Generalized Harish-Chandra Modules

Ivan Penkov ^{1} and Vera Serganova ^{2}

Department of Mathematics  Department of Mathematics
University of California  University of California
Riverside, California 92521  Berkeley, California 94720

            penkov@math.ucr.edu      serganov@math.berkeley.edu

Abstract. Let 𝔤 be a complex reductive Lie algebra and 𝔥 be a Cartan subalgebra of 𝔤. If 𝔨 is a subalgebra of 𝔤, we call a 𝔤-module M a strict (𝔤, 𝔨)-module if 𝔨 coincides with the subalgebra of all elements of 𝔤 which act locally finitely on M. For an intermediate 𝔨, i.e. such that 𝔥 ⊂ 𝔨 ⊂ 𝔤, we construct irreducible strict (𝔤, 𝔨)-modules. The method of construction is based on the D-module localization theorem of Beilinson and Bernstein. The existence of irreducible strict (𝔤, 𝔨)-modules has been known previously only for very special subalgebras 𝔨, for instance when 𝔨 is the (reductive) subalgebra of fixed points of an involution of 𝔤. In this latter case strict irreducible (𝔤, 𝔨)-modules are Harish-Chandra modules.

We give also separate necessary and sufficient conditions on 𝔨 for the existence of an irreducible strict (𝔤, 𝔨)-module of finite type, i.e. an irreducible strict (𝔤, 𝔨)-module with finite 𝔨-multiplicities. In particular, under the assumptions that the intermediate subalgebra 𝔨 is reductive and 𝔤 has no simple components of types B_{n} for n > 2 or F_{4}, we prove a simple explicit criterion on 𝔨 for the existence of an irreducible strict (𝔤, 𝔨)-module of finite type. It implies that, if 𝔤 is simple of type A or C, for every reductive intermediate 𝔨 there is an irreducible strict (𝔤, 𝔨)-module of finite type.

Acknowledgments. We are indebted to Ivan Dimitrov and Alexander Rosenberg for very stimulating discussions at the early stages of the project. We thank also Ilya Zakharevich for help in proving Lemma 5 and Gregg Zuckerman for constant support while writing the paper.

1Research partially supported by the NSF and the Max Planck Institute for Mathematics, Bonn.

2Research partially supported by the NSF and University Lyon 1.

Notational conventions

The ground field is ℂ, however all our results can be easily carried out over an algebraically closed field of characteristic zero. ℝ_{+} (respectively ℤ_{+}) denotes the set of non-negative real numbers (respectively integers), and ⟨·⟩_{ℂ}, ⟨·⟩_{ℝ_{+}}, ⟨·⟩_{ℤ}, or ⟨·⟩_{ℤ_{+}} stands for linear span respectively with coefficients in ℂ, ℝ_{+}, ℤ, or ℤ_{+}. 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 U ⊂ X is an open subset, F_{|U} denotes the restriction of F onto U, and F(U) := Γ(F_{|U}). For v ∈ Γ(F), v_{|U} is the restriction of v to U. 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^{∗} denotes an inverse image functor of O-modules. A multiset is defined as a map from a set Y into ℤ_{+}\{0}, or, more informally, as a set whose elements have finite multiplicities.

1 Origin of the problem

Let 𝔤 be a finite-dimensional Lie algebra and M be a 𝔤-module. S. Fernando proved in [F] that all elements of 𝔤 which act locally finitely on M ^{3}(i.e. all elements g ∈ 𝔤 such that ⟨m, g · m, g^{2} · m, …⟩_{ℂ} is a finite-dimensional subspace of M for any m ∈ M) form a Lie subalgebra 𝔤[M] of 𝔤. ^{4} Let 𝔨 be a subalgebra of 𝔤. We call M a (𝔤, 𝔨)-module, or a generalized Harish-Chandra module for the pair (𝔤, 𝔨), if 𝔨 ⊂ 𝔤[M]. A strict (𝔤, 𝔨)-module, or a strict generalized Harish-Chandra module for the pair (𝔤, 𝔨), is by definition a (𝔤, 𝔨)-module M for which 𝔤[M] = 𝔨. Furthermore, we call 𝔨 a Fernando subalgebra of 𝔤 if there exists an irreducible strict (𝔤, 𝔨)-module M. (Clearly, for every 𝔨 there exist induced 𝔤-modules which are strict (𝔤, 𝔨)-modules but are not necessarily irreducible.)

As we show at the end of this section, not all subalgebras of sl(n) are Fernando subalgebras. The problem of describing all Fernando subalgebras of a given finite-dimensional Lie algebra 𝔤 is, in our opinion, an important problem in the representation theory of Lie algebras. Not much is known for non-reductive Lie algebras 𝔤 and the problem is still open in the reductive case. One of our objectives is to desribe all Fernando subalgebras of a reductive Lie algebra 𝔤 containing a Cartan subalgebra. In what follows we will automatically assume that 𝔤 is reductive and that 𝔥 is a fixed Cartan subalgebra of 𝔤. In Theorem 1 below we prove that any intermediate subalgebra 𝔨, i.e. such that 𝔥 ⊂ 𝔨 ⊂ 𝔤, is a Fernando subalgebra. The corresponding irreducible strict (𝔤, 𝔨)-modules M are necessarily weight modules and, as it turns out, 𝔨 determines certain essential invariants of M. Here is a brief description of relevant known results about weight modules.

3Fernando assumes that M is a finitely generated 𝔤-module but it is easy to see that this assumption is not necessary.

4A combination of arguments due to V. Kac (see [K]) gives a significantly simpler proof of the same result. Moreover, Kac shows that, for any infinite-dimensional Lie algebra 𝔞, the elements of 𝔞 which act locally finitely on an 𝔞-module form a linear subspace of 𝔞.

Recall that a 𝔤-module M is a weight module if

M = ⨁_{ν∈𝔥^{∗}} M^{ν} ,  (1)

where 𝔥^{∗} stands for the dual space of 𝔥 and M^{ν} := {m ∈ M | h · m = ν(h)m ∀ h ∈ 𝔥}. It is well known that, if M is irreducible, (1) is equivalent to 𝔥 being contained in 𝔤[M], see for instance Proposition 1 in [PS]. The spaces M^{ν} are the weight spaces of M, and the set of all weights of M, i.e. linear functions ν ∈ 𝔥^{∗} with M^{ν} ≠ 0 is by definition the support supp M of M. The weight multiplicities are the dimensions dim M^{ν} and they can be finite or infinite. The formal character of M is defined as the formal sum

∑_{ν∈supp M}(dim M^{ν})e^{ν} .

This invariant of M has been studied now for about 70 years: the classical Weyl character formula from 1924 computes the formal character of any irreducible finite-dimensional 𝔤module M as a function of its highest (or extremal) weight(s). In general, the support of an irreducible weight module M carries less information than its formal character, however at the two extremes, when dim M < ∞ or when dim M^{ν} = ∞ for all ν ∈ supp M, the support determines the formal character. Furthermore, it is known that, if M is irreducible, all weight multiplicities are simultaneously finite or infinite. In the recent paper [M], O.Mathieu completed the classification of the irreducible weight modules with finite weight multiplicities (see also [BL]), and in particular Mathieu’s results lead to a formula for the formal character of any such module. The case of infinite multiplicities is also important and interesting, for most Harish-Chandra weight modules have infinite weight multiplicities. This is the case for which Theorem 1 below is of interest.

Let now 𝔥 ⊂ 𝔨 and M be an irreducible strict (𝔤, 𝔨)-module. To explain the relationship between 𝔨 and supp M we need to introduce one more invariant of M, its shadow decomposition. Let Δ ⊂ 𝔥^{∗} denote the root system of 𝔤 and Δ_{𝔨} ⊂ Δ be the set of roots of 𝔨. Define Γ_{𝔨} as the submonoid of ⟨Δ⟩_{ℤ} generated by Δ\Δ_{𝔨}. The M-decomposition of Δ, or the shadow decomposition of Δ corresponding to M,

Δ = Δ_{M}^{−} ⊔ Δ_{M}^{I} ⊔ Δ_{M}^{F} ⊔ Δ_{M}^{+} ,  (2)

is defined by setting

Δ_{M}^{I} := {α ∈ Δ | α ∈ ⟨Γ_{𝔨}⟩_{ℝ_{+}}, −α ∈ ⟨Γ_{𝔨}⟩_{ℝ_{+}}} ,
Δ_{M}^{F} := {α ∈ Δ | α ∉ ⟨Γ_{𝔨}⟩_{ℝ_{+}}, −α ∉ ⟨Γ_{𝔨}⟩_{ℝ_{+}}} ,
Δ_{M}^{+} := {α ∈ Δ | α ∉ ⟨Γ_{𝔨}⟩_{ℝ_{+}}, −α ∈ ⟨Γ_{𝔨}⟩_{ℝ_{+}}} ,
Δ_{M}^{−} := {α ∈ Δ | α ∈ ⟨Γ_{𝔨}⟩_{ℝ_{+}}, −α ∉ ⟨Γ_{𝔨}⟩_{ℝ_{+}}} .

In particular, the M-decomposition of Δ is determined by 𝔨. The decomposition (2) induces a decomposition of 𝔤,

𝔤 = (𝔤_{M}^{I} + 𝔤_{M}^{F}) ⊕ 𝔤_{M}^{+} ⊕ 𝔤_{M}^{−} ,  (3)

where

𝔤_{M}^{F} := 𝔥 ⊕ ( ⨁_{α∈Δ_{M}^{F}} 𝔤^{α}) ,  𝔤_{M}^{I} := 𝔥 ⊕ ( ⨁_{α∈Δ_{M}^{I}} 𝔤^{α}) ,  𝔤_{M}^{±} := ⨁_{α∈Δ_{M}^{±}} 𝔤^{α} .

It follows from the main results of [DMP] that 𝔭_{M} := (𝔤_{M}^{F} + 𝔤_{M}^{I}) ⊕ 𝔤_{M}^{+} is a parabolic subalgebra whose semisimple part is nothing but the direct sum of Lie algebras [𝔤_{M}^{F}, 𝔤_{M}^{F}] ⊕ [𝔤_{M}^{I}, 𝔤_{M}^{I}]. In particular, [𝔤_{M}^{F}, 𝔤_{M}^{F}] and [𝔤_{M}^{I}, 𝔤_{M}^{I}] are two commuting semisimple subalgebras of 𝔭_{M}. Furthermore, if 𝔨 = 𝔤[M], then

𝔨 = (𝔤_{M}^{F} + (𝔨 ∩ 𝔤_{M}^{I})) ⊕ 𝔤_{M}^{+} .

This is a consequence of the inclusion 𝔤_{M}^{F} ⊕ 𝔤_{M}^{+} ⊂ 𝔨 and of the fact that 𝔤_{M}^{−} ∩ 𝔨 = 0. The latter follows easily from some basic representation theory of sl(2).

Note now that the very definition of Γ_{𝔨} implies

supp M = supp M^{𝔨} + Γ_{𝔨} ,  (4)

where M^{𝔨} is any irreducible finite-dimensional 𝔨-submodule of M; see also Proposition 2 in [PS]. Moreover, as Δ_{M}^{I} ⊂ Γ_{𝔨} and (Δ_{M}^{F} + Δ_{M}^{I}) ∩ Δ = ∅, if M^{F} is any irreducible 𝔤_{M}^{F} ⊕ 𝔤_{M}^{+}- submodule of M^{𝔨}, we have also

supp M = supp M^{F} + Γ_{𝔨} .  (5)

M^{F} always exists, is necessarily finite-dimensional, and is irreducible as a 𝔤_{M}^{F}-module. Since 𝔤_{M}^{F} is reductive, supp M^{F} is nothing but the intersection of the convex hull of all extremal weights of M^{F} with the weight lattice of 𝔤_{M}^{F} shifted by any element of supp M^{F}. Furthermore, as 𝔤_{M}^{+} acts by zero on M^{F}, the right-hand side of (5) equals simply supp M^{F} + ⟨Δ_{M}^{I}⟩_{ℤ} + ⟨Δ_{M}^{−}⟩_{ℤ_{+}}. Therefore,

supp M = supp M^{F} + ⟨Δ_{M}^{I}⟩_{ℤ_{+}} + ⟨Δ_{M}^{−}⟩_{ℤ_{+}},  (6)

and supp M^{F} is not determined by 𝔨 (and could be arbitrary for a fixed 𝔨) while ⟨Δ_{M}^{I}⟩_{ℤ_{+}} + ⟨Δ_{M}^{−}⟩_{ℤ_{+}} is determined by 𝔨. In other words, the support of an irreducible strict (𝔤, 𝔨)-module is determined by 𝔨 up to adding the support of an arbitrary irreducible finite-dimensional 𝔤_{M}^{F}-module. (A more general version of this statement for a not necessarily reductive Lie algebra see in [PS].)

The starting point of this paper is the observation that, when M has infinite weight multiplicities, its formal character (or equivalently its support) does not fully determine the Lie algebra 𝔤[M] and more precisely its subalgebra 𝔤[M] ∩ 𝔤_{M}^{I}. In particular, the existing theory of weight modules provides no answer to the following question. Let 𝔨 ⊂ 𝔤 be a subalgebra with 𝔨 ⊃ 𝔥. Is it true that there exists an irreducible strict (𝔤, 𝔨)-module? Theorem 1 below gives an affirmative answer.

Once the existence problem for irreducible strict (𝔤, 𝔨)-modules is resolved, a further natural question arises. For which 𝔨 does there exist an irreducible strict (𝔤, 𝔨)-module M of finite type, i.e. such that considered as a module over the reductive part 𝔨_{red} of 𝔨 M has finite-dimensional isotypic components? Our second objective is to give a partial answer to this latter question. Propositions 4 and 5 give respectively a necessary and a sufficient condition on 𝔨 for the existence of an irreducible strict (𝔤, 𝔨)-module of finite type. For a reasonably large class of subalgebras 𝔨 these conditions are directly verifiable and provide a definitive result, see Corollaries 1 and 2. In particular, if 𝔨 is reductive and 𝔤 has no simple components of type B_{n} for n > 2 or F_{4}, then 𝔤 admits an irreducible strict (𝔤, 𝔨)-module of finite type if and only if the centralizer in 𝔤 of the semisimple part of 𝔨 has simple components only of types A and C.

The 𝔨-finiteness problem has been studied previously in two particular cases: when 𝔨 coincides with the fixed points of an involution on 𝔤, and when 𝔨 is replaced by 𝔥. It is a classical result that in the first case any irreducible (𝔤, 𝔨)-module has finite type. In the second case finite type means nothing but finite weight multiplicities, and the following Proposition summarizes known results.

Proposition 1 Let 𝔨 ⊂ 𝔤 be a subalgebra with 𝔨 ⊃ 𝔥, and let M be a strict irreducible (𝔤, 𝔨)-module.

(a) If 𝔨 ∩ 𝔤_{M}^{I} ≠ 𝔥, then M necessarily has infinite weight multiplicities.

(b) If 𝔨 ∩ 𝔤_{M}^{I} = 𝔥 and M has finite weight multiplicities, then [𝔤_{M}^{I}, 𝔤_{M}^{I}] is isomorphic to a direct sum of simple Lie algebras of types A and C.

(c) If 𝔤_{M}^{I} = 𝔥, then M necessarily has finite weight multiplicities.

Proof. Fernando has proved in [F] that 𝔨 ∩ 𝔤_{M}^{I} = 𝔥 whenever M has finite weight multiplicities. This implies (a). To prove (b), consider a non-zero vector m ∈ M with 𝔤_{M}^{+} · m = 0. Recall that 𝔭_{M} = (𝔤_{M}^{F} + 𝔤_{M}^{I}) ⊕ 𝔤_{M}^{+} is a parabolic subalgebra of 𝔤 and set M^{F I} := U(𝔭_{M}) · m. It is straightforward to verify that M^{F I} is irreducible as a [𝔤_{M}^{F}, 𝔤_{M}^{F}] ⊕ [𝔤_{M}^{I}, 𝔤_{M}^{I}]- module. Therefore M^{F I} ≃ M^{F} ⊠ M^{I} for some irreducible finite- dimensional [𝔤_{M}^{F}, 𝔤_{M}^{F}]- module M^{F} and some irreducible [𝔤_{M}^{I}, 𝔤_{M}^{I}]-module M^{I} with [𝔤_{M}^{I}, 𝔤_{M}^{I}][M^{I}] = 𝔥 ∩ [𝔤_{M}^{I}, 𝔤_{M}^{I}]. Furthermore one notes that, as M has finite weight multiplicities, M^{I} has also finite 𝔥 ∩ [𝔤_{M}^{I}, 𝔤_{M}^{I}]-weight multiplicities. Then the main result of Fernando, [F], implies that [𝔤_{M}^{I}, 𝔤_{M}^{I}] is isomorphic to a direct sum of simple Lie algebras of types A and C only. (b) is proved. Claim (c) follows from the fact that M^{F I} is a finite-dimensional 𝔭_{M}-module when 𝔤_{M}^{I} = 𝔥. Indeed, then the surjection U(𝔤) ⊗_{U(𝔭_{M})} M^{F I} → M gives that M is a highest weight 𝔤module with respect to any Borel subalgebra 𝔟 ⊂ 𝔤 with 𝔥 ⊂ 𝔟 ⊂ 𝔭_{M}. Hence M has finite weight multiplicities. □

We will need the following slightly stronger version of claim (b). If M is a strict (𝔤, 𝔨)- module which is not necessarily irreducible, we call M isotropic if, for every non-zero m ∈ M, 𝔨 coincides with the set of elements g ∈ 𝔤 which act locally finitely on m. Any irreducible 𝔤-module M is automatically isotropic as a strict (𝔤, 𝔤[M])-module.

Lemma 1 Let 𝔨 be solvable, 𝔨 ⊃ 𝔥, and let M be an isotropic strict (𝔤, 𝔨)-module with finite weight multiplicities. Then there exists a parabolic subalgebra 𝔭 ⊂ 𝔤 with 𝔤 = 𝔨+𝔭, 𝔭∩𝔨 = 𝔥, and such that its semisimple part 𝔭_{ss} is a direct sum of simple Lie algebras of types A and C.

Proof. Without loss of generality we can assume that M is generated by some 1dimensional 𝔨-submodule E^{λ} ⊂ M^{λ}. Then (similarly to (4)) supp M = λ + Γ_{𝔨}. We claim that Δ_{𝔨} ∩ Γ_{𝔨} = ∅. Indeed, if α ∈ Δ_{𝔨} ∩ Γ_{𝔨}, the solvability of 𝔨 implies that 𝔤^{α} acts locally finitely and 𝔤^{−α} acts freely on M. Furthermore {λ + ℤα} ⊂ supp M, and therefore there are infinitely many non-zero vectors m^{ν} ∈ M^{ν} for ν ∈ {λ + ℤα} with 𝔤^{α} · m^{ν} = 0. Hence M has infinitely many Verma submodules of the rank one subalgebra 𝔤^{−α}⊕𝔥⊕𝔤^{α}, and consequently M has infinite weight multiplicities. Contradiction. Therefore 𝔭 := 𝔥 ⊕ (⨁_{α∈Δ∩Γ_{𝔨}} 𝔤^{α}) is a parabolic subalgebra with 𝔤 = 𝔨 + 𝔭 and 𝔭 ∩ 𝔨 = 𝔥.

It remains to show that the simple components of 𝔭_{ss} are of types A and C only. Consider E := U(𝔭_{ss}) · E^{λ}. Since for every α ∈ Δ_{𝔭_{ss}} both 𝔤^{α} and 𝔤^{−α} act freely on M, dimE^{μ} = dimE^{μ±α} for any μ ∈ supp M, i.e. all weight multiplicities of E are equal. Then E is admissible as defined in [M], and thus has finite length ([M], Lemma 3.3). As E is obviously a strict (𝔭_{ss}, 𝔥 ∩ 𝔭_{ss})-module, any irreducible submodule of E is also a strict (𝔭_{ss}, 𝔥 ∩ 𝔭_{ss})- module, and by Proposition 1(b) 𝔭_{ss} is isomorphic to a direct sum of simple Lie algebras of types A and C. □

We conclude this introductory section with an example of a subalgebra 𝔨 which is not a Fernando subalgebra. Let 𝔤 = sl(n) for n > 2 , let 𝔟 denote the subalgebra of upper triangular matrices, 𝔫 := [𝔟, 𝔟], and let 𝔨 ⊂ 𝔟 be the subalgebra of all upper triangular matrices with zero first column. Note that 𝔨 contains no Cartan subalgebra. We claim that there is no irreducible strict (𝔤, 𝔨)-module. Indeed, assume to the contrary that M is such a 𝔤-module. Then, since 𝔫 = [𝔨, 𝔨], one can find a non-zero m ∈ M such that 𝔨 · m ⊂ ⟨m⟩_{ℂ} and 𝔫 · m = 0. Choose a basis h_{1}, …, h_{n−2} in 𝔥 ∩ 𝔨. The Casimir operator Ω can be written as

Ω = u^{2} + up(h_{1}, …, h_{n−2}) + q(h_{1}, …, h_{n−2}) + ∑_{i} x_{i}y_{i}

for some non-zero u in the 1-dimensional orthogonal complement of 𝔥 ∩ 𝔨 (with respect to the Killing form) in 𝔥, for certain polynomials p and q, degp = 1, degq = 2, and for certain strictly upper (respectively strictly lower) triangular matrices y_{i} (resp. x_{i}). As Ω · m = μm for some μ ∈ ℂ, and h_{i} · m = λ_{i}m for some λ_{i} ∈ ℂ, we obtain that u^{2} · m ∈ ⟨m⟩_{ℂ} + ⟨u · m⟩_{ℂ}. Therefore u acts locally finitely on M, which is a contradiction because u ∉ 𝔨.

2 Existence theorem

Theorem 1 Any subalgebra 𝔨 ⊂ 𝔤, such that 𝔥 ⊂ 𝔨, is a Fernando subalgebra.

The proof splits naturally into two cases: that of a solvable 𝔨 and that of a general 𝔨. We start with some preliminaries, most of which amount to fixing notations.

2.1 Preliminaries needed in the proof. Let G denote a connected reductive algebraic group with Lie algebra 𝔤. All subgroups of G considered will automatically be assumed to be connected and will be denoted by capital letters B, P, etc. The lower case German letters 𝔟, 𝔭, etc stand for the corresponding Lie subalgebras of 𝔤, and determine the subgroups B, P, etc.

B will always denote a fixed Borel subgroup of G whose Borel subalgebra 𝔟 contains the fixed Cartan subalgebra 𝔥 of 𝔤. Sometimes 𝔟 will satisfy additional requirements which will be stated explicitly. The choice of 𝔟 fixes a triangular decomposition 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫^{+} where 𝔟 = 𝔥⊕𝔫^{+}. By Δ^{+} and Δ^{−} we denote respectively the roots of 𝔟 and 𝔟^{−} := 𝔥⊕𝔫^{−}. N^{−} is the subgroup of G with Lie algebra 𝔫^{−}. The big cell U of the flag variety G/B is defined as the (open) orbit N^{−} · B of the point B ∈ G/B. N^{−} acts freely on U. This enables us to identify N^{−} with 𝔫^{−}, and therefore obtain a set of coordinates {x_{α}}_{α∈Δ^{−}} on U which correspond to coordinates on 𝔫^{−} arising from a basis of roots vectors in 𝔫^{−}. The algebra O_{G/B}(U) is then identified with the polynomial algebra ℂ[x_{α}]_{α∈Δ^{−}}. Note also that 𝔤 acts by derivations on O_{G/B} via its canonical homomorphism into the tangent bundle T_{G/B}.

If i : Q ↪ G/B is the embedding of a (possibly singular) subvariety of G/B, Q̄ will denote the closure of Q in G/B, and Stab_{𝔤} Q ⊂ 𝔤 is defined as the Lie algebra of the subgroup of G which preserves Q.

Throughout the rest of the paper μ will denote a fixed regular 𝔟-dominant weight, sometimes satisfying explicit additional conditions. Following Beilinson and Bernstein, we denote by D^{μ} the sheaf of twisted differential operators (or “tdo” for short) on G/B corresponding to μ, see [BB1]. If Q is an open subvariety we define the tdo D_{Q}^{μ} simply as D_{|Q}^{μ}. If Q is a closed non-singular subvariety then D_{Q}^{μ} is defined as the sheaf of left differential endomorphisms of the inverse image i^{∗}D^{μ}. Clearly, in this way the tdo D_{Q}^{μ} is well-defined for any non-singular locally closed subvariety Q of G/B, and D_{Q}^{μ} = D^{μ} when Q = G/B. If μ = ρ ( where ρ := 1/2 ∑_{α∈Δ^{+}} α), then D := D^{ρ} is nothing but the sheaf of differential operators on G/B. Furthermore, as U is affine, D^{μ}(U) can be identified with the Weyl algebra ℂ[u_{α}, \frac{∂}{∂u_{α}}]_{α∈Δ^{−}} = D(U).

A D_{Q}^{μ}-module F is by definition a sheaf of D_{Q}^{μ}-modules on Q which is quasi-coherent as a sheaf of O_{Q}-modules. The support of F is the closed subvariety of Q consisting of all closed points for which the sheaf-theoretic fiber of F is non-zero. We will make extensive use of the D^{μ}-affinity theorem of A. Beilinson and J. Bernstein, [BB1]. This theorem states that G/B is D^{μ}-affine , i.e. that every D^{μ}-module F is generated over D^{μ} by its global sections Γ(F) and that all higher cohomology groups of the sheaf F vanish. Consequently Γ is an equivalence between the category of D^{μ}-modules and the category of modules over the associative algebra Γ(D^{μ}). Moreover, a further result of Beilinson and Bernstein, [BB1], claims that Γ(D^{μ}) is canonically isomorphic to the quotient U(𝔤)/(Kerθ^{μ}), where U(𝔤) is the universal enveloping algebra of 𝔤 and (Kerθ^{μ}) is the two sided ideal generated by the kernel of the central character θ^{μ} of a module with 𝔟-highest weight μ − ρ.

The following Lemma relates the Fernando subalgebra of the global sections of a D^{μ}- module with its support.

Lemma 2 If Q is the support of a D^{μ}-module F, then 𝔤[Γ(F)] ⊂ Stab_{𝔤} Q.

Proof. Set R := (G/B)\Q. For any x ∈ 𝔤[Γ(F)], g_{x} := exp x is an automorphism of G/B as well as of Γ(F). Furthermore, for any v ∈ Γ(F),

v|_{g_{x}(R)} = g_{x}^{−1}(v)|_{R} = 0 .  (7)

Since F is generated by Γ(F) over D^{μ}, (7) implies F(g_{x}(R)) = 0. But then g_{x}(R) = R and therefore x ∈ Stab_{𝔤}Q. □

Assume that i : Q ↪ G/B is a locally closed embedding of a non-singular subvariety Q. We denote by i_{∗} the D_{Q}^{μ}-module direct image functor. By definition, i_{∗}F := D\underleftarrow{μ} ⊗_{D_{Q}^{μ}} F, where D\underleftarrow{μ} := i^{∗}(D^{μ} ⊗_{O_{G/B}} Ω_{G/B}^{∗}) ⊗_{O_{Q}} Ω_{Q} and Ω stands for volume forms. Furthermore the D^{μ}-module i_{∗}F has a natural O_{G/B}-module filtration with successive quotients

Λ^{max}(N_{Q}) ⊗_{O_{Q}} S^{i}(N_{Q}) ⊗_{O_{Q}} F

for i ∈ ℤ_{+}, where N_{Q} denotes the normal bundle of Q in G/B, S^{i} stands for i^{th} symmetric power, and Λ^{max} stands for maximal exterior power. In the case when i is a closed embedding, Kashiwara’s theorem, [Ka], claims that i_{∗} is an equivalence between the category of D_{Q}^{μ}- modules and the category of D^{μ}-modules with support in Q.

A locally closed embedding i is the composition of a closed embedding j : Q → V and an open embedding l : V → X. Then i_{∗} = l_{∗}j_{∗}, where l_{∗} coincides with the sheaf-theoretic direct image. Under the additional assumptions that F is irreducible and locally free (of finite rank) as O_{Q}-module, i_{∗}F contains a unique irreducible D^{μ}-submodule i_{∗!}F, see [B], [BB1]. Furthermore, i_{∗!}F(V) = i_{∗}F(V). In particular the support of i_{∗!}F is Q̄. If i = j, i.e. if i is closed, then i_{∗!}F simply equals i_{∗}F.

We are now ready to start the proof of the Theorem 1. Throughout the proof 𝔨 is a fixed subalgebra of 𝔤 with 𝔥 ⊂ 𝔨.

2.2 The case of a solvable 𝔨. Assume first that 𝔨 is solvable and let 𝔟 ⊃ 𝔨. Set Δ̃_{𝔨} := Δ^{−}\Δ_{𝔨}.

Lemma 3 There are global coordinates {u_{α}}_{α∈Δ_{−}} on U such that h(u_{α}) = α(h)u_{α} for any h ∈ 𝔥 and any α ∈ Δ_{−}, and k(u_{α}) = 0 for all k ∈ 𝔨 ∩ 𝔫^{−} and all α ∈ Δ̃_{𝔨}.

Proof. Consider the K-orbit K · B ⊂ U. The polynomial algebra O(U) = ℂ[x_{α}]_{α∈Δ^{−}}, considered as a 𝔨-module, is a weight module (with respect to 𝔥). The ideal I_{K·B} := Γ(I_{K·B}) is a 𝔨-submodule of O(U), and splits as a direct summand as an 𝔥-submodule. The quotient O(U)/I_{K·B} is nothing but Γ(O_{K·B}) = ℂ[x_{α}]_{α∈Δ_{𝔨}}, and therefore we have an 𝔥-module decomposition

O(U) = ℂ[x_{α}]_{α∈Δ_{𝔨}} ⊕ I_{K·B} .

Set then u_{α} := x_{α} for α ∈ Δ_{𝔨}, and define u_{α} for α ∈ Δ̃_{𝔨} to be the projection of x_{α} onto I_{K·B}. It is straightforward to check that u_{α} are as required. □

Lemma 4 Set Z := ∪_{α∈Δ̃_{𝔨}}Z_{α} where Z_{α} is the set of zeroes of u_{α} in U, and V := U\Z. Then Stab_{𝔤}V = 𝔨.

Proof. Put Y := (G/B)\V. Then Y is the union of irreducible components

Y = (∪_{α∈Δ̃_{𝔨}} Z̄_{α}) ∪ Z_{1} ∪ ⋯ ∪ Z_{n} ,

where Z_{1}, …, Z_{n} are all Schubert varieties of codimension 1 in G/B. Furthermore, Stab_{𝔤}V = Stab_{𝔤}Y preserves each irreducible component, in particular

Stab_{𝔤}V ⊂ (Stab_{𝔤}Z̄) ∩ 𝔫^{−} = 𝔨 .

On the other hand, the definition of u_{α} implies that Z_{α} is 𝔨-invariant. Therefore V is 𝔨-invariant and

Stab_{𝔤}V = (Stab_{𝔤}Z̄) ∩ 𝔫^{−} = 𝔨 .
□

Fix now a map λ : Δ̃_{𝔨} → ℂ and consider the vector space

F_{λ,μ}^{𝔨,𝔟} := O(V) ⊗_{ℂ} ⟨ ∏_{α∈Δ̃_{𝔨}} u_{α}^{λ(α)}⟩_{ℂ} .

It has a natural structure of a D^{μ}(V)-module and therefore of a Γ(D^{μ})-module (as Γ(D^{μ}) is a subalgebra of D^{μ}(V)). Define F_{λ,μ}^{𝔨,𝔟} as the localization of F_{λ,μ}^{𝔨,𝔟} to a D^{μ}-module, i.e. set

F_{λ,μ}^{𝔨,𝔟} := D^{μ} ⊗_{Γ(D^{μ})} F_{λ,μ}^{𝔨,𝔟} .

Then Γ(F_{λ,μ}^{𝔨,𝔟}) = F_{λ,μ}^{𝔨,𝔟}.

Proposition 2 For almost all pairs λ, μ, F_{λ,μ}^{𝔨,𝔟} is an irreducible D^{μ}-module and 𝔤[F_{λ,μ}^{𝔨,𝔟}] = 𝔨.

Proof. First we show that F_{λ,μ}^{𝔨,𝔟} is an irreducible D^{μ}-module whenever μ is generic and (imλ) ∩ ℤ = ∅. Note that, as I_{Z} is generated by u_{α} for α ∈ Δ̃_{𝔨},

F_{λ,μ}^{𝔨,𝔟} = ℂ[u_{β}]_{β∈Δ_{𝔨}} ⊗_{ℂ} ℂ[u_{α}^{±1}]_{α∈Δ̃_{𝔨}} ⊗_{ℂ} ⟨ ∏_{α∈Δ̃_{𝔨}} u_{α}^{λ(α)}⟩_{ℂ} .

Therefore an immediate verification, using the condition (imλ) ∩ ℤ = ∅, shows that as a D(U)-module F_{λ,μ}^{𝔨,𝔟} is generated by each monomial of the form ∏_{α∈Δ^{−}} u_{α}^{m_{α}}, where m_{α} ∈ ℤ_{+} for α ∈ Δ_{𝔨} and m_{α} ∈ λ_{α}+ℤ for α ∈ Δ̃_{𝔨}. On the other hand, F_{λ,μ}^{𝔨,𝔟} is a semisimple module over the commutative subalgebra generated by u_{α} \frac{∂}{∂u_{α}} for α ∈ Δ^{−}, and the corresponding weight spaces of F_{λ,μ}^{𝔨,𝔟} are 1-dimensional and are spanned precisely by the above monomials. As any submodule of a weight module is also a weight module, every non-zero D(U)-submodule of F_{λ,μ}^{𝔨,𝔟} contains at least one monomial, i.e. F_{λ,μ}^{𝔨,𝔟} is necessarily an irreducible D(U)-module. Thus F_{λ,μ|U}^{𝔨,𝔟} is an irreducible D_{|U}^{μ}-module (as U is affine and thus D_{|U}^{μ}-affine).

To prove the irreducibility of F_{λ,μ}^{𝔨,𝔟}, consider the atlas {U_{w}}_{w∈W} where U_{w} := w(U) (U = U_{id}). It suffices to show that F_{λ,μ}^{𝔨,𝔟}(U_{w}) is an irreducible D(U_{w})-module for each w (as U_{w} is affine and thus D_{G/B|U_{w}}^{μ}-affine). The crucial point here is that, if w is the reflection corresponding to a simple root α of 𝔟,

F_{λ,μ}^{𝔨,𝔟}(U_{w}) = F_{λ′,μ}^{w(𝔨)∩𝔨,w(𝔟)} ,

where

λ′(β) := \begin{cases}λ(β) & if β ≠ α \\ −λ(β) + 2\frac{(μ,β)}{(β,β)} & if β = α and − α ∈ Δ̃_{𝔨} \\ −2\frac{(μ,β)}{(β,β)} & if β = α and − α ∈ Δ_{𝔨} . \end{cases}

The fact that μ is generic ensures that λ′ also satisfies the condition (imλ′) ∩ ℤ = ∅, and thus F_{λ′,μ}^{w(𝔨)∩𝔨,w(𝔟)} is an irreducible D(U_{w})-module. Therefore induction on the length of w with respect to 𝔟 enables us to complete the argument.

Finally, it remains to check that 𝔤[F_{λ,μ}^{𝔨,𝔟}] = 𝔨. Note that the restriction homomorphism Γ(F_{λ,μ}^{𝔨,𝔟}) → F_{λ,μ}^{𝔨,𝔟}(V) is an isomorphism. If x ∈ 𝔤[F_{λ,μ}^{𝔨,𝔟}] and V_{x} := g_{x} · V, the restriction homomorphism Γ(F_{λ,μ}^{𝔨,𝔟}) → F_{λ,μ}^{𝔨,𝔟}(V_{x}) is also an isomorphism. Thus F_{λ,μ}^{𝔨,𝔟}(V) and F_{λ,μ}^{𝔨,𝔟}(V_{x}) are canonically identified. Furthermore, F_{λ,μ}^{𝔨,𝔟}(V_{x}) = F_{λ,μ}^{𝔨,𝔟}(V ∩ V_{x}) by the very definition of F_{λ,μ}^{𝔨,𝔟}. This is sufficient to conclude that V = V_{x}, as otherwise the codimension of V \(V ∩ V_{x}) in V would necessarily be 1 and the restriction map F_{λ,μ}^{𝔨,𝔟}(V) → F_{λ,μ}^{𝔨,𝔟}(V ∩ V_{x}) would not be surjective. Therefore x ∈ Stab_{𝔤}V, and since Stab_{𝔤}V = 𝔨 (Lemma 4), we obtain

𝔤[F_{λ,μ}^{𝔨,𝔟}] ⊂ 𝔨 .

The opposite inclusion 𝔨 ⊂ 𝔤[F_{λ,μ}^{𝔨,𝔟}] is verified directly from the definition of F_{λ,μ}^{𝔨,𝔟}. □

2.3 The case of a general 𝔨. Assume now that 𝔨 is arbitrary. The subalgebra 𝔨_{ss} of 𝔤 generated by all 𝔤^{α} for α ∈ Δ_{𝔨} ∩ −Δ_{𝔨} is a Levi subalgebra of 𝔨. Let 𝔠 denote the centralizer of 𝔨_{ss} in 𝔤. Furthermore, for a basis h_{1}, …, h_{r} of 𝔨_{ss} ∩ 𝔥 arising from a Chevalley basis of 𝔨_{ss}, set h := h_{1} + ⋯ + h_{r}. Then

𝔭^{h} := 𝔥 ⊕ ( ⨁_{α(h)≥0} 𝔤^{α})

is a parabolic subalgebra. We call any Borel subalgebra 𝔟 𝔨-preferable if 𝔥 ⊂ 𝔟 ⊂ 𝔭^{h} for a suitable choice of a Chevalley basis.

Fix a 𝔨-preferable Borel subalgebra 𝔟 and set S := C ·B. Clearly, S is a closed subvariety of G/B isomorphic to the flag variety C/(C ∩ B). As 𝔠 ∩ 𝔨 is a solvable subalgebra we can consider U_{S}, Z_{S}, V_{S} ⊂ S as in section 2.2 for the pair (𝔠, 𝔠 ∩ 𝔨) instead of (𝔤, 𝔨). Furthermore, we define Y_{S} to be the set of all points s ∈ V_{S} for which dim((𝔨 + 𝔠) ∩ 𝔟_{s}) is minimal possible. Obviously, Y_{S} is an affine open K ∩ C-invariant subvariety of V_{S}.

Lemma 5 Q := K · Y_{S} is a non-singular subvariety in G/B and 𝔨 ⊂ Stab_{𝔤}(Q̄) ⊂ 𝔨 + 𝔠.

Proof. To prove that Q is non-singular it is sufficient to show that every s ∈ Y_{S} is a non-singular point in K · S, i.e. that the dimension of the tangent space T_{K·S}(s) is constant for all s ∈ Y_{S}. The latter follows immediately from the equality

dimT_{K·S}(s) = dim(𝔨 + 𝔠) − dim((𝔨 + 𝔠) ∩ 𝔟_{s}),

as by the definition Y_{S} consists of s for which the right-hand side is maximal possible.

It is obvious that 𝔨 ⊂ Stab_{𝔤}Q̄. Furthermore S = V̄_{S}, and thus K · S ⊂ Q̄. If x ∈ Stab_{𝔤}Q̄, then g_{tx} · B ∈ K · S for sufficiently small t > 0 as K · S is locally closed in G/B. This implies that x ∈ 𝔨 + 𝔠 + 𝔟, i.e.

𝔨 ⊂ Stab_{𝔤}Q̄ ⊂ 𝔨 + 𝔠 + 𝔟 .

To show that actually Stab_{𝔤}Q̄ ⊂ 𝔨+𝔠, consider Stab_{𝔤}Q̄ as a 𝔨_{ss}-module. Stab_{𝔤}Q̄ is isomorphic to 𝔨 ⊕ 𝔪 for some 𝔨_{ss}-submodule 𝔪 of 𝔠 + 𝔟. Let α be a 𝔟 ∩ 𝔨_{ss}-minimal weight of 𝔪. Then α(h_{i}) ≤ 0 for i = 1, …, r and, as 𝔪 ⊂ 𝔠 + 𝔟, we have 𝔤^{α} ⊂ 𝔟. Thus, α(h) ≥ 0. This is possible only when α(h_{i}) = 0 for each i = 1, …, r, i.e. when 𝔤^{α} ∈ 𝔠. Since 𝔪 is generated by 𝔤^{α} for all minimal α, we have finally 𝔪 ⊂ 𝔠. Therefore Stab_{𝔤}Q̄ ⊂ 𝔨 + 𝔠. □

Let 𝔨_{r} denote the radical of 𝔨. As an 𝔥-module 𝔨_{r} equals (𝔨∩𝔠)⊕𝔱 for a unique 𝔥-invariant subspace 𝔱 of 𝔨_{r}. Set T:=exp𝔱. Since 𝔨_{r} is solvable and 𝔱 is contained in the nilpotent radical of 𝔨_{r}, one can easily show that the multiplication map m : T × (K ∩ C) → K_{r} is an isomorphism of algebraic varieties. Therefore K_{ss} × T × (K ∩ C) ≃ K, and furthermore K_{ss} × T × C ≃ K · C. Hence one can define a projection K × C → C, and this projection induces the morphism p : K · C · B → S. Obviously p(Q) = Y_{S}.

Recall our construction for the solvable case and apply it to the pair (𝔠, 𝔠 ∩ 𝔨). Under the assumptions that the the restriction of μ to 𝔠∩𝔥 is generic (and 𝔟∩𝔠-dominant) and that the restriction of μ to 𝔨_{ss} ∩ 𝔥 equals the half-sum 𝔨_{ss} ∩ 𝔟-positive roots of 𝔨_{ss}, this construction yields an irreducible D_{S}^{μ}-module F_{S}. Put F := F_{S|Y_{S}}.

Proposition 3 M := Γ(i_{∗!}p^{∗}F) is an irreducible strict (𝔤, 𝔨)-module.

Proof. Obviously p^{∗}F is an irreducible D_{Q}^{μ}-module which is a locally free O_{Q}-module of finite rank. Therefore i_{∗!}p^{∗}F is an irreducible D^{μ}-module, and M is an irreducible 𝔤-module. All it remains to show is that 𝔤[M] = 𝔨.

We have 𝔨 = 𝔨̃ ⊕ (𝔠 ∩ 𝔥) where 𝔨̃ := [𝔨, 𝔨]. As a 𝔨̃-sheaf i_{∗!}p^{∗}F is isomorphic to i_{∗!}O_{Q}, and thus the 𝔨̃-action on i_{∗!}p^{∗}F comes from the action of K̃ ⊂ G on O_{Q}. This implies that 𝔨̃ acts locally finitely on M. As a 𝔠 ∩ 𝔥-sheaf i_{∗!}p^{∗}F is isomorphic to O_{Q} ⊗ L for some one-dimensional 𝔠∩𝔥-module L. Therefore 𝔠∩𝔥 acts also locally finitely on M, and 𝔨 ⊂ 𝔤[M].

To verify the opposite inclusion, note that, by Lemma 2, 𝔤[M] ⊂ Stab_{𝔤}Q̄ ⊂ 𝔨 + 𝔠. Thus we need to check only that 𝔤[M] ∩ 𝔠 ⊂ 𝔨 ∩ 𝔠. By construction, Γ(F) is an isotropic strict (𝔠, 𝔨 ∩ 𝔠)-module. Therefore Γ(p^{∗}F_{|Q}) is an isotropic strict (𝔠, 𝔨 ∩ 𝔠)-module. Furthermore, i_{∗}p^{∗}F has a natural 𝔠-sheaf filtration whith successive quotients

Λ^{max}(N_{Q}) ⊗_{O_{Q}} S^{i}(N_{Q}) ⊗_{O_{Q}} p^{∗}F.

This implies that Γ(i_{∗}p^{∗}F) is also an isotropic strict (𝔠, 𝔨∩𝔠)-module. Since M is a submodule of Γ(i_{∗}p^{∗}F), finally 𝔠[M] = 𝔤[M] ∩ 𝔠 = 𝔨 ∩ 𝔠.

□

Example 1 If 𝔠 is abelian, Q is a (possibly non-closed) K-orbit in G/B. In this case our construction is the same as the Beilinson-Bernstein construction of Harish-Chandra modules, see [B], [BB1], [BB2]. In the classical Harish-Chandra setting 𝔨 coincides with the fixed points of an inner involution and 𝔠 is automatically abelian.

3 On (𝔤, 𝔨)-modules of finite type

Let M be a 𝔤-module and 𝔨 be a Lie subalgebra of 𝔤[M] whose reductive part 𝔨_{red} acts semisimply on M. ( If M is irreducible then 𝔨_{red} automatically acts semisimply.) We say that M is of finite (respectively, infinite) type over 𝔨 if each isotypic 𝔨_{red}-component of M is finite-dimensional (resp., infinite-dimensional). When M is a strict (𝔤, 𝔨)-module we will say simply that M is of finite (or, infinite) type. We define a Fernando subalgebra 𝔨 of 𝔤 to be of finite type if there exists an irreducible strict (𝔤, 𝔨)-module of finite type. Otherwise we say that 𝔨 is of infinite type. The problem of classifying all Fernando subalgebras of finite type is of obvious significance in the structure theory of 𝔤-modules. In this section we give separate necessary and sufficient conditions for a subalgebra 𝔨 with 𝔨 ⊃ 𝔥 (𝔨 is a Fernando subalgebra by Theorem 1) to be of finite type, providing in this way a partial solution of this problem.

Our starting point is the following Lemma which is similar to the fact that the weight multiplicities of an irreducible weight 𝔤-module are either all finite or all infinite.

Lemma 6 For any reductive subalgebra 𝔨 ⊂ 𝔤, an irreducible (𝔤, 𝔨)-module is either of finite or infinite type over 𝔨.

Proof. Any given irreducible (𝔤, 𝔨)-module M is a quotient of an induced module

M′ := U(𝔤)⊗_{U(𝔤)^{𝔨}}M_{0},

where M_{0} is a (any) 𝔨-isotypic component of M and U(𝔤)^{𝔨} := {y ∈ U(𝔤)|ad_{𝔨}(y) = 0}. If M is not of infinite type over 𝔨, we can choose M_{0} to be finite-dimensional. Therefore, to prove that M is then of finite type over 𝔨, it suffices to show that M′ is of finite type over 𝔨.

Note that there is an isomorphism of 𝔨-modules

M′ ≃ (U(𝔤)/(U(𝔤) · U(𝔤)^{𝔨})) ⊗_{ℂ} M_{0}

(U(𝔤) being 𝔨-module via the adjoint action). As M_{0} is finite-dimensional, it is enough to prove that U(𝔤)/(U(𝔤) · U(𝔤)^{𝔨}) is of finite type over 𝔨. The Poincaré-Birkhoff-Witt theorem gives an isomorphism of 𝔨-modules

U(𝔤)/(U(𝔤) · U(𝔤)^{𝔨}) ≃ S^{·}(𝔤)/I(𝔤),

I(𝔤) being the ideal in S^{·}(𝔤) generated by the invariants of the coadjoint action of 𝔨 in 𝔤^{∗}. Furthermore, S^{·}(𝔤)/I(𝔤) ≃grΓ(O_{X}), where X is a generic closed K-orbit in 𝔤^{∗}, O_{X} is the sheaf of regular functions on X, and gr stands for graded ring. Since passing to the graded ring commutes with the 𝔨-action, all it remains to show is that Γ(O_{X}) is of finite type over 𝔨. But, for any x ∈ X, we have X ≃ K/K^{x}, where K^{x} is the stabilizer of x. Thus Γ(O_{X}) can be identified with the subspace of regular functions on K which are invariant with respect to right multiplication by elements from K^{x}. As K is a linear algebraic group, the 𝔨-module of all regular functions on K is of finite type over 𝔨. Therefore Γ(O_{X}) is also of finite type over 𝔨.

□

In the rest of the paper we assume that 𝔨 ⊃ 𝔥 and address the question of when 𝔨 is of finite type. If 𝔨 is solvable, a (𝔤, 𝔨)-module is of finite 𝔨-type if and only if it has finite weight multiplicities. Furthermore there is a simple criterion for a solvable 𝔨 to be of finite type. Namely, 𝔨 is of finite type if and only if 𝔤 = 𝔨 + 𝔭 for some parabolic subalgebra 𝔭 of 𝔤 with 𝔭 ∩ 𝔨 = 𝔥 such that 𝔭_{ss} is a direct sum of simple Lie algebras of types A and C. In one direction this is a direct corollary of Lemma 1, and in the other direction it follows from the fact that, given 𝔭, there always exists an irreducible strict (𝔤, 𝔨)-module M of finite type with 𝔭_{M} = 𝔭. M can be taken as the module induced from a suitable 𝔭_{ss} + 𝔨-module M_{0} with trivial action of [𝔨, 𝔨]. We leave it to the reader to check this.

The above observation leads to the following necessary condition for a general 𝔨 to be of finite type. Recall that the reductive subalgebra 𝔠 ⊂ 𝔤 is defined as the centralizer of 𝔨_{ss} in 𝔤.

Proposition 4 Let 𝔨 be of finite type. Then the solvable subalgebra 𝔨 ∩ 𝔠 is of finite type in 𝔠, i.e. 𝔠 = (𝔨 ∩ 𝔠) + 𝔭 for some parabolic subalgebra 𝔭 of 𝔠 with 𝔭 ∩ 𝔨 = 𝔥 and such that 𝔭_{ss} is a direct sum of simple Lie algebras of types A and C. In addition, [𝔨, 𝔭_{ss}] ⊂ 𝔨.

Proof. Let M be an irreducible strict (𝔤, 𝔨)-module of finite type and M′ be a 𝔨_{ss}- isotypic component of M. Fix a Borel subalgebra 𝔟′ ⊂ 𝔨_{red} with 𝔟′ ⊃ 𝔥 and set M′′ := {m′ ∈ M′|[𝔟′, 𝔟′] · m′ = 0}. An immediate verification shows that M′′ is an isotropic strict (𝔠, 𝔨 ∩ 𝔠)-module of finite type. Therefore 𝔠 = (𝔨 ∩ 𝔠) + 𝔭 for a parabolic subalgebra 𝔭 of 𝔠 as in Lemma 1.

It remains to show that [𝔨, 𝔭_{ss}] ⊂ 𝔨. Assume the contrary, i.e. that there are γ ∈ Δ_{𝔨}\Δ_{𝔨_{red}} and β ∈ Δ_{𝔭_{ss}} with α = β + γ ∈ Δ\Δ_{𝔨}. Moreover, γ can be assumed to be a 𝔟′-maximal weight of the 𝔨_{red}-module 𝔨. Let α_{1}, …, α_{s} be the simple roots of 𝔟′. Then −γ, α_{1}, …, α_{s} is a system of simple roots of a reductive subalgebra 𝔰 containing 𝔨_{red}. The subalgebra 𝔮 := 𝔨 ∩ 𝔰 is a parabolic subalgebra of 𝔰, and its reductive part equals 𝔨_{red}. As 𝔮 ⊂ 𝔨, there is an irreducible finite-dimensional 𝔮-submodule L_{λ} ⊂ M of 𝔟′-highest weight λ. Furthermore, since Δ_{𝔮}\Δ_{𝔨_{red}} ⊂ Γ_{𝔨}, M has an infinite family of finite-dimensional irreducible 𝔮-submodules L_{λ_{k}} with highest weights λ_{k} belonging to {λ + ⟨Δ_{𝔮}\Δ_{𝔨_{red}}⟩_{ℤ_{+}}}. Let M_{k} := U(𝔰) · L_{λ_{k}}. For almost all k, M_{k} contains a 𝔨_{red}-submodule isomorphic to L_{λ}. Thus the multiplicity of L_{λ} in M is infinite. Contradiction. □

Proposition 4 implies the existence of a large class of reductive subalgebras 𝔨 of infinite type.

Corollary 1 Let 𝔤 be simple of types B_{n} for n > 4, D_{n} for n > 4, E_{7}, E_{8}, and let 𝔨 be reductive and such that 𝔠_{ss} has a simple component not of type A or C. Then 𝔨 is of infinite type.

Example 2 The Lie algebra 𝔤 = sp(6) is the simple Lie algebra of smallest dimension which admits a subalgebra 𝔨 of infinite type with non-zero nilpotent radical and non-zero semisimple part. Let 𝔨 = 𝔥⊕𝔤^{α} ⊕𝔤^{−α} ⊕𝔤^{β} where α and β are respectively a long and a short orthogonal roots. Then there is no 𝔭 as in Proposition 5, and hence 𝔨 is of infinite type.

Our final step is to state a sufficient condition for a reductive 𝔨 to be of finite type.

Proposition 5 Let 𝔨 be reductive and such that the simple components of 𝔠_{ss} are of types A and C only. If in addition ⟨Δ_{𝔠}⟩_{ℂ} ∩ Δ = Δ_{𝔠}, then 𝔨 is of finite type.

Proof. Let 𝔥′ be the centralizer of 𝔠_{ss} in 𝔥, and 𝔨′ := 𝔨_{ss}+𝔥′. The condition ⟨Δ_{𝔠}⟩_{ℂ}∩Δ = Δ_{𝔠} enables us to choose a 𝔨-preferable Borel subalgebra 𝔟 for which the projection of Δ^{−}\Δ′_{𝔠} onto (𝔥′)^{∗} is contained in some open half-space of some real subspace of (𝔥′)^{∗}. Put Q_{1} := K′/(K′ ∩ B), Q_{2} := C_{ss}/(C_{ss} ∩ B) and Q := (K′ × C_{ss}) · B. Then Q = Q_{1} × Q_{2} is a closed subvariety of G/B, and Lemma 5 implies that Stab_{𝔤}Q = 𝔨′ ⊕ 𝔠_{ss}. Let F_{2} be an irreducible strict (𝔠_{ss}, 𝔥′′)-module with finite weight multiplicities, where 𝔥′′ := 𝔠_{ss} ∩ 𝔥. The existence of F_{2} is well-known, see for instance [M]. Denote by F_{2} the localization of F_{2} on Q_{2}. Set finally F := O_{Q_{1}} ⊠ F_{2}, M := i_{∗}F and M := Γ(M), i being the closed embedding of Q into G/B.

Arguments similar to ones in the proof of Proposition 3 imply that M is a strict irreducible (𝔤, 𝔨)-module. We will show that M is of finite type. Consider the filtration of M with successive quotients

Λ^{max}(N_{Q}) ⊗_{O_{Q}} S^{i}(N_{Q}) ⊗_{O_{Q}} F,

and choose a finer filtration of M such that its successive quotients are all sheaves O(λ)⊗_{O_{Q}} F, where λ runs over the multiset D of sums of roots from Δ^{−}\Δ_{𝔨+𝔠} and O(λ) stands for the invertible K′ × C_{ss}-sheaf on Q induced by λ. Then O(λ) ⊗_{O_{Q}} F ≃ O(λ′) ⊠ F(λ′′), where λ′ (respectively, λ′′) is the projection of λ on (𝔥′)^{∗} (resp., (𝔥′′)^{∗}) and F(λ′′) := F_{2} ⊗_{O_{Q_{2}}} O(λ′′). Thus M is a 𝔨 + 𝔠-submodule of ⊕_{λ∈D}(Γ(O(λ′)) ⊠ Γ(F(λ′′))). The Borel-Weil-Bott theorem implies that Γ(O(λ′)) ≠ 0 if and only if λ′ is 𝔟∩𝔨-antidominant. In the latter case Γ(O(λ′)) = V (−λ′)^{∗}, where V (−λ′) is the irreducible finite-dimensional 𝔨′-module of 𝔟∩𝔨′-highest weight −λ′. Furthermore Γ(F(λ′′)) is a (𝔠_{ss}, 𝔥′′)-module of finite type over 𝔥′′. Therefore to show that M has finite type it suffices to check that for each ω ∈ (𝔥′)^{∗} the set D_{ω} := {λ ∈ D|λ′ = ω} is finite. The latter is the direct corollary of the fact that the projection of D onto (𝔥′)^{∗} is contained in an open half-space of a real subspace of (𝔥′)^{∗}. □

Corollary 2 Let 𝔨 be reductive.

(a) If 𝔠 is abelian, then 𝔨 is of finite type.

(b) Assume that 𝔤 has no simple components of type B_{n} for n > 2 or F_{4}. Then 𝔨 is of finite type if and only if all simple components of 𝔠_{ss} are of types A and C.

Proof. (a) is obvious. In proving (b) one can assume that 𝔤 is simple. If 𝔤 is not of type C_{n}, B_{n} for n > 2 or F_{4}, the condition that ⟨Δ_{𝔠}⟩_{ℂ} ∩ Δ = Δ_{𝔠} is automatic, and therefore Propositions 4 and 5 imply the statement. To check the condition ⟨Δ_{𝔠}⟩_{ℂ} ∩ Δ = Δ_{𝔠} for 𝔤 of type C_{n} we will use the fact that 𝔠 is the centralizer 𝔨_{ss}. Assume, to the contrary, that ⟨Δ_{𝔠}⟩_{ℂ}∩Δ ≠ Δ_{𝔠}. Then there are two orthogonal long roots α, β ∈ Δ_{𝔠} such that (α+β)/2 ∉ Δ_{𝔠}. Furthermore, for γ ∈ Δ_{𝔨_{ss}}, α ± γ, β ± γ ∉ Δ_{𝔠} but (α + β)/2 + γ ∈ Δ_{𝔠}. This is impossible. □

In the classical Harish-Chandra setting, when 𝔨 coincides with the fixed points of an inner involution, every irreducible strict (𝔤, 𝔨)-module is of finite type. The latter is not true just under the assumption that 𝔨 is of finite type. It is an open problem to characterize all subalgebras 𝔨 for which any irreducible strict (𝔤, 𝔨)-module is of finite type.

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