernando in [F].) An important particular case of the above open problem is the problem of characterizing all primal subalgebras which are Fernando-Kac subalgebras of finite type.

We conclude this section with an application to the classical theory of subalgebras of a semisimple Lie algebra. In the fundamental paper [D], an important role is played by subalgebras 𝔰 ⊂ 𝔤 which are not contained in a proper root subalgebra. We propose the term stem subalgebra. By Theorem 7.3 and 7.4 of [D], a stem subalgebra is necessarily semisimple with zero centralizer. Here are the best known examples of stem subalgebras.

(1) A principal 𝔰𝔩(2)-subalgebra is a stem subalgebra.

(2) If 𝔤 = 𝔰𝔩(n), a proper subalgebra 𝔰 ∈ 𝔤 is a stem subalgebra if and only if the defining representation of 𝔤 is irreducible over 𝔰.

(3) In general, any semisimple maximal subalgebra is either a stem subalgebra or a root subalgebra. For instance, if n ≥ 3 is odd, 𝔬(n) ⊕ 𝔬(n) is a stem subalgebra of 𝔬(2n). (This is moreover a symmetric pair.) Another well known example of a stem subalgebra is G_{2} ⊕ F_{4} in E_{8}.

(4) If 𝔤 is an exceptional simple Lie algebra over ℂ, Table 39 of [D] gives a complete catalog of the stem subalgebras of 𝔤.

Theorem 4.4 combined with Theorems 7.3 and 7.4 in [D] imply Corollary 4.6. If 𝔤 = 𝔤_{ss}, any stem subalgebra is a Fernando-Kac subalgebra of finite type.

Finally, we have Corollary 4.7. If 𝔤 = 𝔤_{ss}, every maximal proper subalgebra 𝔩 ⊂ 𝔤 is a Fernando-Kac subalgebra of finite type.

Proof. By a theorem of F. Karpelevic, [Kar], 𝔩 is a parabolic subalgebra or a semisimple subalgebra. If 𝔩 is parabolic the statement is obvious as any module induced from a finite-dimensional 𝔩-module has finite 𝔩-multiplicities. Let 𝔩 be semisimple. Then C(𝔩) = 0, and thus 𝔩 is primal by Corollary 4.5. But as 𝔩 is maximal, any irreducible infinite-dimensional (𝔤, 𝔩)-module of finite type is strict, i.e. 𝔩 is a Fernando-Kac subalgebra of finite type. □

5. The case 𝔤 = 𝔤𝔩 (n)

5.1. Description of reductive Fernando-Kac subalgebras of finite type. Theorem 5.1. A reductive in 𝔤 = 𝔤𝔩 (n) subalgebra 𝔨 is a Fernando-Kac subalgebra of finite type if and only if it is primal, or equivalently, if and only if C (𝔨) = Z (𝔨).

Proof. By Theorem 3.1 it suffices to prove that if C (𝔨) = Z (𝔨), then 𝔨 is a Fernando-Kac subalgebra of finite type. We will modify the argument in the proof of Theorem 4.4 under the assumption that 𝔤 = 𝔤𝔩 (n).

Let h = h′ + h′′, 𝔭 and 𝔪 be as in the proof of Theorem 4.4. In particular 𝔪 = C(𝔨_{ss}) ∩ 𝔭. We claim that h′′ can be chosen so that, in addition, there is a decomposition 𝔭 = 𝔞′⊂^{+} 𝔞 and an isomorphism p : C(𝔨_{ss}) → 𝔞. Here is how this claim implies the Theorem. Note that C(𝔨_{ss}) is a direct sum of an abelian ideal and simple ideals of type A. Choose now L to be a strict irreducible (C(𝔨_{ss}), Z(𝔨))-module of finite type over Z(𝔨). Define a 𝔭-module structure on L by putting 𝔞′·L = 0 and letting 𝔞 act on L via the isomorphism p. One can see immediately that L is an irreducible (𝔭, 𝔥_{𝔨})-module of finite type over 𝔥_{𝔨} with 𝔭[L] = 𝔞′ + 𝔥. Apply the construction in Theorem 4.3 to the triple (𝔨, 𝔭, L) to obtain a (𝔤, 𝔨)-module M of finite type over 𝔨. As 𝔪 ⊂ C(𝔨_{ss}), we have 𝔪_{L} = Z(𝔨) and consequently 𝔤[M] = 𝔨.

It remains to prove our claim about the choice of h′′. We will consider the parabolic subalgebra 𝔭′ defined via (4.1) by the fixed element h′ and then we will choose h′′ so that 𝔭 is a certain subalgebra of 𝔭′. Let E be the defining (n-dimensional) 𝔤-module. There is an isomorphism of 𝔨_{ss} ⊕ C (𝔨_{ss})-modules

(5.1)  E ≅ ⊕_{i} (E_{i} ⊗ V_{i}) ,

where the E_{i}’s are pairwise non-isomorphic irreducible 𝔨_{ss}-modules and the V_{i}’s are irreducible C (𝔨_{ss})-modules. We have

(5.2)  C (𝔨_{ss}) ≅ ⊕_{i} End (V_{i}) .

One can check that

(5.3)  𝔭′_{red} = C (𝔥_{𝔨_{ss}}) ≅ ⊕_{λ∈𝔥_{𝔨ss}^{∗}} End (E^{λ}) ,

where E^{λ} denotes the 𝔥_{𝔨_{ss}}-weight space of weight λ. Furthermore, by (5.1),

(5.4)  E^{λ} ≅ ⊕_{i} (E_{i}^{λ} ⊗ V_{i}) .

Put E_{ij}^{λ} := Hom (E_{i}^{λ}, E_{j}^{λ}) ⊗ Hom (V_{i}, V_{j}) and E^{λ} := ⊕_{i,j}E_{ij}^{λ}. Then combining (5.3) and (5.4) one obtains that 𝔭′_{red} ≅ ⊕_{λ}E^{λ}. Note that E_{+}^{λ} := ⊕_{i≤j}E_{ij}^{λ} is a parabolic subalgebra of E^{λ}.

We now choose h′′ ∈ Z (𝔨) so that the parabolic subalgebra 𝔭 associated to h′ + h′′ by (4.1) is precisely (⊕_{λ}E_{+}^{λ})⊃^{+} 𝔫_{𝔭′}. Note that 𝔭_{red} = ⊕_{i,λ}E_{i,i}^{λ}. For each 𝔭 ∩ 𝔨_{ss}-singular weight λ of 𝔨 in E there is a unique index i_{λ} such that the 𝔭 ∩ 𝔨_{ss}-highest weight of E_{i_{λ}} equals λ. Let 𝔞 := ⊕_{λ}E_{i_{λ}i_{λ}}^{λ} and 𝔞′ be the ideal complementary to 𝔞. Since E_{i_{λ}}^{λ} is one-dimensional and E_{i_{λ}i_{λ}}^{λ} ≅ End(V_{i_{λ}}), equation (5.2) enables us to conclude that C (𝔨_{ss}) is isomorphic to 𝔞. □

Corollary 5.2. A reductive in 𝔤 = 𝔤𝔩(n) subalgebra 𝔨 is a Fernando-Kac subalgebra of finite type if and only if the defining 𝔤-module is multiplicity free as a 𝔨-module.

5.2. A combinatorial set-up. Let 𝔥 be a Cartan subalgebra of 𝔤 = 𝔤𝔩(n) and let 𝔩 ⊃ 𝔥 be a root subalgebra of 𝔤. Then 𝔩 is defined by its subset of roots Δ (𝔩) ⊂ Δ, where Δ ⊂ 𝔥^{∗} is the root system of 𝔤. Recall that Δ = {ε_{i} − ε_{j} | 1 ≤ i ≠ j ≤ n} for an orthonormal basis ε_{1}, …, ε_{n} of 𝔥^{∗}. Set 𝔨 := 𝔩_{red} and 𝔫 := 𝔫_{𝔩}. Then 𝔩 = 𝔨⊃^{+} 𝔫. Fix an arbitrary Borel subalgebra 𝔟 ⊂ 𝔤 containing 𝔥, and let S_{𝔨}(𝔤) ⊂ Δ be the set of weights of all 𝔨 ∩ 𝔟-singular vectors in 𝔤. For any α ∈ S_{𝔨}(𝔤) denote by 𝔤 (α) the irreducible 𝔨-submodule in 𝔤 with highest weight α. Obviously any α, β ∈ Δ satisfy the condition

(5.5)  α + β ∈ Δ for α, β ∈ S_{𝔨}(𝔤) ⇒ α + β ∈ S_{𝔨}(𝔤).

More generally, let for any 𝔨-submodule 𝔣 of 𝔤, S_{𝔨}(𝔣) denote the set of all weights of 𝔨 ∩ 𝔟-singular vectors in 𝔣. As 𝔨 and 𝔫 are subalgebras, S_{𝔨}(𝔫) and S_{𝔨}(𝔨) satisfy the analog of condition (5.5).

The following lemma is an easy consequence of the description of root subalgebras in 𝔤𝔩(n) and we leave its proof to the reader. Lemma 5.3. There exist pairwise non-intersecting subsets I, J, K ⊂ {1, …, n} such that |I| = |J| and

S_{𝔨}(𝔤) = {ε_{i} − ε_{j} | i ∈ I ∪ K, j ∈ J ∪ K}.

Let C_{𝔨}(𝔣) denote the set of all linear combinations of vectors from S_{𝔨}(𝔣) with coefficients in ℤ_{+}.

Lemma 5.4. Let 𝔤 = 𝔤𝔩(n). If C_{𝔨}(𝔤/𝔫) ∩ C_{𝔨}(𝔫) ≠ {0}, one of the following relations holds

(1) α_{1} + α_{2} = β_{1} + β_{2},

(2) α_{1} + α_{2} = β for some α_{1}, α_{2} ∈ S_{𝔨}(𝔤/𝔩), β_{1}, β_{2}, β ∈ S_{𝔨}(𝔫), where (α_{1}, α_{2}) = (β_{1}, β_{2}) = 0 in the case of 1.

Proof. If C_{𝔨}(𝔤/𝔫) ∩ C_{𝔨}(𝔫) ≠ {0}, there is a non-trivial relation

(5.6)  α_{1} + ... + α_{k} = β_{1} + ... + β_{l}

for α_{i} ∈ S_{𝔨}(𝔤/𝔫) and β_{i} ∈ S_{𝔨}(𝔫). Among all such relations we fix one with minimal k and minimal l for given fixed minimal k. Consider first the case when α_{1} +α_{p} ∈ S_{𝔨}(𝔤) for some p ≤ k. We claim that then α_{1} + α_{p} ∈ S_{𝔨}(𝔫). For, if α_{1} + α_{p} ∈ S_{𝔨}(𝔤/𝔫), one can reduce k in (5.6) by the substitution β = α_{1} + α_{p}, which contradicts our assumption. Thus β := α_{1} + α_{p} ∈ S_{𝔨}(𝔫), and to show that α_{1} + α_{p} = β is a relation of type 2 we need only verify that α_{1}, α_{p} ∉ S_{𝔨}(𝔨). But the assumption α_{1} ∈ S_{𝔨}(𝔨) (and similarly α_{p} ∈ S_{𝔨}(𝔨)) is obviously contradictory, as then −α_{1} ∈ Δ(𝔨) and α_{p} = α_{1} + α_{p} − α_{1} = β − α_{1} ∈ Δ(𝔫). Therefore α_{1}, α_{p} ∈ S_{𝔨}(𝔤/𝔩), β ∈ S_{𝔨}(𝔫) and α_{1} + α_{p} = β.

In the remainder of the proof we assume that α_{1} + α_{p} ∉ S_{𝔨}(𝔤) for all p ≤ k. If α_{1} = ε_{i} − ε_{j}, then ε_{i} and −ε_{j} appear in α_{1} + … + α_{k} with positive coefficients. Therefore, there exist a and b such that β_{a} = ε_{i} − ε_{r} and β_{b} = ε_{s} − ε_{j}, s ≠ r by minimality. By Lemma 5.3 γ := ε_{s} − ε_{r} ∈ S_{𝔨}(𝔤). We claim that γ ∈ S_{𝔨}(𝔤/𝔫). Indeed, assume to the contrary that γ ∈ S_{𝔨}(𝔫). Then one can modify (5.6) by removing α_{1} and replacing β_{a} + β_{b} by γ. Since (5.6) is minimal, the new relation must be trivial. Thus α_{1} = β_{1} + … + β_{l}. Since β_{1} + … + β_{l} ∈ Δ, β := β_{1} + … + β_{l} ∈ S_{𝔨}(𝔫), and hence α = β ∈ S_{𝔨}(𝔫). Contradiction. Therefore indeed γ ∈ S_{𝔨}(𝔤/𝔫), and we have a relation α_{1} + γ = β_{a} + β_{b}, where α_{1}, γ ∈ S(𝔤/𝔫), β_{a}, β_{b} ∈ S_{𝔨}(𝔫). Obviously, (α_{1}, γ) = (β_{a}, β_{b}) = 0. To complete the proof we need to show that α_{1}, γ ∈ S_{𝔨}(𝔤/𝔩). But the assumption α_{1} ∈ S_{𝔨}(𝔨) (and similarly γ ∈ S_{𝔨}(𝔨)) is contradictory as it implies β_{b} − α_{1} ∈ Δ(𝔫). Hence γ = β_{a} + (β_{b} − α_{1}) ∈ Δ(𝔫). □

Corollary 5.5. C_{𝔨}(𝔤/𝔫) ∩ C_{𝔨}(𝔫) = {0} if and only if C_{𝔨}(𝔤/𝔩) ∩ C_{𝔨}(𝔫) = {0}.

Proof. As C_{𝔨}(𝔤/𝔩) ⊂ C_{𝔨}(𝔤/𝔫), C_{𝔨}(𝔤/𝔫) ∩ C_{𝔨}(𝔫) = {0} implies C_{𝔨}(𝔤/𝔩) ∩ C_{𝔨}(𝔫) = {0}. To prove the converse assume that C_{𝔨}(𝔤/𝔩)∩C_{𝔨}(𝔫) = {0} but C_{𝔨}(𝔤/𝔫)∩C_{𝔨}(𝔫) ≠ {0}. Then by Lemma 5.4 one has a relation 1 or 2 with α_{1}, α_{2} ∈ S_{𝔨}(𝔤/𝔩). Hence C_{𝔨}(𝔤/𝔩)∩C_{𝔨}(𝔫) ≠ {0}. Contradiction. □

Lemma 5.6. Let 𝔰 = 𝔤𝔩(m), 𝔮 ⊂ 𝔰 be a maximal parabolic subalgebra, and 𝔨 := 𝔮_{red} . Let V_{κ} be the irreducible 𝔰-module with highest weight κ and V_{μ}(𝔨) be the irreducible 𝔨-module with highest weight μ. If λ is a dominant 𝔨-integral weight and β is the highest root of 𝔰, then, for large enough q ∈ ℤ_{+}, the multiplicity of V_{λ+qβ}(𝔨) in V_{λ+pβ} is one for any p ∈ ℤ_{+}, p ≥ q.

Proof. Set μ := λ + pβ, ν := λ + qβ for fixed p ≥ q ∈ ℤ_{+}. Note that μ and ν are 𝔨 -dominant and hence V_{ν}(𝔨) is finite-dimensional. If M_{μ} denotes the Verma module over 𝔰 and M_{μ}(𝔨) denotes the Verma module over 𝔨, then M_{μ} is isomorphic to M_{μ}(𝔨) ⊗ S(𝔮/𝔨)^{∗} as a 𝔨-module. Thus M_{μ} admits a filtration by 𝔨-submodules such that the associated graded 𝔨-module is a direct sum of Verma modules over 𝔨, each appearing with finite multiplicity. As the multiplicity of the weight (q − p)β in S(𝔮/𝔨)^{∗} is one, the multiplicity of M_{ν}(𝔨) in M_{μ} is one. Therefore the multiplicity of V_{ν}(𝔨) in M_{μ} is also one.

Now let N ≠ V_{μ} be an irreducible subquotient of M_{μ}. We will show that, for q large, the multiplicity of V_{ν}(𝔨) in N is zero. It is known (see for example Theorem 7.6.23 [Dix]) that N is a subquotient of M_{w_{α}(μ+ρ)−ρ} for some positive root α such that (μ, α) ∈ ℤ_{+}. Therefore it suffices to prove that the multiplicity of M_{ν}(𝔨) in M_{w_{α}(μ+ρ)−ρ} is zero. This is equivalent to showing that w_{α}(μ + ρ) − ρ − ν is not a weight of S(𝔮/𝔨), i.e. that w_{α}(μ + ρ) − ρ − ν does not belong to the convex hull C of Δ(𝔮/𝔨).

Choose q so that (ν, α) > 0 for any positive α satisfying (α, β) = 1, and assume p ≥ q. First consider the case when (α, β) = 0. Here w_{α}(μ + ρ) − ρ − ν = w_{α}(ν + ρ) − ρ − ν + (p − q)β. But w_{α}(ν + ρ) − ρ − ν = aα for some negative a, which implies that w_{α}(μ + ρ) − ρ − ν = (p − q)β + aα does not belong to C. Next consider the case when (α, β) = 1. Here w_{α}(μ + ρ) − ρ − ν = w_{α}(ν + ρ) − ρ − ν + (p − q)w_{α}(β) = −(b + 1 + p − q)α + (p − q)β, where b = (ν, α) is positive by our choice of q. One can see that −(b + 1 + p − q)α + (p − q)β is not in C. Finally, the case α = β is obvious. □

Corollary 5.7. Let 𝔰 = 𝔰_{1} ⊕ ⋯ ⊕ 𝔰_{j} where each 𝔰_{i} is isomorphic to 𝔤𝔩(m_{i}), 𝔮_{i} be a maximal parabolic subalgebra of 𝔰_{i} and β_{i} be the highest root of 𝔰_{i}. Let 𝔮 = 𝔮_{1}⊕…⊕𝔮_{l} ⊕𝔰_{l+1}⊕…⊕𝔰_{j} for l ≤ j, 𝔨 be the reductive part of 𝔮, and β = β_{1}+…+β_{l}. If λ is a dominant 𝔨-integral weight, then there is a positive integer q such that such that the multiplicity of V_{λ+qβ}(𝔨) in V_{λ+pβ} is one for any p ≥ q.

Proof. The statement follows easily from Lemma 5.6 as an irreducible highest weight module over a direct sum of reductive Lie algebras is isomorphic to the tensor product of irreducible modules over the components. □

5.3. Description of Fernando-Kac root subalgebras of finite type. Theorem 5.8. A root subalgebra 𝔩 = (𝔨⊃^{+} 𝔫) ⊂ 𝔤 = 𝔤𝔩(n) is a Fernando-Kac subalgebra of finite type if and only if C_{𝔨}(𝔤/𝔩) ∩ C_{𝔨}(𝔫) = {0}.

Proof. First, we will show that if C_{𝔨}(𝔤/𝔩) ∩ C_{𝔨}(𝔫) ≠ {0}, then 𝔩 is not a Fernando-Kac subalgebra of finite type. If C_{𝔨}(𝔤/𝔩) ∩ C_{𝔨}(𝔫) ≠ 0, Lemma 5.4 provides us with a relation of type 1 or 2. Assume that the relation is of type 2, i.e. α_{1} + α_{2} = β for some α_{1}, α_{2} ∈ S_{𝔨}(𝔤/𝔩), β ∈ S_{𝔨}(𝔫). Let 𝔰 be the subalgebra generated by 𝔨 and 𝔤^{±β}, and 𝔮 be the subalgebra generated by 𝔨 and 𝔤^{β}. Then the triple (𝔰, 𝔮, β) satisfies the hypothesis of Corollary 5.7 with l = 1. Moreover, 𝔤(β) commutes with 𝔤^{α_{i}}.

Let M be an irreducible strict (𝔤, 𝔩)-module. There exists a 𝔟 ∩ 𝔨-singular vector v ∈ M such that 𝔤(β)v = 0. Let λ denote the weight of v. For any positive integer t, set v_{t} = (g^{α_{1}})^{t}(g^{α_{2}})^{t}v for 0 ≠ g^{α_{i}} ∈ 𝔤^{α_{i}}. As g^{α_{i}} acts freely on M, we have v_{t} ≠ 0. Furthermore v_{t} is 𝔟 ∩ 𝔨-singular and 𝔤(β)v_{t} = 0. Hence v_{t} generates an 𝔰-submodule M(v_{t}) ⊂ M of highest weight λ + tβ. By Corollary 5.7, for a fixed large r ∈ ℤ_{+}, the multiplicity of V_{λ+rβ} in M(v_{t}) is not zero for any t ≥ r. Therefore the multiplicity of V_{λ+rβ} in M is infinite. Contradiction.

In the case of a relation of type 1, α_{1} + α_{2} = β_{1} + β_{2}, let 𝔰 ⊂ 𝔤 be the subalgebra generated by 𝔨, 𝔤^{±β_{1}} and 𝔤^{±β_{2}}, and 𝔮 ⊂ 𝔰 be the subalgebra generated by 𝔨, 𝔤^{β_{1}} and 𝔤^{β_{2}}. The reader can check that the triple (𝔰, 𝔮, β) satisfies the conditions of Corollary 5.7 with l = 2, and moreover that 𝔤(β_{1}) ⊕ 𝔤(β_{2}) commutes with 𝔤^{α_{i}}. Therefore, an argument similar to that in the case of a relation of type 2, leads to a contradiction.

It remains to prove that 𝔩 is a Fernando-Kac subalgebra of finite type whenever C_{𝔨}(𝔤/𝔩) ∩ C_{𝔨}(𝔫) = {0}. Using Theorem 4.3 we will construct an irreducible strict (𝔤, 𝔩)-module M of finite type over 𝔩.

Note first that C_{𝔨}(𝔤) consist of 𝔨-dominant roots, and therefore C_{𝔨}(𝔤) ∩ −C_{𝔨}(𝔤) = C_{𝔨}(C(𝔨_{ss})) and S_{𝔨}(𝔤) ∩ −S_{𝔨}(𝔤) = S_{𝔨}(C(𝔨_{ss})). Furthermore, as 𝔫 is nilpotent, C_{𝔨}(𝔫) ∩ −C_{𝔨}(𝔫) = {0}. Let C_{0} = C_{𝔨}(𝔤/𝔫)∩−C_{𝔨}(𝔤/𝔫), and Δ_{0} = S_{𝔨}(𝔤/𝔫)∩−S_{𝔨}(𝔤/𝔫). The above implies immediately that Δ_{0} ⊂ S_{𝔨}(C(𝔨_{ss})) and Δ_{0} generates C_{0}. By Corollary 5.5, C_{𝔨}(𝔤/𝔫) ∩ C_{𝔨}(𝔫) = {0}. Therefore one can find h ∈ 𝔥 such that all eigenvalues of ad_{h} : 𝔤 → 𝔤 are rational and

(5.7)

α(h) > 0 for α ∈ S_{𝔨}(𝔫),
α(h) = 0 for α ∈ Δ_{0},
α(h) < 0 for α ∈ S_{𝔨}(𝔤/(𝔫 + C(𝔨_{ss}))).

One can easily verify that, in additon, h can be chosen so that

(5.8)  α(h) < 0 for all α ∈ Δ(𝔟 ∩ 𝔨_{ss}).

Let 𝔭 be defined by (4.1). Then Δ(𝔭_{red}) = Δ_{0}, and 𝔫 ⊂ 𝔫_{𝔭}.

Let L be an irreducible (𝔭, 𝔥)-module of finite type over 𝔥 with trivial action of 𝔫_{𝔭} and such that 𝔭_{red}[L] = 𝔥. Such L exists as 𝔭_{ss} is a sum of ideals of type A. Let M be as in Section 4.3. Then by Theorem 4.3, M is an irreducible (𝔤, 𝔨)-module of finite type over 𝔨. Let 𝔤 [M] = 𝔨⊃^{+} 𝔫′. We claim that 𝔫′ = 𝔫. Indeed 𝔤 (α) ⊂ 𝔫′ if and only if 𝔤 (α) ⊂ 𝔭 [L]. In particular, α(h) ≥ 0. If α (h) > 0, then by (5.7) and (5.8) 𝔤 (α) ⊂ 𝔫 ⊂ 𝔫_{𝔭} ⊂ 𝔭 [L]. If α (h) = 0, then α ∈ Δ_{0}. As 𝔭_{red} (L) = 𝔥, we have 𝔤 (α) ⊄ 𝔭 [L]. Thus 𝔫 = 𝔫′. Theorem 5.8 is proven. □

Corollary 5.9. A root subalgebra 𝔩 = (𝔨⊃^{+} 𝔫) ⊂ 𝔤𝔩(n) with 𝔫 ⊂ C(𝔨_{ss}) is a Fernando-Kac subalgebra of finite type if and only if 𝔫 is the nilradical of a parabolic subalgebra in C(𝔨_{ss}).

Proof. For the necessity see Theorem 3.1 5. For the sufficiency we use Theorem 5.8. By hypothesis 𝔫 is the nilradical of a parabolic subalgebra in C(𝔨_{ss}). We will show that C_{𝔨}(𝔤/𝔩) ∩ C_{𝔨}(𝔫) = {0}. Suppose not. Then there exist roots α_{1}, ..., α_{k} ∈ C_{𝔨}(𝔫/𝔩) and roots β_{1}, ..., β_{l} ∈ C_{𝔨}(𝔫) such that (5.6) holds. Restrict both sides of (5.6) to 𝔥_{𝔨_{ss}} and write γ̃ for the restriction of a weight γ to 𝔥_{𝔨_{ss}}. Because 𝔫 ⊂ C(𝔨_{ss}), β̃_{i} = 0 for all i and hence α̃_{1} + ... + α̃_{l} = 0. But the α̃_{j}’s are dominant weights for 𝔨_{ss}. Therefore α̃_{j} = 0 for all j, and each α_{j} ∈ C_{𝔨}(C(𝔨_{ss})) = Δ(C(𝔨_{ss})). Equation (5.6) becomes a nontrivial relation among roots in Δ(𝔫) and Δ(C(𝔨_{ss})) \ Δ(𝔫). Contradiction. □

Example. Let 𝔤 = 𝔤𝔩(4), 𝔥 be the diagonal subalgebra, and 𝔩 ⊃ 𝔥 be a root subalgebra of 𝔤. The rank of 𝔩_{ss} can be 0, 1 or 2. In the first case 𝔩 is solvable, and, by Proposition 3.2, 𝔩 is of finite type if and only if 𝔫_{𝔩} is the nilradical of a parabolic subalgebra. In the third case 𝔩_{red} equals the fixed points of an involution θ : 𝔤 → 𝔤 and 𝔩 is always a Fernando subalgebra of finite type: the corresponding strict (𝔤, 𝔩)-modules are Harish-Chandra modules.

In the case when 𝔩_{ss} ≅ 𝔰𝔩(2) we can fix the roots of 𝔩_{ss} to be ±(ε_{1} − ε_{2}). To determine 𝔩 we need to specify the roots of 𝔫_{𝔩} . Up to automorphisms of 𝔤 that stabilize 𝔩_{ss} there are eight choices for 𝔫_{𝔩} (including the possibility 𝔫_{𝔩} = 0). A direct checking based on Theorem 5.8 and Corollary 5.9 shows that there is a single choice of 𝔫_{𝔩} for which 𝔩 is not a Fernando-Kac subalgebra of finite type. We may normalize this 𝔩 so that the roots in 𝔫_{𝔩} are ε_{1} − ε_{3} and ε_{2} − ε_{3}. Furthermore, the so defined 𝔩 = 𝔩_{red}⊃^{+} 𝔫_{𝔩} satisfies conditions 1–5 in Theorem 3.1. This shows in particular that the conditions in Theorem 3.1 are not sufficient for a subalgebra 𝔩 to be a Fernando-Kac subalgebra of finite type.

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Dept. of Mathematics, University of California at Riverside, Riverside, CA 92521

E-mail address: penkov@math.ucr.edu

Dept. of Mathematics, University of California at Berkeley, Berkeley, CA 94720

E-mail address: serganov@math.berkeley.edu

Dept. of Mathematics, Yale University, New Haven, CT 06520

E-mail address: gregg@math.yale.edu