Classification of irreducible weight modules Olivier Mathieu*

Summary: Introduction

0. General conventions and definitions.

1. Reduction of the classification to cuspidal modules.

2. Generalities about weight modules.

3. Support of admissible modules.

4. Coherent families.

5. Components of coherent families.

6. Admissible highest weight modules.

7. Computations in rank two.

8. Classification of coherent families for 𝔰𝔩(n + 1).

9. Classification of coherent families for 𝔰𝔭(2n).

10. Classification of cuspidal representations.

11. Explicit realization of coherent families for 𝔰𝔩(n + 1).

12. Explicit realization of coherent families for 𝔰𝔭(2n).

13. Character formula for simple weight modules. Appendix: Generalized Enright functors.

Introduction: Let 𝔤 be a reductive algebra and let 𝔥 be a Cartan subalgebra. A 𝔤module M is called a weight module if and only if M = ⊕_{λ}M_{λ}, where each weight space M_{λ} is finite dimensional (in the litterature, the last condition is sometimes dropped). The main result of this paper is the classification of all simple weight 𝔤-modules.

We will now describe the strategy to get such a classification. For any proper parabolic subalgebra 𝔭⊃𝔥 with unipotent radical 𝔲, and any simple weight 𝔭/𝔲-module L, the induced module ind(𝔭, 𝔤,L) has a unique simple quotient denoted by L(𝔭,L). The simple weight module L(𝔭,L) is called parabolically induced, and a simple weight module which is not parabolically induced is called cuspidal. By a result due to Fernando, any simple weight module is parabolically induced from a cuspidal module. Hence the general classification reduces to the classification of cuspidal modules (see more details in section 1). Moreover, for simple Lie algebras 𝔤, there are cuspidal modules only if 𝔤 has type A or C. Thus the main result of the paper is the explicit classification of cuspidal modules for Lie algebras of type A and C.

The idea of this classification is based on the notion of coherent families. A coherent family of degree d is a weight module M such that dim M_{λ} = d, for all λ ∈𝔥^{∗} and such that the function λ ↦ T r(u|M_{λ}) is polynomial in λ, for all u in the centralizer of 𝔥 in U(𝔤). For any Q-coset t, M[t] = ⊕_{λ∈t} M_{λ} is a 𝔤-submodule of M. We show that all cuspidal modules occurs as a module M[t] for an unique t and a unique coherent family M which is irreducible and semi-simple (as it is defined in Section 4).

It turns out that semi-simple irreducible coherent families M are completely classified by the list of all infinite dimensional highest weight modules that occur in M. Moreover the highest weight modules L(λ) occuring in M are admissible, i.e. the multiplicity of

* Research supported by UA 1 du CNRS weights in L(λ) is uniformly bounded. In order to classify coherent families, we first classify all simple admissible highest weight modules. This classification can be done by using Kazhdan-Lusztig combinatorics, but we follow a more elementary approach because it gives quicker proofs. From this we deduce the theorems 8.5 and 9.3 which state the classification of semi-simple irreducible coherent families for type A and C respectively.

The last step of the classification is Theorem 10.2 which determine in each irreducible coherent family M the full list of its cuspidal submodules, i.e. to determine for which coset t the module M[t] is cuspidal. In the sections 11 and 12, we give a more concrete version of the classification of semi-simple coherent families and we describe explicitly all such families. For 𝔤=𝔰𝔩(n), irreducible coherent families are classified by some finite dimensional simple 𝔤𝔩(n)-module and they can be described as a space of tensors or as a difference of two spaces of tensors (Theorem 11.5). For 𝔤=𝔰𝔭(2n), irreducuble coherent families are classified by odd pairs of Spin(2n)-modules and they are closely related to the Shale-Weil representation (Theorem 12.2).

In the last section, we give a character formula for all simple weight modules (Theorem 13.4). Beside Fernando’s Theorem, there were two main results toward the classification of all simple weight modules in the previous litterature. For 𝔰𝔩(2), the classification is usually attributed to Gabriel [Gab] and can be found in [Mi] and in [D] (7.8.16). More recently, Britten and Lemire classified all simple multiplicity free weight modules [BL], see also [BHL] [BFL]. Using our methods, we recover the result of Britten and Lemire (see the two corollaries 11.5 and 12.3).

Acknowledgements: We thank heartily V.M. Futorny, T. Joseph, V. Mazorchuk for their helpful conversations, and W. Soergel for his comments concerning the appendix.

0. General conventions and definitions.

(0.1) An important convention Let x, y ∈ K be scalars. The notation x ≥ y means that x − y is a positive integer (we use the same conventions with the operator ≥). A half integer is a scalar of the form 1/2+n, where n ∈ Z. So an half integer x is never an integer and ±x cannot be ≥ 0. The only exception concerning this convention will be the lemmas 3.2 and 3.3, where real numbers are involved.

(0.2) General definitions: For a reductive Lie algebra 𝔤, we will always fix a Cartan subalgebra 𝔥. We will denote by Δ the set of roots, by Q the root lattice and for α ∈ Δ we will denote by h_{α} the corresponding coroot, by e_{α} a corresponding root vector.

For be any 𝔥-module M and any λ ∈𝔥^{∗}, we denote by M_{λ} the weight space of weight λ, i.e. the subspace of all m ∈ M such that h.m = λ(h).m for any h ∈𝔥, and for a subset t of 𝔥^{∗}, we set M[t] = ⊕_{λ∈t}M_{λ}.

Recall that any basis B is a graph, whose edge are pairs of distinct roots α, β with < α|β >≠ 0. We will denote by Δ_{B}^{+} (respectively Δ_{B}^{−}, P_{B}^{+}) the set of positive roots (respectively the set of negative roots roots, the set of dominant integral weights) relative to B. Similarly, we will use the notation L_{B}(λ) to denote the simple B-highest weight simple module with highest weight λ ∈𝔥^{∗}. When there will be no ambiguity on B, the index B will be dropped.

1. Reduction of the classification to cuspidal modules.

In this section we will recall why the general classification of simple weight modules can be reduced to the classification of cuspidal modules of Levi factors. This idea is mostly due to Fernando [F].

Let 𝔤 be a reductive algebra, 𝔥 be a Cartan subalgebra. Let 𝔭 be a parabolic subalgebra with unipotent radical 𝔲 (we always assume that all parabolic subalgebras contain 𝔥). A basis B of Δ is called adapted to 𝔭 if all roots in B are roots of 𝔭. Let L be a simple weight 𝔭/𝔲-module. The induced module ind(𝔭,𝔤,L) has a unique simple quotient denoted by L(𝔭, L) (see e.g. [DMP] or the end of this section). A simple weight module which is not isomorphic to some L(𝔭, L) for some proper parabolic subalgebra 𝔭 is called cuspidal.

Let L be a simple weight 𝔭/𝔲-module as before, let B be an adapted basis of Δ and let B^{⊥} ⊂ B the subset of all roots which are orthogonal to all roots of 𝔭/𝔲. The set B^{⊥} is independent of the choice of the adapted basis B. Moreover for any α ∈ B^{⊥}, h_{α} acts on L as some scalar l_{α}. Let B_{F} the set of all roots α ∈ B^{⊥} such that l_{α} ≥ 0. The root system generated by B_{F} will be denoted by Δ(𝔭,L) and its Weyl group by W (𝔭, L).

THEOREM 1.1 (i) (Fernando) Any simple weight 𝔤-module is isomorphic to L(𝔭, L) for some parabolic subalgebra 𝔭 and some cuspidal 𝔭-module L.

(ii) Let L, L′ be cuspidal representations of two parabolic subalgebras 𝔭, 𝔭'. Then L(𝔭, L) and L(𝔭', L′) are isomorphic if and only if there exists some w ∈ W (𝔭, L) such that 𝔭'=𝔭^{w} and L′ = L^{w}.

The main part of the Theorem is due to Fernando [F]. A quick proof can be found in [DMP], [Jo]. Assertion (ii) is quite trivial, see [DMP].

Set 𝔤=𝔷⊕(⊕_{1≤i≤n}𝔤_{i}) where 𝔷 is the center of 𝔤 and where 𝔤_{i} are its simple components. Any simple weight module M decomposes as Z ⊗ (⊗_{1≤i≤n}M_{i}), where Z is a one dimensional representation of 𝔷 and each M_{i} is a simple weight 𝔤_{i}-module. Thus the general classification reduces to the problem of determining all cuspidal representations for simple Lie algebras, which is the aim of the present paper. So from now on we can assume that 𝔤 is simple. Moreover, Fernando also proved the next statement (see [F]).

PROPOSITION 1.2 (Fernando) Assume that 𝔤 is a simple Lie algebra. Then 𝔤 has at least one cuspidal module if and only if it is of type A or C.

Let M be a simple weight 𝔤-module and let α ∈ Δ. It is easy to prove that the action of e_{α} on M is either injective or locally nilpotent. Following [DMP], we decompose Δ into four parts Δ_{M}^{I}, Δ_{M}^{F}, Δ_{M}^{+} and Δ_{M}^{−} as follows:

Δ_{M}^{I} = {α ∈ Δ; e_{α}|_{M} and e_{−α}|_{M} are injective}

Δ_{M}^{F} = {α ∈ Δ; e_{α}|_{M} and e_{−α}|_{M} are locally nilpotent}
Δ_{M}^{+} = {α ∈ Δ; e_{−α}|_{M} is injective but e_{α}|_{M} is locally nilpotent }
Δ_{M}^{−} = −Δ_{M}^{+}

We will call cone any finitely generated submonoid of Q containing 0. By definition, the cone of M is the monoid C(M) generated by Δ_{M}^{I} ∪ Δ_{M}^{−}. Its support Supp(M) is the set of all its weights. The following statement is proved in [DMP].

PROPOSITION 1.3 Let M be a simple weight 𝔤-module.

(i) There is a finite set Ω such that Supp(M) = Ω + C(M).

(ii) Let M ≃ L(𝔭,L) for some parabolic subalgebra 𝔭 and some cuspidal 𝔭-module L. Then Δ_{M}^{I} is the root system of 𝔭/𝔲, Δ_{M}^{F} = Δ(𝔭,L) and Δ_{M}^{+} is the set of roots of the unipotent radical 𝔳 of some parabolic subalgebra.

(iii) In particular, Δ_{M}^{I} and Δ_{M}^{F} are orthogonal.

Let M be a simple weight module. A basis B is called adapted to M if B is adapted to some parabolic subalgebra 𝔭 such that M ≃ L(𝔭,L), with L cuspidal. By the previous proposition, a basis is adapted if and only if B ∩ Δ_{M}^{I} is a basis of Δ_{M}^{I} and Δ_{M}^{+} ⊂ Δ_{B}^{+} (moreover B ∩ Δ_{M}^{F} is a basis of Δ_{M}^{F}).

It is obvious that the support of a simple weight module lies inside a single Q-coset. The following proposition, which is also a consequence of Proposition 1.3, was first proved by Fernando, see [F].

Proposition 1.4 (Fernando)For a simple weight 𝔤-module M, the following assertion are equivalent:

(i) M is cuspidal,

(ii) e_{α} acts injectively on M for all α ∈ Δ, (iii) The support of M is exactly one Q-coset. Remark: A module satisfying (i) is called torsion-free and a module with property (iii)

is called dense. By the previous proposition, cuspidal, torsion-free and dense are equivalent properties, and in the litterature the three terminologies are used simultaneously (for clarity we will only use the terminology ”cuspidal”).

Let 𝔭 be a parabolic, let 𝔲 be its radical, let Q(𝔭/𝔲) be the root lattice of 𝔭/𝔲. The cone of 𝔭, denoted by C(𝔭), is the cone generated by the opposite of the roots of 𝔲. For any Q(𝔭/𝔲)-coset u, consider the category C_{u} (respectively D_{u}) of all weight 𝔭-modules, or equivalently 𝔭/𝔲-modules, (respectively weight 𝔤-modules) whose support is included in u (respectively in u + C(𝔭)). Let L ∈ C_{u} and let M = ind(𝔭,𝔤,L) be the induced module. As Q(𝔭/𝔲)∩C(𝔭)= 0, the space M decomposes as M = L ⊕ M^{+}, where M^{+} = ⊕_{λ∉u}M_{λ}. Set L(𝔭,L)=M/Z, where Z is the biggest 𝔤-submodule of M included in M^{+}. It is obvious that L ∈ C_{u} ↦ L(𝔭,L) ∈ D_{u} is a functor, which preserves injective and surjective maps (however this functor is not exact). In particular:

(i) if L′ is a suquotient of L, L(𝔭,L′) is a subquotient of L(𝔭,L),

(ii) L(𝔭,L) is simple if and only if L is simple.

More details can be found in [DMP]. As we will repetively use these remarks, let state them as a lemma:

LEMMA 1.5 Let 𝔭 be a parabolic subalgebra with radical 𝔲, let u be a Q(𝔭/𝔲)-coset and let M be a weight 𝔤-module with support inside u + C(𝔭). Set L = M[u].

(i) L is a 𝔭-submodule on which 𝔲 acts trivially,

(ii) for any 𝔭-subquotient L′ of L, L(𝔭,L′) is a subquotient of M.

2. Generalities about weight modules. From now on, 𝔤 denotes a simple Lie algebra and 𝔥 denotes a fixed Cartan subalgebra. Set U = U(𝔤) , let A = U_{0} be the commutant of 𝔥 and let B = ⊕_{λ≠0}U_{λ} be its complement in U. Let λ ∈ 𝔥^{∗}. By definition, a A-module of weight λ is a finite dimensional A-module X such that X = X_{λ} as 𝔥-module. Set I(X) = U ⊗_{A} X and I^{+}(X) = B ⊗_{A} X. We have I(X) = ⊕_{μ∈Q}I(X)_{μ+λ}, where I(X)_{μ+λ} = B_{λ} ⊗_{A} X and I(X)_{λ} = X. Denote by Z(X) the biggest U-submodule contained in I^{+}(X) and set I_{!}(X) = I(X)/Z(X).

LEMMA 2.1 Let X be a A-module of weight λ.

(i) I(X) is a weight module generated by X.

(ii) Let Y be a U-module generated by Y_{λ}. Any morphism of A-modules π : Y_{λ} → X extends to a unique morphism of U-modules Y → I_{!}(X).

Proof: By Hilbert’s Theorem, the right A-module U_{μ−λ} is finitely generated (see e.g. [DMP]), hence I(X)_{μ} is finite dimensional. This proves (i). Let Y be a U-module generated by Y_{λ} and let π^{e} : I(Y_{λ}) → I(X) be the morphim extending π. We have Y ≃ I(Y_{λ})/Z for some submodule Z of I(Y_{λ}) with Z_{λ} = 0. We have π^{e}(Z) ⊂ Z(X) and π^{e} induces a morphism Y → I_{!}(X). Q.E.D.

LEMMA 2.2 Let M be a simple weight module and let λ ∈ Supp(M).

(i) The A-module M_{λ} is simple.

(ii) We have M ≃ I_{!}(M_{λ}).

Proof: The previous proposition is a immediate corollary of Lemma 2.1. Q.E.D. Let M be a weight module. Its trace T r^{M} is the function (λ, u) ∈𝔥^{∗} ×A ↦ T r_{λ}^{M}(u) =

tr(u|_{M_{λ}}). A weight module is called semi-simple if it is a finite or infinite direct sum of simple modules.

LEMMA 2.3 Let M, N two semi-simple weight modules. If T r^{M} = T r^{N}, then M ≃ N. Proof: Let S be a simple weight module, and let λ ∈ Supp(S). By proposition 1.3,

we have we have S ≃ I_{!}(S_{λ}). Hence the mutiplicity of S in M as U-modules is the same as the mutiplicity of S_{λ} in M_{λ} as A-modules. The last mutiplicity being determined by T r_{λ}^{M}, the proposition follows.

3. Support of admissible modules. A weight module M is called admissible if dim M_{λ} is uniformly bounded and Supp(M) is included in finitely many Q-cosets . The degree of an admissible module is the maximal dimension of its weight spaces. For an admissible module M of degree d, denote by Supp_{ess}(M) its essential support, i.e. the set of λ ∈ Supp(M) with dim (M_{λ}) = d. Let M be a cuspidal 𝔤-module. By Fernando’s proposition 1.4, e_{α} and e_{−α} act injectively on M. Hence they act bijectively, and the dimensions of the weight spaces of M are all the same. In particular a cuspidal representation is admissible.

LEMMA 3.1 Let M be an infinite dimensional simple weight 𝔤-module. Then its cone C(M) generates Q as a group.

Proof: First assume that Δ_{M}^{±} = 0. As Δ is the disjoint union of the orthogonal set Δ_{M}^{I} and Δ_{M}^{F}, we have Δ = Δ_{M}^{I} and C(M) = Q. So we can assume that Δ_{M}^{±} ≠ 0. By Proposition 1.3, Δ_{M}^{±} are the roots of the unipotent radicals 𝔳^{±} of two opposed parabolic subalgebras. We have 𝔤=𝔳^{+}+𝔳^{−} +[𝔳^{+},𝔳^{−}], hence ±Δ_{M}^{−} generates Q. As C(M) contains Δ_{M}^{−}, it generates Q.

For x ≥ 0, set ρ(x) = card(B(x) ∩ Q), where B(x) is the closed ball of 𝔥^{∗} with radius x. For a weight module, define its density δ(M) as as the infimum limit (when x → +∞) of ρ(x)^{−1} ∑_{μ∈B(x)} dim M_{λ}. In the following two lemmas and their proofs, the quantities δ(M), ϵ … are real numbers and the symbols > is used with its usual meaning.

LEMMA 3.2 There exists ϵ > 0 such that δ(M) > ϵ, for any infinite dimensional simple weight module M.

Proof: For any cone C, the number δ(C) = inflim_{x→∞}ρ(x)^{−1} card ((λ + C) ∩ B(x)) is independent of λ and is non zero if C generates Q. For any infinite dimensional simple weight module M, we have Supp(M) ⊃ λ + C(M), for some any λ ∈ Supp(M). Thus we get δ(M) ≥ δ(C(M)) and δ(C(M)) > 0 by Lemma 3.1. By proposition 1.3, there are only finitely many different cones attached to all simple weight modules. Hence we can find ϵ > 0 with δ(C(M)) > ϵ for any infinite dimensional simple weight module M, i.e. δ(M) > ϵ for any M. Q.E.D.

LEMMA 3.3 Any admissible weight module M has finite lenght.

Proof: As M is admissible, we have δ(M) < ∞. For any exact sequence 0 → X′ → X → X” → 0, we have δ(X) ≥ δ(X′) + δ(X”). Let X be the set of all integral weights μ such that μ(h_{α}) ≤ 1 for any α ∈ Δ. For any simple module of finite dimension L, we have L_{λ} ≠ 0 for some λ ∈ X, see [B1]. It follows easily that the lenght of M is finite, and bounded by A + δ(M)/ϵ, where the constant ϵ is defined by Lemma 3.2. and A = ∑_{λ∈X} dim M_{λ}. Q.E.D.

LEMMA 3.4 Let C ⊂ Q be a cone generating Q and let S_{1} ⊂ S_{2} be non-empty subsets of 𝔥^{∗} contained in a single Q-coset. Assume that S_{2} = C + Ω for some finite set Ω and that C + S_{1} ⊂ S_{1}. Then S_{2} \ S_{1} is contained in a finite union of affine hyperplanes. In particular S_{1} is Zariski dense but S_{2} \ S_{1} is not Zariski dense.

Proof: Choose α ∈ S_{1}. For any ϕ ∈ Ω, we have α − ϕ ∈ Q, hence α + ϕ_{1} = ϕ + ϕ_{2} for some ϕ_{1}, ϕ_{2} ∈ C. Set α′ = α + ∑_{ϕ∈Ω} ϕ_{1}. We have α′ = ϕ + ϕ_{2} + ∑_{ψ≠ϕ} ψ_{1}, i.e. α′ − ϕ ∈ C for any ϕ. So we have S_{2} \ S_{1} ⊂ ∪_{ϕ∈Φ} ϕ + (C \ ((α′ − ϕ) + C)). Hence we can assume S_{2} = C and S_{1} = β + C for some β ∈ C.

Let Γ be a finite set of generators of C. Let b (respectively c) the set of all subsets of Γ which are a basis of 𝔥^{∗} (respectively the subsets of rank < the rank of Q) and for any T ∈ c denote by H_{T} an hyperplane containing T. For any T ∈ b, there is an identity m_{T} β = ∑_{γ∈T} m_{γ}^{T} .γ, where m_{T} > 0 and all m_{γ}^{T} are integers. Fix a constant N > 0 with m_{γ}^{T} ≤ N for all T ∈ b, γ ∈ T and denote by Γ′ the set of all combinations ∑_{γ∈Γ} m_{γ}γ with 0 ≤ m_{γ} < N for all γ.

Any τ ∈ C can be written as ∑_{γ∈Γ} m_{γ}γ where all m_{γ} are ≥ 0. For such a decomposition of τ, set T = {γ|m_{γ} ≥ N}. If T contains a basis T′, then τ − m_{T′}β ∈ C and τ belongs to S_{2}. Otherwise τ belongs to Γ′ +H_{T}. Hence S_{2} \S_{2} is contained in ∪_{T ∈c} Γ′ +H_{T}. Q.E.D.

PROPOSITION 3.5 Let M be a infinite dimensional admissible simple module.

(i) For any λ ∈ Supp_{ess}(M), we have λ + C(M) ⊂ Supp_{ess}(M),

(ii) The essential support of M is Zariski dense, but Supp(M) \ Supp_{ess}(M) is contained in a finite union of affine hyperplanes.

Proof: Let λ ∈ Supp_{ess}(M) and let α ∈ Δ_{M}^{I} ∪ Δ_{M}^{−}. As e_{α} acts injectively on M, we have dim M_{λ+α} ≥ dimM_{λ}. Hence λ + α ∈ Supp_{ess}(M), and Assertion (i) is proved. Set C = C(M), S_{2} = Supp(M) and S_{1} = Supp_{ess}(M). By Proposition 1.3, Lemma 3.1, and Assertion (i), the triple (C, S_{1}, S_{2}) satisfies the hypotheses of Lemma 3.4. Hence Assertion (ii) follows from Lemma 3.4. Q.E.D.

4. Coherent families. After the preparatory lemmas of Sections 2 and 3, we introduce in this section the main tool used in the classification, namely the coherent families, which will be investigated in the next three sections. A coherent family of degree d is a weight 𝔤-module M such that

(i) dim M_{λ} = d for any λ ∈𝔥^{∗}.

(ii) for any u ∈ A, the function λ ∈𝔥^{∗} ↦ T r(u|M_{λ}) is polynomial in λ.

Define the dual torus T^{∗} of 𝔤 by T^{∗}=𝔥^{∗}/Q. We can decompose any weight module M as M = ⊕_{t∈T^{∗}}M[t]. For a coherent family M, its submodules M[t] are admissible.

Example of a coherent family for 𝔰𝔩(2): Denote by A the vector space with basis (x^{α})_{α∈K}. Denote by e, f, h the usual basis of 𝔰𝔩(2). For any a ∈ K, denote by M(a) the structure of 𝔰𝔩(2)-module on the vector space A given by the following formulas:

f ↦ −d/dx + a/x, h ↦ 2xd/dx and e ↦ x^{2}d/dx + ax,

where x denotes the operator x^{α} ↦ x^{α+1} and d/dx denotes the operator x^{α} ↦ α.x^{α−1}. Clearly, M(a) is an example of coherent family.

A commuting set of roots is a subset Σ ⊂ Δ such that [e_{α}, e_{β}] = 0 for any α, β ∈ Σ. For any subset I of a basis B of Δ, denote by Q_{I}, Δ_{I}, Δ_{I}^{+} the subgroup of Q generated by I, the root system generated by I, the set of positive roots of Δ_{I} with respect to its basis I.

LEMMA 4.1 Let B be a basis of Δ, let I ⊂ B.

(i) For each connected component of c of I, let β_{c} ∈ c. There exists a set of commuting roots Σ′ ⊂ Δ_{I}^{+} such that Σ′ is a basis of Q_{I} and such that each β_{c} belongs to Σ′.

(ii) Let F be a proper subset of B with F ∩ I = ∅, and let Σ′ ⊂ Δ_{I}^{+} be a set of commuting roots which is a basis of Q_{I}. There exists a set of commuting roots Σ which is a basis of Q such that Σ′ ⊂ Σ ⊂ Δ_{B}^{+} \ Δ_{F}^{+}.

Proof: Assertion (i) follows easily from Assertion (ii), so we only prove Assertion (ii). We can write B = {α_{1}, …, α_{n}} in a such way that I = {α_{1}, …, α_{p}}, and any connected component of {α_{1}, …, α_{s}} contains at least one element outside F for any s > 0. For s > p, set γ_{s} = ∑_{j∈J_{s}} α_{j}, where J_{s} is the connected component of {α_{1}, …, α_{s}} containing α_{s}. By [B2], γ_{s} is a root, and by definition γ_{s} ∉ Δ_{F}

Define the commuting set of roots Σ = {β_{1}, …, β_{n}} as follows. First set {β_{1}, …, β_{p}} = Σ′. Then define the roots β_{p+1}, …, β_{n} by induction. For s ≥ p, assume that the set of commuting roots {β_{1}, ..., β_{s}} is already defined. Set 𝔪_{s} = ⊕_{1≤j≤s}K e_{β_{j}}. By assumption, 𝔪_{s} is a commutative ad-nilpotent Lie subalgebra, so there is a non zero root vector e in Ad(U(𝔪_{s}))(e_{γ_{s+1}}) with [𝔪_{i}, e] = 0. Thus define β_{s+1} to be the root of e, i.e. e = e_{β_{s+1}}. By definition, Σ = {e_{β_{j}}|1 ≤ j ≤ n} is a set of commuting roots. Moreover for s > p, we have β_{s} = α_{s} + ∑_{k<j} m_{k,j}α_{k} for some integers m_{k,j}. Hence Σ is a basis of Q.

LEMMA 4.2 Let R be an associative algebra without zero divisor and let S be a multiplicative subset generated by locally ad-nilpotent elements. Then S satisfy Ore’s conditions for localizability.

Proof: Let Γ be a set of locally ad-nilpotent generators of S. To check Ore’s conditions it suffices to prove that for any s ∈ Γ and r ∈ R, we can find s′, s” ∈ S and r′, r” ∈ R with s.r′ = r.s′ and r”.s = s”.r.

Let r ∈ R and s ∈ Γ. We have ad^{N}(s)(r) = 0 for some N > 0. Then the identity ∑_{0≤i≤N}(−1)^{i}(_{i}^{N})s^{i}.r.s^{N−i} = 0 can be written as s.r′ = r.s^{N} and r”.s = s^{N}.r for some r′, r” ∈ A. Hence Ore’s conditions are satisfied. Q.E.D.

For S and R as previously, we will denote by R_{S} the localization of R relative to S and for any R-module M we will set M_{S} = R_{S} ⊗_{R} M. Recall that the map M → M_{S} is injective if and only if any s ∈ S acts injectively on M. When S is generated by a single element s, we will set R_{s} = R_{S} and M_{s} = M_{S}.

LEMMA 4.3 Let R be an associative K-algebra, and let S be a multiplicative subset generated by q commuting and locally ad-nilpotent elements s_{1}, …, s_{q} Then there exists a unique q-parameters family of automorphisms Θ_{(x_{1}...,x_{q})} : R_{S} → R_{S} such that we have Θ_{(x_{1}...,x_{q})}(r) = s_{1}^{x_{1}} … s_{q}^{x_{q}}.r.s_{q}^{−x_{q}} … s_{1}^{−x_{1}} if all x_{i} are integers and the map (x_{1} …, x_{q}) ∈ K^{q} ↦ Θ_{(x_{1}...,x_{q})}(r) is polynomial in (x_{1} …, x_{q}), for any r ∈ R_{S}.

Proof: By Lemma 4.2, the localized ring R_{S} is well-defined. As S is commutative, the elements s_{i} are locally ad-nilpotent on R_{S}. For r ∈ R_{S}, there is an integer N ≥ 0 such that ad(s_{i})^{N+1}(r) = 0, for all i. Using the general identity, e^{m}u = ∑_{0≤i≤m}(−1)^{i}(_{i}^{m}) ad(e)^{i}(u)e^{m−i}, we get: Θ_{(x_{1}...,x_{q})}(r) = ∑_{0≤i_{1},...,i_{q}≤N}(−1)^{i_{1}+...+i_{q}}(_{i_{1}}^{x_{1}}) … (_{i_{q}}^{x_{q}}) ad(s_{1})^{i_{1}} … ad(s_{q})^{i_{q}}(r) s_{1}^{−i_{1}} … s_{q}^{−i_{q}}, for any x_{1}, …, x_{q} ≥ 0. As the binomial coefficient (_{i}^{x}) = x(x − 1) … (x − i + 1)/i! can be extended to a polynomial function defined for any x ∈ K, there is a polynomial extension of Θ_{(x_{1}...,x_{q})}(r) for any (x_{1} …, x_{q}) ∈ K^{q}. By unicity of the polynomial extension, Θ_{(x_{1}...,x_{q})} is a q-parameters family of automorphisms of R_{S}. Q.E.D.

For a set of commuting roots Σ, we will denote by F_{Σ} the multiplicative set of U generated by (e_{−α})_{α∈Σ}.

LEMMA 4.4 Let M be an infinite dimensional admissible simple module of degree d, let Σ ⊂ Δ_{M}^{I} ∪ Δ_{M}^{+} be a set of commuting roots which is a basis of Q and set M′ = M_{F_{Σ}}.

Then M′ is a weight-module containing M, Supp(M′) = Q+Supp(M) and dim M′_{μ} = d for all μ ∈ Supp(M′).

Proof: By Lemma 4.2, F_{Σ} satisfies Ore’s conditions and we can define the localized module M′. For any α ∈ F_{Σ}, the elements e_{−α} acts injectively on M, so we have M ⊂ M′. It is clear that M′ = ⊕_{λ∈t}M′_{λ}, where t = Q + Supp(M).

We claim that each weight space M′_{λ} is finite dimensional of dimension d. So let X ⊂ M′_{λ} be a finite dimensional subspace. There exists s ∈ F_{Σ} such that s.X ⊂ M. We have s.X ⊂ M_{λ+β}, where β is the weight of s. As s acts injectively, we have dim X ≤ d, i.e dim M′_{λ} ≤ d. As Σ is a basis of Q, the dimension of all weight spaces M′_{λ} are the same. It follows that dim M′_{μ} = d for all μ ∈ t. Q.E.D.

Let Σ be a set of commuting roots which is a basis of Q, and let Σ′ ⊂ Σ. Let Q_{Σ′} the subgroup of Q generated by Σ′ and set 𝔥_{Σ′}^{∗} = K ⊗ Q_{Σ′}. For any ν =∈𝔥_{Σ′}^{∗}, denote by F_{Σ′}^{ν} the formal expression ∏_{α∈Σ′} e_{−α}^{x_{α}}, where the scalar x_{α} are defined by the identity ν = − ∑_{α∈Σ′} x_{α} α. By Lemma 4.3, there is a well defined automorphism Θ_{ν} : U_{F_{Σ′}} → U_{F_{Σ′}} which is formally defined by u ↦ F_{Σ′}^{−ν} u F_{Σ′}^{ν}. For any U_{F_{Σ′}} -module M, we denote by F_{Σ′}^{ν}.M the module M twisted by the automorphism Θ_{ν}.

LEMMA 4.5 Let M be an infinite dimensional admissible simple module of degree d.

(i) Let Σ ⊂ Δ_{M}^{I} ∪ Δ_{M}^{+} be a set of commuting roots which is a basis of Q and let Σ′ ⊂ Σ. Then M is contained in a coherent family M of degree d such that M contains F_{Σ′}^{ν}.(M_{F_{Σ′}}), for any ν ∈ 𝔥_{Σ′}^{∗}.

(ii) In particular, let B be an adapted basis and α ∈ B with α ∉ Δ_{M}^{F}. Then M is contained in a coherent family M of degree d such that e_{−α} acts bijectively and M contains e_{−α}^{x}.(M_{e_{−α}}), for all x ∈ K.

Proof: Set M′ = M_{F_{Σ}} and let t be its support. In order to define a coherent family M, we will define each component M[u] by M[u] = F_{Σ}^{ν}.M′, for any ν ∈𝔥^{∗} with ν + t = u. For any μ ∈ Q, the automorphism Θ_{Σ}^{μ} is inner. Thus the module M[u] is well defined up to isomorphism.

Using the automorphisms Θ_{μ}(u), we can identify all weight spaces M_{ξ} and by Lemma 4.3, the action of the elements of A depends polynomially on ξ. So M is a coherent family of degree d, which contains M.

By definition, M contains F_{Σ′}^{ν}.(M_{F_{Σ′}}), for any ν ∈ 𝔥_{Σ′}^{∗}. Assertion (i) is proved, and Assertion (ii) follows from Assertion 1 and Lemma 4.1. Q.E.D.

Remark: We can easily show that there is a morphism ρ :𝔤→𝔤𝔩(d, Diff(T)), such that the module M occuring in the proof of the lemma is the restriction of the natural tensor representation of 𝔤𝔩(d, Diff(T)), where T is the adjoint torus of 𝔤.

LEMMA 4.6 Let M be an infinite dimensional admissible simple module of degree d. For any u ∈ A, the map λ ∈ Supp_{ess}(M) ↦ tr(u|_{M_{λ}}) is polynomial (recall that Supp_{ess}(M) is Zariski dense).

Proof: By Lemma 4.5, there is a coherent family M of degree d containing M. As T r(u|_{M_{λ}}) = T r(u|_{M_{λ}}) for any u ∈ A and any λ ∈ Supp_{ess}(X), the restriction of the trace to Supp_{ess}(M) is polynomial. Q.E.D.

A coherent family M is called irreducible if the A-module M_{λ} is simple for some λ. LEMMA 4.7 Let M be an irreducible coherent family. Then the set of all weights μ

such that the A-module M_{μ} is irreducible is a nonempty Zariski open subset of 𝔥^{∗}.

Proof: Let λ be any weight such that M_{λ} is a simple A-module. The A-module M_{μ} is simple if and only if the bilinear map B_{μ} : (u, v) ∈ A × A ↦ T r_{μ}(uv) has maximal rank d^{2}, where d is the degree of M. There is a subspace E ⊂ A of dimension d^{2} such that the restriction of B_{λ} to E has rank d^{2}. The set of all weights μ such that B_{μ}|E has maximal rank d^{2} is Zariski open, and contains λ. Hence Ω is Zariski open. Q.E.D.

Let M be a coherent family. By Lemma 3.3, the 𝔤-module M[t] is of finite lenght for any t ∈ T^{∗}. Denote by M^{ss} the semi-simple coherent family such that for any t ∈ T^{∗}, M^{ss}[t] has the same composition series as M[t]. Roughly speaking, M^{ss} is the ”semisimplification” of M. Let M be an infinite dimensional admissible simple module of degree d. A coherent extension of M is a coherent family M of degree d containing M as a subquotient.

PROPOSITION 4.8 Let M be an infinite dimensional admissible simple module of degree d.

(i) There exists a unique semi-simple coherent extension EX T (M) of M.

(ii) The coherent family EX T (M) is irreducible and any infinite dimensional submodule of EX T (M) is admissible of degree d.

(iii) the central character of all simple submodules of EX T (M) are the same. (iv) If M is any coherent extension of M, then M^{ss} is isomorphic to EX T (M).

Proof: First prove (i) and (iv). By Lemma 4.4, M is contained in a coherent family M of degree d. Hence M is contained in the coherent extension M^{ss}. As Supp_{ess}(M) is Zariski dense (Lemma 3.4), the trace of any coherent extension is determined by the trace of M. The unicity of semi-simple coherent extensions follows from Lemma 2.3. Assertion (iv) follows as well.

Prove Assertion (ii): let M′ be a infinite dimensional simple submodule of EX T (M). Note that for any λ ∈ Supp_{ess}(M), EX T (M)_{λ} ≃ M_{λ} is a simple A-module (Lemma 2.2). Hence the coherent family EX T (M) is irreducible. By Lemma 4.7, EX T (M)_{μ} is a simple A-module for all μ in some non empty Zariski open subset Ω and by Lemma 3.4, Supp_{ess}(M′) is Zariski dense. Hence Supp_{max}(M′) ∩ Ω is not empty, and M′ has same degree d.

Prove Assertion (iii): Let u ∈ U be a central element. Note that u ∈ A. Hence the coefficients of the characteristic polynomial of u|_{EX T (X)_{λ}} depends polynomially in λ. However these coefficients are constant on Supp_{ess}(M), which is Zariski dense. So all simple components of EX T (M) have the same central character. Q.E.D.

5. Components of coherent families. In this section, we will prove that any semi-simple coherent family contains an infinite dimensional highest weight module.

LEMMA 5.1 (for 𝔤=𝔰𝔩(2)) Let M be a coherent family of degree d. Then the function λ → det(ef|M_{λ}) is a non-zero polynomial of degree 2d.

Proof: Let Ω = ef + h^{2}/4 − h/2 be the Casimir operator. It is easy to see that the map e : M_{nλ} → M_{(n+1)λ} is bijective for n ≥ N_{0} for some N_{0} ∈ Z. So all the eigenvalues of Ω|M_{nλ} are the same for n ≥ N_{0}. Let a_{1}, …, a_{d} be these eigenvalues. Then we have det(ef|M_{λ}) = ∏_{1≤i≤d}(a_{i} + λ(h)/2 − λ(h)^{2}/4), for all λ of the form nα, n ≥ N_{0}. The two sides of the equality being polynomials in λ, it holds for all λ. Q.E.D.

From now on in the section, 𝔤 will denote an arbitrary simple Lie algebra. For any open subset Ω of 𝔥^{∗}, set T (Ω) = ∩_{μ∈Q}(μ + Ω). For any two open subset Ω, Ω′ we have T (Ω) ∩ T (Ω′) = T (Ω ∩ Ω′). The collection of sets T (Ω)/Q is a basis of a topology of the dual torus T^{∗}.

LEMMA 5.2 (i) Let Ω be a non empty open subset of 𝔥^{∗}. Then T (Ω)is not empty. (ii) Any non empty open subset of T^{∗} is dense. Proof: We can assume that Ω is the open set {f ≠ 0} for some regular function

f. Choose a basis of Q, e.g. a basis B of Δ. Relatively to B, f is identified with a polynomial f(x_{1}, …, x_{n}) of degree d with coefficients in some some finitely generated subfield k of K. Choose some generators ξ_{1}, … ξ_{n} of n disjoint extensions of degree > d of k. The monomials ξ_{1}^{m_{1}} … ξ_{n}^{m_{n}}, for 0 ≤ m_{i} ≤ d, being linearly independent over k, the monomials (ξ_{1} +y_{1})^{m_{1}} … (ξ_{n} +y_{n})^{m_{n}}, for 0 ≤ m_{i} ≤ d, are also independant over k, for any y_{1}, …, y_{n} ∈ Z. Hence we have f(ξ_{1} +y_{1}, …, ξ_{n} +y_{n}) ≠ 0, i.e. the element ξ of coordinates (ξ_{1}, ..., ξ_{l}) belongs to T (Ω). Hence Assertion (i) is proved. Assertion (ii) follows from the fact that T (Ω) ∩ T (Ω′) = T (Ω ∩ Ω′) is non empty whenever Ω, Ω′ are non empty. Q.E.D.

For a coherent family M, let Ω_{M} be the set of all t ∈ T^{∗} such that e_{α}e_{−α}|M_{μ} is invertible for all α ∈ Δ and all μ ∈ t. In Section 10, we will prove that Ω_{M} is the complement of finitely many codimension one subtori. At this stage of the paper, we can only prove a weaker statement about Ω_{M}:

LEMMA 5.3 (i) For any coherent family, Ω_{M} is open and dense in T^{∗}.

(ii) If there is at least one coherent family, the Lie algebra 𝔤 is of type A or C.

Proof: Let α ∈ B. By Lemmas 5.1 and 5.2, the set Ω_{α} of all t ∈ T^{∗} such that e_{α}e_{−α}|M_{μ} is invertible for all μ ∈ t is dense and open. Hence Ω_{M} is open and dense in T^{∗}. Assume that there is a coherent family M, and let t ∈ Ω_{M}. By lemma 3.3, M[t] contains a simple submodule M. By assumption, e_{α} acts injectively on M for all α ∈ Δ, hence by Proposition 1.4 M is cuspidal. By Proposition 1.2, the type of 𝔤 is A or C (of course, it is possible to use directly Fernando’s proof to get Assertion (ii)). Q.E.D.

PROPOSITION 5.4 Let M be a irreducible coherent family, and let t ∈ T^{∗}. The following assertions are equivalent:

(i) M[t] is simple, (ii) t belongs to Ω_{M}, (iii) M[t] is cuspidal. Proof: The facts that (i) and (iii) are equivalent, and that (iii) implies (i) follows from

Propositions 1.4. Assume that t ∈ Ω_{M}. By Lemma 3.3, M[t] contains a simple submodule M. As e_{α}e_{−α}|M is bijective, M is cuspidal. By Proposition 4.8, the degree of M equals the degree of M. Hence M = M[t] and M[t] is simple. Q.E.D.

LEMMA 5.5 Let M be a semi-simple coherent family. Then M is a direct sum of irreducible coherent families, and such a decomposition is unique up to isomorphism.

Proof: By Lemma 5.3, we can choose an element t in Ω_{M}. The module M[t] is a finite sum of cuspidal representations, say N_{1} ⊕ … ⊕ N_{k} and this decomposition is unique up to isomorphism. It follows that M and EX T (N_{1}) ⊕ … ⊕ EX T (N_{1}) have the same trace. By Lemma 2.3, these two modules are isomorphic. Q.E.D.

LEMMA 5.6 Let M be a simple weight module, and let α ∈ Δ. The following assertions are equivalent:

(i) β ∈ Δ_{M}^{I},

(ii) for some λ ∈ Supp(M), we have λ + Zβ ⊂ Supp(M).

Proof: See e.g. [DMP].

PROPOSITION 5.7 Let M be a semi-simple irreducible coherent family. Then M contains an infinite dimensional simple highest weight module relative to some basis B of Δ.

Proof: By Proposition 1.3, it suffices to find an infinite dimensional simple submodule M ⊂ M with Δ_{M}^{I} = ∅. Thus we will prove that for any simple submodule M with Δ_{M}^{I} ≠ ∅, we can find another infinite dimensional simple submodule M′ ⊂ M such that Δ_{M′}^{I} is properly included in Δ_{M}^{I}.

Let B be a basis adpted to M, let d be its degree and let α ∈ Δ_{M}^{I} ∩ B. Let 𝔞≃ 𝔰𝔩(2) be the Lie algebra generated by e_{±α}. Set e = e_{α}, f = e_{−α}. By Lemma 4.5, there exists a coherent family N of degree d on which f acts injectively and wich contains the module X = ⊕_{x∈K/Z}f^{x}.M (note that M = M_{f}). Let λ ∈ Supp_{ess}(M) and set P = ⊕_{x∈K} f^{x} M_{λ}. For any x ∈ K, P_{λ+x.α} has dimension d, so P_{λ+x.α} = N_{λ+x.α}. In particular, P is a coherent 𝔞-family and the function x ↦ det(fe|P_{λ+x.α}) is a non zero polynomial of degree 2d (Lemma 5.1). So there is a non zero v ∈ P_{λ+xα} ⊂ X with fe.v = 0 for some x ∈ K. As f acts bijectively on N, we have e.v = 0.

Set V = {m ∈ X[t]|e^{N}.m = 0 for some N >> 0}, where t = λ + xα + Q. As ad(e) is nilpotent, V is a submodule and v ∈ V. As V is admissible, it has finite lenght by Lemma 3.3. Thus it contains a simple submodule M′. As f acts injectively on N, the submodule M′ is infinite dimensional.

Next we claim that Δ_{M′}^{I} is a proper subset of Δ_{M}^{I}. We have X[t] = f^{−x}.M, hence Supp(M′) ⊂ Supp(X[t]) = xα + Supp(M). For any β ∈ Δ_{M′}^{I}, the support of M′ is an union of (Zβ)-cosets, so the support of M contains at least one (Zβ)-coset. By Lemma 5.6, β belongs to Δ_{M}^{I}, hence Δ_{M′}^{I} ⊂ Δ_{M}^{I}. However ±α ∉ Δ_{M′}^{I}. Hence Δ_{M′}^{I} is properly included in Δ_{M}^{I}. Moreover N^{ss} = M. Hence M′ is also a submodule of M. Q.E.D.

6. Admissible highest weight modules. It follows from the previous section that in order to classify the semi-simple coherent family, we need to classify all the simple admissible highest weight modules, and to determine when two such modules occur in the same irreducible family. This section is devoted to preparatory results to solve these two questions.

LEMMA 6.1 Let λ be a weight, let B a basis of Δ and let α ∈ B. Assume λ ∉ P_{B}^{+}.

(i) The module L_{s_{α}B}(s_{α}λ) belongs to EX T (L_{B}(λ)).

(ii) Moreover assume that λ(h_{α}) ∉ Z_{≥0}. Then L_{B}(s_{α}(λ + ρ) − ρ) and L_{s_{α}B}(λ + α) belong to EX T (L_{B}(λ)).

Proof: If λ(h_{α}) ≥ 0, we have L_{s_{α}B}(s_{α}λ) ≃ L(λ) and Assertion (i) is obvious. Also we will assume that λ(h_{α}) ∉ Z_{≥0}. Set f = e_{−α}, e = e_{α}, and let v = v_{λ} be the highest weight vector of L_{B}(λ). By definition B is adapted to L(λ), and α ∉ Δ_{M}^{F}. By Lemma 4.5, there is a coherent extension M of L_{B}(λ) such that f|M is bijective and M contains the submodule X = ⊕_{x∈K/Z}f^{x}.L(λ)_{f}. Let C be the cone generated by α and Δ_{B}^{−}. Set 𝔫_{B}^{+} = ⊕_{β∈Δ_{B}^{+}}𝔤_{α}, 𝔫_{s_{α}B}^{+} = s_{α} 𝔫_{B}^{+}, 𝔲=𝔫_{B}^{+}∩ 𝔫_{s_{α}B}^{+}. The support of f^{x}.L(λ)_{f} lies in (λ−x.α)+C, hence we have 𝔲f^{x}.v = 0 for any x.

Set v′ = f^{−λ(h_{α})−1}.v. By the standard 𝔰𝔩(2)-computations, we have fe.v′ = 0. As f is invertible, we have e.v′ = 0, hence 𝔫_{B}^{+}.v′ = 0. Note that v′ is a weight vector of weight s_{α}(λ + ρ) − ρ. Hence L_{B}(s_{α}(λ + ρ) − ρ) is a quotient of the 𝔤-module V′ generated by v′. Hence L_{B}(s_{α}(λ + ρ) − ρ) belongs to EX T (L_{B}(λ)).

Set v” = f^{−λ(h_{α})}.v and let V ” be the 𝔤-module generated by v”. We have f.v” = v′ and e.v′ = 0. Hence V ” contains V′, but v” ∉ V′. So we have 𝔫_{s_{α}B}^{+}.v” = 0 modulo V′. As v” is a weight vector of weight s_{α}λ, L_{s_{α}B}(s_{α}λ) is a quotient of V ”/V′. Hence L_{s_{α}B}(s_{α}λ) belongs to EX T (L_{B}(λ)).

Set v′′′ = f^{−1}.v and let be the 𝔤-module generated by v′′′. One proves similarly that L_{s_{α}B}(λ − α) is a quotient of V′′′/L_{B}(λ). Hence L_{s_{α}B}(λ + α) belongs to EX T (L_{B}(λ)). Q.E.D.

PROPOSITION 6.2 (i) Let M be a semi-simple coherent family. Then for any w ∈ W, we have M^{w} ≃ M.

(ii) Assume that Mis irreducible. Then for any basis B of Δ, there exists an infinite dimensional B-highest weight module L such that M ≃ EX T (L).

Proof: By Lemma 5.5, we can assume that M is irreducible. By Proposition 5.7, there exists a basis B_{0} of Δ such that M ≃ EX T (L_{B_{0}}(λ)) for some weight λ ∉ P_{B}^{+}. By Lemma 6.1, EX T (L_{B_{0}}(λ)) ≃ EX T (L_{B_{0}}(λ)^{s_{α}}) for any α ∈ B_{0}. Hence the isomorphism class of M is W -invariant and point (i) is proved. Point (ii) follows from the fact that any basis B is W -conjugated to B_{0}. Q.E.D.

LEMMA 6.3 Let M be an irreducible coherent family, let t ∈ T^{∗} and let M, N two distinct infinite dimensional simple submodules of M[t]. Then C(M) ∩ C(N) is contained in a hyperplane.

Proof: Assume that C(M) ∩ C(N) is not contained in a hyperplane. Then the group Q′ generated by C(M) ∩ C(N) is a finite index subgroup of Q. Let t′ ⊂ t be a Q′ coset. It follows from Lemma 3.1 and Proposition 3.5 (i) that Supp_{ess}(M) ∩ t′ ≠ ∅ and Supp_{ess}(N) ∩ t′ ≠ ∅. Hence we can find μ ∈ Supp_{ess}(M), ν ∈ Supp_{ess}(N), β, γ ∈ C(M) ∩ C(N) such that μ − ν = β − γ. It follows that Supp_{ess}(M) ∩ Supp_{ess}(N) contains μ + γ = ν + β. By Proposition 4.8, M and N are admissible of the same degree than M. Hence for any λ ∈ Supp_{ess}(M) ∩ Supp_{ess}(N), we have M_{λ} = M_{λ} = N_{λ}, i.e. we have M = N. Q.E.D.

LEMMA 6.4 Let B a basis, let λ ∈𝔥^{∗} such that L_{B}(λ) is admissible and let A be the set of all α ∈ B such that λ(h_{α}) ∉ Z_{≥0}.

(i) If λ(h_{α}) < 0, for some α ∈ B, then A = {α}. In particular λ is integral.

(ii) For any α, β ∈ A, we have < α|β >≠ 0. In particular A contains at most two elements.

Proof: For any γ ∈ B, denote by D(γ) the set of all positive roots of the form = ∑_{δ∈B} m_{δ}δ with m_{γ} ≠ 0 (i.e. D(γ) is the set of roots of the minimal unipotent radical containing e_{γ}) and let C(γ) be the cone generated by −D(γ). Let C be the cone of L_{B}(λ), and let α ≠ β ∈ B. Note that s_{α} C(β) = C(β), and that C(β) is not contained in some hyperplane.

First assume that λ(h_{α}) < 0. By Lemma 6.1, L(λ) and L_{s_{α}B}(s_{α}λ) belong to the same irreducible coherent family. Moreover their support is in the same Q-coset. By Lemma 6.3, C ∩ s_{α}C lies in a hyperplane. Hence C does not contains C(β), i.e. λ(h_{β}) ≥ 0. This proves Assertion (i).

Next let α, β ∈ A. The cone of L_{B}(λ) contains C(β). As A contains more than one element, it follows from Assertion (i) that λ(h_{α}) and λ(h_{β}) are not integers. We claim that < α|β >≠ 0. Assume otherwise. Then β is a simple root of Δ_{s_{α}B}^{+}. As (λ+α)(h_{β}) ∉ Z, the cone of L_{s_{α}B}(λ+α) also contains C(β). By Lemma 6.1, and L_{B}(λ) and L_{s_{α}B}(λ+α) belongs to the same irreducible coherent family, what contradicts Lemma 6.3. Hence < α|β >≠ 0. As B is simply connected, A contains at most two elements. Q.E.D.

LEMMA 6.5 Let M be an irreducible coherent family, let B be a basis, let α ∈ B. Then M contains at most one module L_{B}(λ) with λ(h_{α}) < 0.

Proof: Let L_{B}(μ), L_{B}(λ) be two such submodules in M. It follows from Lemma 6.4 that λ and μ are integral and their cones are the same. By Proposition 4.8, their central character is the same. It follows that λ and μ are in the same Q-coset. By Lemma 6.3, we have λ = μ. Q.E.D.

7. Rank two computations. In order to classify all admissible simple highest weight modules, we first need to classify all of them for rank two Lie algebras. For rank two Lie algebras, character formulas have been computed by Jantzen [Ja] well before the general Kazhdan-Lusztig conjecture, and the computations are based on his formulas. In this section, 𝔤 will be a rank two Lie algebra of type A_{2} or C_{2}. We will fix a basis B. By ch(M), we denote the character of the weight module M. We will denote by M(λ) the Verma module with highest weight λ.

LEMMA 7.1 (𝔤 of type A_{2}) Let λ ∉ P^{+} be a weight. The module L(λ) is admissible if and only if (λ + ρ)(h_{α}) > 0, for at least one root α ∈ Δ^{+}.

Proof: It is easy to show that the Verma modules are not admissible. So if L(λ) is admissible, (λ + ρ)(h_{α}) > 0 for some α ∈ Δ^{+}. Conversely, assume that n = λ + ρ(h_{α}) > 0 for some α ∈ Δ^{+}. Let β, γ be the other two roots in Δ^{+} and set L = M(λ)/M(λ − nα). We have

ch(L) = ch M(λ) − ch M(λ − nα)

= e^{λ}(1 + e^{−α} + ... + e^{−nα})/(1 − e^{−β})(1 − e^{−γ}).

Hence L is admissible. As L(λ) is a quotient of L, it is also admissible. Q.E.D.

LEMMA 7.2 (𝔤 of type C_{2}) Let λ ∉ P^{+} be a weight. The module L(λ) is admissible if and only if we have (λ + ρ)(h_{α}) > 0 for the two short positive roots α and λ(h_{β}) ∈ 1/2 + Z for any long root β.

Proof: Assume that L(λ) is admissible. The Verma module M(λ), nor any quotient M(λ)/M(μ) are admissible. It follows from [Ja] that M(λ) contains two disctinct maximal Verma submodules. Thus λ+ρ is regular and there exists at least two positive roots γ ≠ γ′ with (λ + ρ)(h_{γ}) > 0 and (λ + ρ)(h_{γ′}) > 0.

First prove that λ + ρ is not integral. Assume otherwise. Then λ = x(μ + ρ) − ρ for some dominant and integral weight μ and some x ∈ W, x ≠ 1. By using Lemma 6.1, we get that L(s(μ + ρ) − ρ)) is admissible for some for some simple reflexion s, and let s′ be the other simple reflexion. Following [Ja], we get:

ch(L(s(μ + ρ) − ρ)) = − ∑_{x≥s} ϵ(x) ch M(x(λ + ρ) − ρ) Any element x ∈ W different from 1 or s′ is ≥ s, we get: ch(L(s(μ + ρ) − ρ)) = ch(M(μ)) − ch(M(s′(μ + ρ) − ρ)) − ∑_{x∈W} ϵ(x) ch M(x(λ + ρ) − ρ) = ch(M(μ)/M(s′(μ + ρ) − ρ)) − ch(L(μ)).

Hence L(λ) is not admissible. As λ + ρ is not integral but takes at least two positive integral values on some positive coroots, it is clear that (λ + ρ)(h_{α}) > 0 for any short root α, and (λ + ρ)(h_{β}) ∈ 1/2 + Z for any long root β, and the conclusion of the lemma holds.

Conversely assume that λ satisfy the conditions. Let α_{1}, α_{2} the two short roots, let β_{1}, β_{2} the two long roots. Following [Ja], we have

ch(L(λ))

= ch(M(λ))−ch(M(s_{α_{1}}(λ+ρ)−ρ))−ch(M(s_{α_{2}}(λ+ρ)−ρ))+ch(M(s_{α_{1}}s_{α_{2}}(λ+ρ)−ρ)) = e^{λ}(1 + e^{−β_{1}} + … + e^{−m_{1}α_{1}})(1 + e^{−α_{2}} + … + e^{−m_{2}α_{2}})/(1 − e^{−β_{1}})(1 − e^{−β_{2}}), where m_{i} = (λ + ρ)(h_{α_{i}}). Hence (L(λ)) is admissible. Q.E.D.

8. Classification of coherent families for 𝔰𝔩(n + 1). In the section, we will classify all semi-simple irreducible families for 𝔰𝔩(n + 1) by describing all highest weight modules occuring in such a family. By Proposition 6.2, all semi-simple families are invariant by W, thus the list of highest weight modules is independent upon a choice of a basis of Δ. A more concrete version of the classification will be given in Section 11.

In this section, 𝔤=𝔰𝔩(n + 1). We will denote by h_{1}, …, h_{n} the simple coroots in a such way that two consecutive coroots are connected in the Dynkin diagram, i.e. < h_{i}|h_{j} >= 0 if |i − j| > 1. Denote by α_{1}, …, α_{n} the corresponding roots. There exist a spanning set ϵ_{1}, …, ϵ_{n} of 𝔥 such that ∑_{i} ϵ_{i} = 0 and α_{i} = ϵ_{i} − ϵ_{i+1}.

LEMMA 8.1 (for 𝔤=𝔰𝔩(n + 1)) Assume that the 𝔤-module L(λ) is infinite dimensional and admissible, and let A = {i|(λ + ρ)(h_{i}) ∉ Z_{>0}}. Then one of the following three assertions holds:

(i) A = {1} or A = {n}.

(ii) A = {i} for some 1 < i < n and (λ + ρ)(h_{i−1} + h_{i}) > 0 or (λ + ρ)(h_{i} + h_{i+1}) > 0. (iii) A = {i, i + 1} for some 1 ≤ i < n and (λ + ρ)(h_{i} + h_{i+1}) > 0.

Proof: First assume that A contains a unique element. Then we can assume that A = {i} for some 1 < i < n. Set λ′ = s_{i}(λ + ρ) − ρ and A′ = {j|(λ′ + ρ)(h_{j}) ∉ Z_{>0}}. By Lemma 6.1, the module L(λ′) is admissible. Hence by Lemma 6.4, A′ cannot contain both i−1 and i+1. So we have (λ′+ρ)(h_{i−1}) > 0 or (λ′+ρ)(h_{i+1}) > 0, i.e. (λ+ρ)(h_{i−1}+h_{i}) > 0 or (λ + ρ)(h_{i} + h_{i+1}) > 0. Hence Assertion (ii) holds.

Assume now that A contains at least two elements. By Lemma 6.4, we have A = {i, i + 1} for some 1 ≤ i < n. Let 𝔰 be the rank two Lie subalgebra of 𝔤 corresponding to the simple roots α_{i}, α_{i+1}. Denote by L(λ,𝔰) be the simple highest weight 𝔰-module with higest weight λ. As L(λ,𝔰) is a subquotient of L(λ) (indeed a submodule), it is admissible. By Lemma 7.1, Assertion (iii) holds. Q.E.D.

LEMMA 8.2 (for 𝔤=𝔰𝔩(n + 1)) Let λ ∉ P^{+} be a weight such that λ(h_{i}) ≥ 0 for all i ≠ 1 or for all i ≠ n. Then the module L(λ) is admissible.

Proof: By symmetry of the Dynkin diagram, we can assume that λ(h_{i}) ≥ 0 for all for all i ≠ n. Let 𝔭 be the parabolic subalgebra whose Levi component is 𝔰𝔩(n) (i.e. corresponds with the n first simple roots) and let 𝔲 be its radical. As L(λ) is locally finite under 𝔰𝔩(n), it is finitely generated as 𝔲-module. The roots of 𝔲 are the roots β_{i} = ϵ_{i} −ϵ_{n+1}, so they are linearly independant and the 𝔥-module U(𝔲) is multiplicity free. Hence L(λ) is admissible.

A 𝔰𝔩(n + 1)-sequence is a n + 1-tuple m = (m_{1}, …, m_{n+1}) of scalars such that ∑_{1≤i≤n+1} m_{i} = 0. A sequence is ordered if we have m_{1} > m_{2} > … m_{n+1} (recall our convention: x > y means x − y is a positive integer). Any weight λ of 𝔰𝔩(n + 1) is identified with the sequence m(λ) defined by (λ+ρ)(h_{i}) = m_{i}(λ)−m_{i+1}(λ). Thus dominant integral weights are identified with ordered sequence. Let P the set of all 𝔰𝔩(n + 1) non ordered sequences which become ordered after removing one term. Note that a weight λ satisfies one of the assertion (i) (ii) (iii) of Lemma 8.1 if and only if m(λ) ∈ P.

The Weyl group Σ_{n+1} acts over the set of sequences by permuting the indices. By definition a central character is a 𝔰𝔩(n + 1)-sequence modulo the action of Σ_{n+1}. A central character m is called integral (respectively regular, singular) if all differences m_{i} − m_{j} are integers (respectively are not zero, at least one is zero).

We define a structure of oriented graph on P as follows. By definition, there is an arrow m → m′ between two elements m, m′ ∈ P if there is an index i such that m_{i} − m_{i+1} ∉ Z_{>0} and m′ = r_{i,i+1}.m, where r_{k,l} is the transposition exchanging k and l. When m_{i}−m_{i+1} ∉ Z there will be also an oriented edge from m′ to m and we will denote these two opposite edges by m ↔ m′.

Define P^{+} (respectively P^{−}) as the subset of all sequences m ∈ P such that the subsequence m_{2}, m_{3}, … m_{n+1} is ordered (respectively m_{1}, m_{2}, … m_{n} is ordered).

Clearly, there are three types of central characters occuring in P:

(i) the regular and integral central characters,

(ii) the singular central characters: in such case the character is integral, and there is exactly one pair of indices i, j such that m_{i} = m_{j} (i ≠ j),

(iii) the non integral characters: in such case the character is regular and there exists one index i such that m_{j} − m_{k} ∈ Z for any j, k ≠ i.

Let χ be a regular and central character. We can write χ as m_{1} > m_{2} … > m_{n+1} (however this sequence is not in P). For any i with 1 ≤ i, k ≤ n set χ(i, k) = r_{i+1,k}.m if k ≤ i and χ(i, k) = r_{i,k+1}.m if k > i.

Let χ be a singular character. We can write χ as m_{1} > m_{2} … > m_{i} = m_{i+1} > m_{+2} … > m_{n+1}. For any k with 1 ≤ k ≤ n, set χ(k) = r_{i,k}.m if k < i, χ(i) = m and χ(k) = r_{i+1,k+1}.m if k > i.

Let χ be a non integral character. If n + 1 ≥ 3, we can write χ as a sequence m with m_{2} > m_{3} … > m_{n+1}. Then set χ(1) = m and χ(k) = r_{i,k}.m for any 1 < k ≤ n + 1. In the special case of 𝔰𝔩(2) there is no canonical choice for m_{1} and m_{2}: in such a case we should make an arbitrary choice.

Each connected component of P belongs to the same central character. It is easy to compute the connected components of each central character, and we get:

LEMMA 8.3 Let χ be a central character occuring in P.

(i) If χ is regular and integral, it contains exactly n connected components. More precisely χ(i, k) and χ(i′, k′) are in the same connected component if and only if i = i′. The connected component of χ(i, i) is as follows:

χ(i, 1) → χ(i, 2) → … → χ(i, i) ← χ(i, i + 1) ← … ← χ(i, n).

Moreover χ(i, 1) is in P^{+} and χ(i, n) is in P^{−}.

(ii) If χ is singular, it contains one connected component which is as follows:

χ(1) → χ(2) → … → χ(i) ← χ(i + 1) ← … ← χ(n).

Moreover χ(1) is in P^{+} and χ(n) is in P^{−}.

(iii) If χ is non integral, it contains one connected component which is as follows: χ(1) ↔ χ(2) ↔ … ↔ χ(i) ↔ χ(i + 1) ↔ … ↔ χ(n + 1). Moreover χ(1) is in P^{+} and χ(n + 1) is in P^{−}. PROPOSITION 8.4 (for 𝔤=𝔰𝔩(n + 1)). Let λ ∉ P^{+}. The module L(λ) is admissible if

and only if m(λ) ∈ P, (i.e. if and only if λ satisfy one of the three conditions of Lemma 8.1.)

Proof: By Lemma 8.1, if L(λ) is admissible then m(λ) belongs to P. Conversely, assume that m(λ) ∈ P. By Lemma 8.3, there is always a weight μ such that m(μ) ∈ P^{+} ∪ P^{+} and an oriented path from m(μ) to m(λ). By lemma 8.2, L(μ) is admissible. Hence by Lemma 6.1, L(λ) is admissible.

For any semi-simple irreducible coherent family M, denote by m(M) the set of all m(λ) where such that λ ∉ P^{+} and L(λ) occurs in M.

THEOREM 8.5 (for 𝔤=𝔰𝔩(n + 1))

(i) For any semi-simple irreducible coherent family M, m(M) is one connected component of P.

(ii) The map M ↦ m(M) is a bijection from the set of all irreducible semi-simple coherent families to the set of connected components of P.

Proof: Let M semi-simple irreducible coherent family. Let m′ ∈ P which is connected to an element m ∈ m(M), i.e. such that there is an edge m′ → m or an edge m′ ← m, and let λ, λ′ be the weights such that m(λ) = m, m(λ′) = m′. By Proposition 4.8 and Lemma 6.1, we have M = EX T (L(λ)) = EX T (L(λ′)). Hence m′ ∈ m(M). Hence m(M) is an union of connected components of P.

We claim that m(M) contains exactly one connected component. By Proposition 6.2, m(M) is not empty, i.e it contains at least one connected component. As M admits a central character, all connected components of m(M) are in the same central character. By Lemma 8.3, all central characters are connected except the regular integral one. Thus we can assume that the central character of M is integral. By lemma 6.5, m(M) contains at most one module L(λ) with λ(h_{1}) < 0, i.e. m(M) contains at most one element in P^{+}. However, each connected component of P meets P^{+}. Hence Assertion (i) is proved.

Assertion (ii) follows easily, because for non-isomorphic M and M′, the sets m(M) and m(M′) are disjoint (Proposition 4.8). Q.E.D.

9. Classification of coherent families for 𝔰𝔭(2n). In this section, 𝔤=𝔰𝔭(2n) (in the notation 𝔰𝔭(2n), we always mean n ≥ 2). Similarly we will classify all semi-simple irreducible 𝔤-coherent families by giving the list of all highest weight modules occuring in such a family. It turns out that the result is much more easy than in the 𝔰𝔩(n + 1)-case, because there are always exactly two highest weight modules. A concrete version of the classification theorem will be given in Section 12.

There is an orthonormal basis ϵ_{1}, …, ϵ_{n} of 𝔥^{∗} and a basis α_{1}, …, α_{n} of the root system such that α_{i} = ϵ_{i} − ϵ_{i+1} for i < n and α_{n} = 2ϵ_{n}, see e.g. [B2]. The corresponding coroots are h_{i} = ϵ_{i}^{∗} − ϵ_{i+1}^{∗} for i < n and h_{n} = ϵ_{n}^{∗}.

LEMMA 9.1 (for 𝔤=𝔰𝔭(2n)) Let λ ∉ P^{+}. If L(λ) is admissible, we have:

(i) λ(h_{i}) ≥ 0 for any i ≠ n,

(ii) λ(h_{n}) ∈ 1/2 + Z, (iii) λ(h_{n−1} + 2h_{n}) ≥ −2. Proof: Let i < n. Let 𝔰 be the rank two subalgebra whose simple positive roots are

α_{i} = ϵ_{i} − ϵ_{i+1} and β_{i} = 2ϵ_{i+1} and let ρ′ be the half sum of positive roots of 𝔰. The Lie algebra 𝔰 is of type C_{2}. Let L(λ,𝔰) be the simple highest weight 𝔰-module with higest weight λ. As L(λ,𝔰) is a subquotient of L(λ), it is admissible. Hence by Lemma 7.2, we have λ + ρ′(h_{i}) > 0. As α_{i} is a simple root in 𝔰, we have ρ′(h_{i}) = ρ(h_{i}) = 1. Hence λ(h_{i}) ≥ 0. Assume now i = n − 1. In such a case, β_{i} is the simple root α_{n}, i.e. the restriction of ρ and ρ′ to corrots of 𝔰 are the same. Set β = ϵ_{n−1} + ϵ_{n}. By lemma 7.2, (λ + ρ)(h_{n}) is an half integer, so is λ(h_{n}). Moreover we have (λ + ρ)(h_{β}) > 0, i.e. λ(h_{n−1} + 2h_{n}) + 3 > 0 what proves Assertion (iii). Q.E.D.

The Shale-Weil representation V is an explicit action of 𝔰𝔭(2n) on K[x_{1}, …, x_{n}]. As a 𝔥-module, V is multiplicity-free. As a 𝔰𝔭(2n)-module, we have V = L(ω^{+}) ⊕ L(ω^{−}), where ω^{+}(h_{n−1}) = 0, ω^{+}(h_{n}) = −1/2, ω^{−}(h_{n−1}) = 1, ω^{−}(h_{n}) = −3/2 and ω^{±}(h_{i}) = 0 for i ≠ n − 1, n. Thus L(ω^{+}) and L(ω^{−}) are admissible. Note that the conditions (i), (ii) and (iii) of Lemma 9.1 are equivalent to (λ + ρ)(h_{α}) > 0 for any short positive root and (λ + ρ)(h_{β}) is an half integer for any long root β.

LEMMA 9.2 (for 𝔤=𝔰𝔭(2n)) Let λ ∉ P^{+}. The module L(λ) is admissible if and only if λ satisfies the conditions (i),(ii) and (iii) of Lemma 9.2.

Proof: In view of the previous lemma, it is enough to prove that for any weight λ satisfying condidions, the module L(λ) is admissible. First assume λ(h_{n}) ≥ −1/2. We have λ = μ + ω^{+}, where μ ∈ P^{+}. The module L(μ) ⊗ L(ω^{+}) is admissible and L(λ) is a subquotient (indeed a direct factor). Hence L(λ) is admissible.

Assume now that λ(h_{n}) < −1/2. Set λ′ = s_{n}(λ + ρ) − ρ. We have s_{n} h_{n−1} = 2 h_{n} + h_{n−1}, hence λ′ satisfies (i), (ii) and (iii). Moreover λ′(h_{n} ≥ −1/2. Hence L(λ′) is admissible. By Lemma 6.1, L(λ) belongs to EX T (L(λ′)), hence L(λ) is admissible. Q.E.D.

A 𝔰𝔭(2n) sequence is an arbitrary sequence (m_{1}, …, m_{n}). Any weight λ of 𝔰𝔭(2n) is identified with the sequence m(λ) defined by m_{i}(λ) =< λ + ρ|ϵ_{i} >. The Weyl group Σ_{n} × (Z/2Z)^{n} acts on the set of sequence by permuting the indices and multiplying by ±1 the m_{i}. A central character is a sequence modulo the action of the Weyl group.

Let Q be the set of sequences (m_{1}, …, m_{n}) with m_{1} > m_{2} > … > m_{n−1} > ±m_{n} and each m_{i} is a half integer. Then λ satisfy the conditions (i), (ii) and (iii) of Lemma 9.1 if and only if m(λ) ∈ Q. Any central character χ occuring in Q can be represented by a sequence m with m_{1} > m_{2} > … > m_{n−1} > m_{n} ≥ 1/2. For such a χ, define the two sequences χ^{±} by χ^{+} = m, and χ^{−} = (m_{1}, …, m_{n−1}, −m_{n}). There is a structure of oriented graph on Q whose only arrows are χ^{+} ↔ χ^{−}.

For an irreducible semi-simple coherent family M, denote by m(M) the set of all m(λ) where such that L(λ) occurs in M.

THEOREM 9.3 (for 𝔤=𝔰𝔭(2n))

(i) For any irreducible semi-simple coherent family M, m(M) is one connected component of Q.

(ii) The map M → m(M) is a bijection from the set of all irreducible semi-simple coherent families to the set of connected components of Q.

Proof: The result follows easily from Proposition 4.8, Lemma 6.1, 6.2 and 9.2.

10. Classification of cuspidal representations. In the previous sections, we proved that any cuspidal module M is isomorphic to M[t] for a unique semi-simple irreducible coherent family M and a unique t ∈ T^{∗}. By Lemma 5.3, there are coherent families only for Lie algebras of type A or C and the coherent families have been classified for these two types in Sections 8 and 9. To finish the classification of cuspidal representations, we only need to determine when M[t] is cuspidal, or equivalently when M[t] is simple (proposition 5.4). This is the aim of Theorem 10.2 proved in this section. For the simplicity of the statement, we will assume 𝔤≠ 𝔰𝔩(2). The 𝔰𝔩(2)-case being easy, this restriction is not a loss of generalty.

Let M be an irreducible coherent family. Choose a basis B of Δ and denote by X the list of all weights λ such that L_{B}(λ) occurs as submodules of M^{ss}. Its image in T^{∗} will be denoted by Sing(M).

LEMMA 10.1 The set Sing(M) is W -invariant. In particular, it is does not depend on a choice of B.

Proof: Prove that Sing(M) is invariant by any simple reflexion s_{α}. Let λ ∈ X. If λ(h_{α}) ∈ Z, its image in Sing(M) is s_{α}-invariant. Otherwise by Lemma 6.1, s_{α}(λ + ρ) − ρ also belongs to X. However, s_{α}(λ+ρ)−ρ) = s_{α}λ modulo Q, hence Sing(M) is s_{α}-invariant. Moreover M ≃ M^{w} for any w ∈ W (Lemma 6.2). Hence Sing(M) is independant on B.

Remark: It follows from the classification of coherent families, that Sing(M) contains two elements if 𝔤=𝔰𝔭(2n). If 𝔤=𝔰𝔩(n + 1), Sing(M) contains one element if the central character is integral, and n + 1-elements otherwise.

Let B be a basis of Δ and let n be the rank. Let B′ ⊂ B be a connected subset of cardinality n − 1, such that B′ contains a long root if 𝔤=𝔰𝔭(2n), let Q′ be the sugroup of Q generated by B′. Denote by T the set of all sublattices of the form w.Q′, where w runs over W. The set T is independent of the choices of a basis B and of its subset B′ and T contains exactly n + 1-elements if 𝔤=𝔰𝔩(n + 1) (and n ≥ 2) and n elements if 𝔤=𝔰𝔭(2n). Roughly speaking, T is the set of all sublattices of type X_{n−1} if 𝔤 is of type X_{n} (X = A or C). For any sublaticce Q” ⊂ Q such that Q/Q” is torsion-free, we denote by T_{Q”}^{∗} = (K ⊗ Q”)/Q” the corresponding subtorus in T^{∗}.

Let τ ∈ T^{∗} and let Q′ ∈ T . Say that τ is not Q′-integral if there is a root α ∈ Δ ∩ Q′ such that λ(h_{α}) is not integral for some (or equivalentely for all) λ ∈ τ.

THEOREM 10.2 (for 𝔤≠𝔰𝔩(2)) Let M be an irreducible coherent family, and let t ∈ T^{∗}. Then M[t] is cuspidal if and only if none of the following two conditions hold:

(i) the central character of M is integral, and t − τ ∈ T_{Q′}^{∗} for some Q′ ∈ T , where τ is the unique element of Sing(M).

(ii) the central character of M is not integral, and we have t − τ ∈ T_{Q′}^{∗} for some Q′ ∈ T and some τ ∈ Sing(M) such that τ is not Q′-integral.

Proof: By Proposition 5.4, we can assume that M is semi-simple. Let d be its degree. First assume that M[t] is not cuspidal, i.e. M[t] is not simple and let M be an infinite

dimensional simple submodule. By Theorem 1.1, there is a proper parabolic subalgebra 𝔭 with unipotent radical 𝔲 and a cuspidal 𝔭/𝔲-module N such that M ≃ L(𝔭,N)). Let B be a a basis of the root system which is adapted to 𝔭 and let B” the corresponding basis of the root system of 𝔭/𝔲.

First assume that B” is empty. We have M = L_{B}(λ) for some λ ∈ t. If λ is integral then (i) holds for any Q′ ∈ T . If λ is not integral and 𝔤=𝔰𝔩(n + 1), let α ∈ B such that λ(α) ∉ Z. Then condition (ii) holds for any of the n − 1 sublattices Q′ containing α. If 𝔤=𝔰𝔭(2n), note that L_{B}(λ) is admissible, hence λ(h_{α}) ∉ Z for any long root α and (ii) holds for any Q′ ∈ T .

So we can assume that B” is not empty. Let C be the set of connected components of B” and for any c ∈ C, let 𝔞_{c} be the corresponding simple component in 𝔭/𝔲. We have 𝔭/𝔲=𝔷⊕(⊕_{c∈C}𝔞_{c}), where 𝔷 is the center of 𝔭/𝔲, and N = Z ⊗ (⊗_{c∈C}N_{c}), where each N_{c} is a cuspidal representation of 𝔞_{c} and Z is a one dimensional 𝔷-module. By Lemma 4.1, there is a set of commuting roots Σ” which is a basis of the root lattice Q” of 𝔭/𝔲. By Lemmas 4.1 and 4.5, there exists a coherent extension M′ of M which contains X = ⊕_{ν∈T_{Q”}^{∗}} F_{Σ”}^{ν}.M as a submodule. Set N = ⊕_{ν∈T_{Q”}^{∗}} F_{Σ”}^{ν}.N. Similarly N = Z ⊗ (⊗_{c∈C}N_{c}), where each N_{c} is a coherent 𝔞_{c}-family. By Proposition 6.2, each 𝔞_{c}-family N_{c}^{ss} contains an infinite dimensional c-highest weight module L_{c}. Thus there exists ν ∈ T_{Q”}^{∗} such that F_{Σ”}^{ν}.N has a 𝔭/𝔲 subquotient L, where L = Z ⊗ (⊗_{c∈C}L_{c}).

We have Supp(F_{Σ”}^{ν}.N) = ν + Supp(N), and Supp(F_{Σ”}^{ν}.M) = ν + Supp(M), i.e. Supp(F_{Σ”}^{ν}.M) ⊂ Supp(F_{Σ”}^{ν}.N) + C(𝔭). By Lemma 1.5, L(𝔭, L) is a subquotient of F_{Σ”}^{ν}.M. We have L(𝔭, L) = L_{B}(λ) for some λ ∈ ν + t. Moreover for each connected component c of B”, there is α ∈ c such that λ(h_{α}) ∉ Z_{≥0}. As L_{B}(λ) is admissible, it follows from Lemma 6.4 that the set A = {α ∈ B|λ(h_{α}) ∉ Z_{≥0}} is connected and contains at most two roots. Hence B” is connected. Moreover, we have:

(C1) if 𝔤 is of type C, any root α ∈ A is a long root, hence B” contains a long root (this follows from Lemma 9.1).

(C2) if λ is not integral, for any root α ∈ A, we have λ(h_{α}) ∉ Z, hence B” contains a root α such that λ(h_{α}) ∉ Z (this follows from Lemma 6.4).

As M is not cuspidal, B” is a proper subset of B, so we can find a connected subset B′ of B of cardinality n − 1 with B” ⊂ B′. Let Q′ the sublattice generated by B′. By Proposition 4.8, the degree of M is d and we have M^{′ss} = M. Hence τ = λ+Q belongs to Sing(M). By definition of λ, we have τ ∈ t + T_{Q”}^{∗}, hence t − τ ∈ T_{Q′}^{∗}. Moreover it follows from (C1) and (C2) that Assertion (i) or (ii) holds.

Conversely, assume that Assertion (i) or Assertion (ii) holds. We have t = τ + ν for some Q′ ∈ T , ν ∈ T_{Q′}^{∗}, some τ ∈ Sing(M) and τ is not Q′-integral if the central character of M is not integral. Choose a basis B of Δ such that B′ = B ∩ Q′ is connected. By definition, there is λ ∈ τ such that L_{B}(λ) is an infinite dimensional submodule of M and λ is not Q′-integral if the central character of M is not integral.

We claim that we can always choose λ ∈ τ such that there is α ∈ B′ with λ(h_{α}) ∉ Z_{≥0}. If the central character is not integral, the existence of such α follows the fact that τ is not Q′-integral. Otherwise, the central character is integral, i.e. Sing(M) = {τ}. In such a case, 𝔤≃𝔰𝔩(n + 1) and it follows from Lemma 8.3 and Proposition 8.4 that there are n-integral weights λ ∈ Q \ P_{B}^{+} such that L_{B}(λ) ∈ M. There are n − 1 among them which satisfy λ(h_{α}) < 0 for some α ∈ B′. As n ≥ 2, the claim is proved.

Set M = L_{B}(λ) and B” = {β ∈ B′|λ(h_{β}) ≥ 0}. As B′ is connected and B” is a proper subset of B′, there is a set of commuting roots Σ′ ⊂ ΔB′^{+} \ Δ_{B”}^{+} such that Σ′ is a basis of Q′. By Lemma 4.5, there is a coherent family M′ of degree d which contains the module X = F_{Σ′}^{ν}.(M_{F_{Σ′}}). We have Supp(X) = ν + Supp(M) + Q′, hence the support of X is strictly included in the Q-coset t. As any subquototient of X is a submodule of M[t], the representation M[t] is not cuspidal. Q.E.D.

11. Explicit realization of coherent families for 𝔰𝔩(n + 1). In this section, we will describe explicitely the irreducible coherent families for 𝔰𝔩(n + 1), and compute their degree. We will reformulate the classification theorem, and we will use a more concrete set of parameters than the set of connected components of the graph P. Indeed, we will show that the irreducible coherent families are parametrized by some finite dimensional representations of 𝔤𝔩(n).

Let X be a smooth affine manifold of dimension n over K, let Ω_{X}^{∗} be the space of forms and let W_{X} be the Lie algebra of vector fields on X. In order to quickly describe the tensor reprensentations of W_{X}, we will make the simplifying assumption that the tangent bundle of X is trivial, i.e. W_{X} is a free K[X]-module. Let (ξ_{1}, …, ξ_{n}) be a basis of W_{X} and let (β_{1}, …, β_{n}) be the dual basis of Ω_{X}^{1}.

The current algebra is the semi-direct product Cur(X) = W_{X} × (𝔤𝔩(n, K[X]). The elements in Cur(X) will be denoted by (ξ, g) (ξ ∈ W_{X}, g ∈𝔤𝔩(n, K[X])), and the elements in 𝔤𝔩(n, K[X]) will be denoted by ∑_{a} g_{a} ⊗ f_{a} (g_{a} ∈𝔤𝔩(n), f_{a} ∈ K[X], a runs over some set of indices). For any closed one-form ω, define j_{ω} : W_{X} → Cur(X), ξ ↦ (ξ, c_{ω}(ξ)), where c_{ω} is the non-abelian one-cocycle defined by

c_{ω}(ξ) = ∑_{1≤i,j≤n} E_{i,j} ⊗ L_{ξ_{i}}(i_{ξ}β_{j}) + 1 ⊗ i_{ξ}ω,

where (E_{i,j})_{1≤i,j≤n} is the usual basis of 𝔤𝔩(n), 1 is its central element, L_{ξ} denotes the Lie derivative of ξ ∈ W_{X} and i_{ξ} is the contraction by the vector field ξ. Using its basis, we identify Ω_{X}^{1} and K[X] ⊗ K^{n}. Then, the operator L_{ξ} + i_{ξ}ω is identified with L_{ξ} ⊗ 1 + 1 ⊗ c_{ω}(ξ). As ω is closed, j_{ω} is a morphism of Lie algebras. For any finite dimensional simple representation L of 𝔤𝔩(n), L ⊗ K[X] is a representation of the current algebra Cur(X) and we denote by T ens_{X}(L, ω) the W_{X}-module j_{ω}^{∗}(K[X] ⊗ L). The modules T ens_{X}(L, ω) are called tensor representations of W_{X}. There are two general remarks about tensor modules:

(i) For any invertible function f ∈ K[X], the multiplication by f induces an isomorphism T ens(L, ω) → T ens(L, ω + df/f). Hence the isomorphism class of T ens_{X}(L, ω) depends only on ω modulo logarithmic differential.

(ii) Let V = K^{n} be the tautological representation of 𝔤𝔩(n). Then T ens_{X}(∧^{i}V, ω) is isomorphic to Ω_{X}^{i} as a vector space, and any ξ ∈ W_{X} acts by L_{ξ} + i_{ξ}ω. The map Ω_{X}^{i} → Ω_{X}^{i+1}, α ↦ dα + ω ∧ α is a W_{X}-equivariant map T ens(∧^{i}V, ω) → T ens(∧^{i+1}V, ω).

Assume now that T is a n-dimensional torus, with Lie algebra 𝔥, and choose a K-basis of 𝔥 as a K[T]-basis of W_{T}, and let ω ∈𝔥^{∗}. Then any h ∈𝔥 acts on T ens_{T} (L, ω) ≃ K[T]⊗L as L_{h} ⊗ 1 + 1 ⊗ ω(h). Hence T ens_{T} (L, ω) is admissible. As the logarithmic differentials are the characters of T, we get a family of representations indexed by the dual torus T^{∗}. Set T EN S(L) = ⊕_{ω∈T^{∗}} T ens_{T} (L, ω).

We will now define the tensor families for 𝔰𝔩(n + 1). Let T be the adjoint torus of 𝔰𝔩(n + 1). Identify T with its open orbit in the projective space PU, where U = K^{n+1} is the tautological representation of SL(n+1). Any g ∈𝔰𝔩(n+1) defines a vector field on PU, and its restriction to the open subset T gives rise to an embedding 𝔰𝔩(n + 1) ⊂ W_{T}. To get an explicit description of it, write K[T] = K[x_{1}^{±}, …, x_{n}^{±}]. Recall that we can decompose 𝔰𝔩(n + 1) as 𝔤_{−1}⊕𝔤_{0}⊕𝔤_{1}, where 𝔤_{0}=𝔤𝔩(n), 𝔤_{−1} = V, 𝔤_{1} = V^{∗}. When we identify 𝔰𝔩(n + 1) with a Lie algebra of vector fields on T, then 𝔤_{−1} has basis (∂/∂x_{i})_{1≤i≤n}, 𝔤_{0} has basis (x_{j}∂/∂x_{i})_{1≤i,j≤n} and 𝔤_{1} has basis (x_{i}.E)_{1≤i≤n}, where E = ∑_{1≤i≤l} x_{i}∂/∂x_{i}. The subspace with basis (x_{i}∂/∂x_{i})_{1≤i≤l} is the Cartan subalgebra of 𝔰𝔩(n + 1). The tensor families are the 𝔰𝔩(n + 1)-coherent families T EN S(L).

Example of a tensor family: For a ∈ K, let K_{a} be the one-dimensional 𝔤𝔩(n)-module on which the central element 1 acts as a. Denote by A the vector space with basis (x_{1}^{α_{1}} … x_{n}^{α_{n}})_{α_{1}...α_{n}∈K} and let any ξ ∈ W_{T} act on A as L_{ξ} + a div(ξ), where div is the usual divergence. Endowed with this action, A is the tensor family associated with the one dimensional module K_{a}. Denote again by M(a) this coherent family.

In order to describe more precisely these families, we need to precise a basis of the root system. The Lie algebras 𝔤𝔩(n)=𝔤_{0} contains the Cartan subalegebra of 𝔰𝔩(n + 1). Denote by E_{i,i} = x_{i} ∂/∂x_{i} the usual basis of 𝔥. The element E = ∑_{1≤i≤n} E_{i,i} can be identified with the element 1 of 𝔤𝔩(n). In this basis, the set of simple positive coroots for 𝔰𝔩(n + 1) are h_{i} = E_{i,i} − E_{i+1,i+1} for i < n and h_{n} = E_{n,n} + E. The coroots (h_{i})_{1≤i<n} are the simple positive coroots for 𝔤𝔩(n). To describe the action of the Weyl group Σ_{n+1} on 𝔥, it is convenient to set E = −E_{n+1,n+1}. Then Σ_{n+1} acts on the generating set (E_{i,i})_{1≤i≤n+1} by permuting the coordinates.

We will denote by S the set of all finite dimensional simple 𝔤𝔩(n)-module, which is identical to the set of all weights λ ∈𝔥^{∗} such that λ(h_{i}) ≥ 0 for all 1 ≤ i < n. Set 𝔭=𝔤_{0}⊕𝔤_{1}, and for L ∈ S set M(𝔭, L) = ind(𝔭,𝔤,L) and let L((𝔭, L) be its unique simple quotient. Indeed L(𝔭, L) = L(λ), where λ is the highest weight of L. It will be convenient to also use these notations for L = 0, in this case M(𝔭, L) and L((𝔭, L) are the zero modules.

For k ≤ n, denote by H^{k} the set of all simple 𝔤𝔩(n)-modules which occur in H^{k}(𝔤_{1}, M) for some finite dimensional simple 𝔰𝔩(n+1)-module. Let λ ∈ P^{+}. By a theorem of Kostant [K], H^{k}(𝔤_{1}, L(λ)) is the simple 𝔤𝔩(n)-module with highest weight w_{k}(λ + ρ) − ρ, where w_{k} ∈ Σ_{n+1} is defined by w_{k}(i) = i for i < n+1−k, w_{k}(n+1−k) = n+1 and w_{k}(i) = i−1 for i > n + 1 − k. Hence the sets H^{i} are disjoint. Moreover, if L ∈ H^{k}, there is a unique finite dimensional simple 𝔰𝔩(n + 1)-module M such that L = H^{k}(𝔤_{1}, M). So for any i ∈ Z, we can set L[i] = H^{k+i}(𝔤_{1}, M). Set H^{∗} = ∪_{0≤i≤n}H^{i}

LEMMA 11.1

(i) Let L ≠ L′ ∈ S, and let λ′, λ′ be their highest weights. If λ + ρ and λ′ + ρ are in the same W -orbit, then L and L′ belongs to H and L′ = L[i] for some i ≠ 0.

(ii) Let L ∈ S. The central character of L(𝔭, L) is integral and regular if and only if L belongs to H^{∗}.

Proof: Lemma 11.1 is a reformulation of Lemma 8.3. Q.E.D.

LEMMA 11.2 Let L ∈ S.

(i) If L ∉ H^{∗}, the 𝔰𝔩(n + 1)-module M(𝔭, L) is simple, and its degree is dim L.

(ii) If L ∈ H^{∗}, there is a 𝔰𝔩(n + 1) exact sequence … → M(𝔭, L[1]) → M(𝔭, L[0]) → L(𝔭, L) → 0. If L ∉ H^{0}, the degree of L(𝔭, L) is ∑_{i≥0}(−1)^{i} dim L[i].

Proof: Both assertions follow from Kazdhan-Luzstig character formulas. However, the simplest way to prove it is by an elementary approach. As the 𝔤𝔩(n)-module M(𝔭, L) is locally finite, any simple submodule is isomorphic to L(𝔭, L′), for some L′ ∈ S. Hence M(𝔭, L) is simple by Lemma 11.1, whenever L ∉ H^{∗}.

Let X = K^{n}. Note that 𝔰𝔩(n + 1) ⊂ W_{X}, and the truncated De Rham complex 0 → Z_{X}^{k} → Ω_{X}^{k} → … → Ω_{X}^{n} → 0 is exact. For i < n, Ω_{X}^{i} has lenght two, with composition factors d Ω_{X}^{i−1} and d Ω_{X}^{i} (see [Gai]) and Ω_{X}^{n} is simple. Moreover, it is easy to identify Ω_{X}^{i} as the O-dual of M(𝔭, K[i]). Hence the existence of the resolution (ii) is proved for L = K[i], and the general case follows from translation principle.

It follows that we can compute the character of L(𝔭, L). We have

ch(L(𝔭, L)) = ch(L) ch(S𝔤_{−1}), if L ∉ H^{∗}, and

ch(L(𝔭, L)) = ∑_{i≥0}(−1)^{i} ch(L[i]) ch(S𝔤_{−1}), if L ∈ H^{∗}.

The set roots of 𝔤_{−1} is a basis of Q, hence the degree of L(𝔭, L) is as asserted. Q.E.D. LEMMA 11.3 Let λ ∈ P^{+} and let L be the simple 𝔤𝔩(n)-module with highest weight

λ. There is a natural complex C(λ) : 0 → T EN S(L[0]) → T EN S(L[1]) → …, and its homology is H_{DR}(T) ⊗ L(λ).

Proof: First consider the case λ = 0, i.e. L = K. Then K[i] ≃ ∧^{i}V, where V is the natural n-dimensional representation of 𝔤𝔩(n)-module. We can identify ⊕_{k≥0}T EN S(L[k]) with A ⊗ ∧𝔥^{∗}, where A is the vector space defined in the previous example. We can define the De Rham differential d as usual. By definition the complex C(0) is A ⊗ ∧𝔥^{∗} with d as differential. We have d ∘ i_{E} + i_{E} ∘ d = E, where i_{E} is the contraction by E, hence the cohomology of C(0) is the same as the cohomology of the subcomplex C(0)^{E} = ∧𝔥^{∗}. Hence the cohomology of C(0) is H_{DR}^{∗}(T).

The general case follows from translation principle. It should be noted that the differential d of C(0) is W_{T} -invariant, but for λ ≠ 0 the corresponding differential on C(λ) is only 𝔰𝔩(n + 1)-invariant.

For n ≥ 2, by Theorem 8.5, any semi-simple irreducible coherent 𝔰𝔩(n + 1)-family M contains a unique infinite dimensional submodule L(λ) with λ(h_{i}) ≥ 0 for all i < n and λ(h_{n}) ∉ Z_{≥0} (this result fails for 𝔰𝔩(2), because it is not possible to distinguish P^{+} from P^{−} for non integral central characters). Set L(M) = H^{0}(𝔤_{1}, L(λ)). Thus L(M) is a finite dimensional simple 𝔤𝔩(n)-module and L(M) belongs to S^{+}, where S^{+} = S \ H^{0}.

It follows from Lemma 5.5 that we can consider formal coherent families, i.e. elements in a suitable K_{0}-group. This is the meaning of the formal sum in Assertion (ii) of the following Theorem. The case 𝔰𝔩(2) being easy, there is no loss loss generality to assume that n ≥ 2 in the next Theorem:

THEOREM 11.4 (for 𝔤=𝔰𝔩(n + 1), 𝔤≠𝔰𝔩(2)) The map M ↦ L(M) is a bijection from the set of semi-simple irreducible coherent 𝔰𝔩(n + 1)-families to the set S^{+}. Moreover, let M be a semi-simple irreducible coherent 𝔰𝔩(n + 1)-family.

(i) If its central character is singular, or is not integral, M is isomorphic to T EN S(L(M))^{ss}, and its degree is dim L(M). (ii) Otherwise M is isomorphic to the effective coherent family ∑_{i≥0}(−1)^{i}T EN S(L[i])^{ss}, where L = L(M) and its degree is ∑_{i≥0}(−1)^{i}dim L(M)[i]. Moreover for t ∉ ∪_{Q′∈T} (Sing(M) + T_{Q′}^{∗}) the following sequence:

0 → M[t] → T EN S(L[0]) → T EN S(L[1]) → …

is a finite resolution of the cuspidal representation M[t].

Proof: The fact that the map M ↦ L(M) is a bijection is a reformulation of Theorem 8.5. Let M be a semi-simple irreducible coherent 𝔰𝔩(n + 1)-family, and set L = L(M). Set X = K^{n}. Note that 𝔰𝔩(n + 1) ⊂ W_{X}, and the coherent family T EN S(L) contains T ens_{X}(L, 0) as a submodule. Moreover T ens_{X}(L, 0) is the O-dual of M(𝔭, L).

First assume that L ∈ H^{∗}, or that L ∈ H^{n} (in such case L[1] = 0). By Lemma 11.2, the module T ens_{X}(L, 0) is simple and its degree is the same as the degree of the coherent family T EN S(L). Hence by Proposition 4.8, the coherent family is irreducible, and the assertions of the lemma follow.

Next assume that L ∈ H^{i} with i ≠ n. By definition of L, we also have i ≠ 0. By lemma 11.2, the module T ens_{X}(L, 0) has two composition factors, namely the O duals of L(𝔭,L) and L(𝔭, L[1]). These two modules are admissible and the sum of their degree is the degree of T EN S(L). As their supports are in the same Q-coset, it follows from Proposition 4.8 that T EN S(L)^{ss} = EX T (L(𝔭,L)) ⊕ EX T (L(𝔭, L[1])). Hence the formal coherent family ∑_{i≥0}(−1)^{i}T EN S(L[i])^{ss} is effective and equals to EX T (L(𝔭,L)) = M. Moreover for t ∈ T^{∗} \ ∪_{Q′∈T} T_{Q′}^{∗}, M[t] is cuspidal (Theorem 10.2). The t-component of the sequence of Lemma 11.3 is exact, and the last assertion follows. Q.E.D.

Remark: Set M = L(𝔭, L), where L ∈ S^{+}. Of course, it is possible to determine when M is multiplicity-free by analyzing the formula giving its degree. However, there is a quickest way. Set V =𝔤_{−1} as before. The torsion free SV -module M has rank one, i.e. as a SV -module M can be identified with an ideal of SV. Moreover, we can embbed M in SV in a such way that the elements in M have no commun divisors. It follows that the ideal M defines a subscheme of codimension ≥ 2. Moreover 𝔰𝔩(n + 1) acts over M by differential operators. Hence the action of 𝔰𝔩(n + 1) on M extends to an action on SV. Then it is easy to identify SV with a generalized Verma module M(𝔭, K_{a}) for a certain a ∈ K. It follows that M = M(𝔭, K_{a}), if a ∉ Z_{≥0} or M is the unique simple submodule of M(𝔭, K_{a}), if a ≥ 0. In both case, EX T (M) is the coherent family M(a)^{ss}. Hence any multiplicity free semi-simple coherent family are the coherent families M(a)^{ss} (for 𝔤≠𝔰𝔩(2) they are distinct), and we recover the following result [BL]:

COROLLARY 11.5 (Britten and Lemire) Any infinite dimensional simple multiplicity free weight 𝔰𝔩(n + 1)-module is a submodule of M(a)^{ss}, for some a ∈ K.

12. Explicit realization of coherent families for 𝔰𝔭(2n). In this section, we will describe and compute the degree of irreducible coherent families for 𝔰𝔭(2n). Also, we will reformulate the classification theorem, by using a more concrete description of the set of conected components of Q. Indeed semi-simple irreducible coherent families for 𝔰𝔭(2n) are exactly classified by odd pairs of Spin(2n)-modules.

The notations specific to 𝔰𝔭(2n) have been introduced in Section 9. Let X be a smooth affine manifold of dimension n and let Diff(X) the ring of differential operators on X. Recall that Diff(X) is generated by the differential operators of order ≤ 1, i.e. by the functions and the vector fields. Any closed one form ω induces an automorphism Θ_{ω} of Diff(X) defined by Θ_{ω}(f) = f if f ∈ K[X] and Θ_{ω}(ξ) = ξ + i_{ξ}ω if ξ ∈ W_{X}. For any invertible function f, Θ_{df/f} is the conjugacy by f. Hence the class of Θ_{ω} in the outer automorphism group of Diff(X) depends only on ω modulo logarithmic differentials.

Let consider now X = T_{sc} be the maximal torus of SP (2n), and denote by T_{sc}^{∗} be the dual torus of T_{sc}. Any ω ∈ T_{sc}^{∗} gives rise to a well define element Θ_{ω} in the outer automorphism group of Diff(T), so we can define the Diff(T)-module V = ⊕_{ω∈T_{sc}^{∗}}K[T]^{Θ_{ω}}.

There is a natural embbeding 𝔰𝔭(2n) in Diff(T). To get an explicit description of it, set K[T_{sc}] = K[x_{1}^{±1}, …, x_{n}^{±1}] and recall that we can decompose 𝔰𝔭(2n) as 𝔤_{−2}⊕𝔤_{0}⊕𝔤_{2}, with 𝔤_{0} =𝔤𝔩(n), 𝔤_{−2} = S^{2}V^{∗} and 𝔤_{2} = S^{2}V, where V is the natural representation of 𝔤𝔩(n). As a Lie subalgebras of Diff(T), 𝔤_{−2} has basis (∂^{2}∂x_{i}∂x_{j})_{1≤i≤j≤n}, 𝔤_{2} has basis (x_{i}x_{j})_{1≤i≤j≤n} and 𝔤_{0} has basis (x_{i}∂/∂x_{j})_{1≤i≠j≤n} ∪ (1/2 + x_{i}∂/∂x_{i})_{1≤i≤n}.

The Shale-Weil coherent family is the 𝔰𝔭(2n)-module V. As its degree is one, V is irreducible. Any element of T_{sc}^{∗} is a union of two Q-cosets, so each component K[T]^{Θ_{ω}} of V is a union of two cosets components. In particular, V contains K[x_{1}, …, x_{n}] as a submodule. It is clear that K[x_{1}, …, x_{n}] ≃ L(ω^{+}) ⊕ L(ω^{−}), where the two components correspond with the space of polynomials of even and odd degree.

Let M be a semi-simple irreducible coherent family, and let χ be its central character.

By Theorem 9.3, M contains exactly two highest weight modules L(λ^{±}) (to define λ^{±}, we require λ^{+}(h_{n}) ≥ −1/2 and λ^{−}(h_{n}) ≤ −3/2). The weight Λ = λ^{+} − ω^{+} is integral and dominant. For any 𝔰𝔭(2n)-module M, denote by M^{(χ)} its maximal submodule with generalized central character χ.

PROPOSITION 12.1 (for 𝔤=𝔰𝔭(2n)) Let M be as before. We have M ≃ (V^{ss} ⊗ L(Λ))^{(χ)}.

Proof: Let χ_{0} be the central character of V, and let T be the translation functor from the category of 𝔰𝔭(2n)-modules with generalized central character χ_{0} to the category of 𝔰𝔭(2n)-modules with generalized central character χ. We have T (V^{ss}) = V^{ss} ⊗ L(Λ))^{(χ)}. By Theorem 9.3, M is the unique semi-simple irreducible coherent family with central character χ. As T is an equivalenve of categories, we have T (V)^{ss} = M. Q.E.D.

Set α′_{i} = α_{i}, h′_{i} = h_{i} for 1 ≤ i < n and α′_{n} = α_{n} + α_{n−1}, h′_{n} = h_{n−1} + 2h_{n}. Then B′ = (α′_{i})_{1≤i≤n} is the basis of the root system of Spin(2n), and (h′_{i})_{1≤i≤n} is the set of corresponding coroots. For any weight λ which is integral and dominant with respect to B′, we will denote by L(λ, Spin(2n)) the corresponding finite dimensional simple Spin(2n)- module with highest weight λ. Denote by σ the involution of B′ which exchanges α′_{n} and α′_{n−1} and denote again by σ the corresponding outer automorphism of Spin(2n). A pair {L, L′} of finite dimensional simple Spin(2n)-module is called an odd pair of Spin(2n)- modules if and only if L′ = L^{σ} and L, L′ are not SO(2n)-modules (i.e. the distinguished central element of Spin(2n) acts non trivially). Equivalently, their highest weights λ, λ′ satisfy λ′ = λ^{σ} and λ(h′_{n−1} + h′_{n}) is odd.

Let L(λ^{±}) the two highest weights of an irreducible coherent 𝔰𝔭(2n)-family M. Then L(λ^{±} + ϵ, Spin(2n)) is an odd pair of Spin(2n)-modules, where ϵ = ∑_{1≤i≤n} ϵ_{n}. This pair will be denoted by L(M). The common dimension of the modules L(λ^{±} + ϵ, Spin(2n)) is denoted by dim L(M).

THEOREM 12.2

(i) The map M ↦ L(M) is a bijection from the set of semi-simple irreducible coherent 𝔰𝔭(2n)-families and the set of odd pairs of Spin(2n)-modules.

(ii) Let M be an irreducible coherent 𝔰𝔭(2n)-family. Its degree is 1/2^{n−1} dim L(M). (iii) Let t = ∑_{1≤i≤n} a_{i}ϵ_{i} ∈ T^{∗}. The module M[t] is cuspidal if and only if a_{i} ∉ 1/2+Z

for all 1 ≤ i ≤ n.

Proof: Assertions (i) and (iii) are just reformulations of Theorems 9.3 and 10.2. Let us prove Assertion (ii). Let L(λ^{±}) be the two highest weight representations of M.

We claim that ch(L(λ^{±})) = ∑_{w∈W′} ϵ(w)e^{w(λ^{±}+ρ)−ρ}/ ∏_{α∈Δ^{+}}(1 − e^{−α}), where W′ is the Weyl group generated by the reflexions associated to short roots. Indeed this follows from [JG]. However the authors of the paper rely their proof on some facts about highest weight representations which are been proved later by Soergel [S]. It is worth to mention than in the special case considered here, there is an elementary and direct proof: indeed it is enough to check the character formula in the case where λ^{±} = ω^{±} and to use the translation fuctor.

Next note that W′ is the Weyl group of Spin(2n). Let Δ^{′+} be the set of positive roots of Spin(2n) and let ρ′ = 1/2 ∑_{α∈Δ^{′+}} α. We have Δ^{+} = Δ^{′+} ∪ {2ϵ_{i}|1 ≤ i ≤ n} and ρ = ρ′ + ϵ. By using Weyl’s character formula for Spin(2n), we get ch(L(λ^{±})) = ch(L(λ^{±} + ϵ, Spin(2n)))/ ∏_{1≤i≤n}(1 − e^{−2ϵ_{i}}). As the set {2ϵ_{i}|1 ≤ i ≤ n} is the basis of a lattice of index 2^{n−1} in Q, the admissible

modules L(λ^{±}) have degree 1/2^{n−1} dim L(M). Thus Assertion (iii) follows from Proposition 4.8. Q.E.D.

Remark: There is only one odd pair of Spin(2n)-modules of dimension 2^{n−1}, namely the pair consisting of the two fundamental spin representations. Hence V^{ss} is the unique semi-simple coherent family of degree one. Thus we recover the following result:

COROLLARY 12.3 (Britten and Lemire) Any infinite dimensional simple multiplicity free weight 𝔰𝔭(2n)-module is a submodule of V^{ss}.

13. Character formula for simple weight modules. In this section, we will describe a character formula for any simple weight modules, or more precisely to reduce the general problem to a solved problem of the category O. This reduction is based on relative coherent families. It would have been possible to treat coherent families in the relative context at once: however notations and statements would have become much more heavy. Also the lemmas in the section could have been stated in a much more general context.

In this section we will fix a simple Lie algebra 𝔤 and a Levi subalgebra 𝔞 containing 𝔥. We will denote by Δ′ the root sytem of 𝔞 and by Q′ its root lattice. For any weight module, we define its relative character rch(M) and its relative degree rdeg(M) to be the functions λ ↦ Sup_{μ∈λ+Q′} dimM_{μ} and λ ↦ Sup_{μ∈λ+K⊗Q′} dimM_{μ}. A relative coherent family is a weight 𝔤-module M such that

(i) the function λ ↦ dim M_{λ} is constant on each K ⊗ Q′-coset,

(ii) for any u ∈ A, the function λ ↦ T r(u|M_{λ}) is polynomial on each K ⊗ Q′-coset.

LEMMA 13.1 Let M be a simple weight module such that Δ_{M}^{I} = Δ′. There exists a relative coherent family M containing M such that rdeg(M) = rdeg(M).

Proof: By Lemma 4.1, there is a set Σ ⊂ Δ′ of commuting roots which is a basis of Q′. For any α ∈ Σ, e_{−α} acts bijectively on M, so we have M = M_{F_{Σ}}. Set M = ⊕_{ν∈T_{Q′}^{∗}} F_{Σ}^{ν}.M. It follows from Lemma 4.3 that M is a relative coherent family, which contains M and of same relative degree. Q.E.D.

A weight 𝔤-module M is called relatively admissible if for any K ⊗ Q′-coset t, the 𝔞-module M[t] is admissible. Let B′ be a basis of Δ′, and λ ∈𝔥^{∗}. Denote by L_{B′}(λ) the simple 𝔞-module with B′-highest weight λ.

LEMMA 13.2 Let B be a basis of Δ containing B′. If the module L_{B′}(λ) is admissible, then L_{B}(λ) is relatively admissible.

Proof: Set 𝔲^{−} = ⊕_{α∈Δ_{B}^{−}\Δ_{B′}^{′−}} 𝔤_{α}. The natural map U(𝔲^{−}) ⊗ L_{B′}(λ) → L_{B}(λ) is onto and for each K ⊗ Q′-coset t the 𝔞-module U(𝔲^{−})[t] is finite dimensional. Hence L_{B}(λ) is admissible. Q.E.D.

The module L_{B′}(λ) is called strictly infinite dimensional if any connected component of B′ contains a root α such that λ(h_{α}) ∉ Z_{≥0}. We can decompose 𝔞 as 𝔷⊕(⊕_{c}𝔞_{c}), where 𝔷 is the center of 𝔤, c runs over the connected components of B′ and 𝔞_{c} is the corresponding simple ideal. Then L_{B′}(λ) decomposes as Z ⊗ (⊗_{c}L_{c}), where Z is a character of 𝔷, and L_{c} is a simple c-highest weight module for 𝔞_{c}. Then L_{B′}(λ) is strictly infinite dimensional means that each L_{c} is infinite dimensional.

LEMMA 13.3 Let B′ be a basis of Δ′, let B ⊃ B′ be a basis of Δ and let λ ∈𝔥^{∗}, such that L_{B′}(λ) is admissible and strictly infinite dimensional. There exists a relative coherent family M containing L_{B}(λ) such that rdeg(M) = rdeg(L_{B}(λ)).

Proof: Set L = L_{B}(λ), L′ = L_{B′}(λ). By Lemma 4.1, there is a set Σ ⊂ Δ_{L′}^{′+} of commuting roots which is a basis of Q′. Set M = L_{F_{Σ}}. By Lemma 13.3, L is admissible, thus its relative character rch(L) only takes finite values.

We claim that ch(M) = rch(L) (in particular, we claim that each weight space in M is finite dimensional). As e_{−α} acts bijectively on M, for any α ∈ Σ, the character of M is Q′-invariant, i.e. ch(M) = rch(M). As (Δ′)_{L′}^{+} ⊂ Δ_{L}^{′+}, L is a submodule of M and thus ch(M) ≥ rch(L). Let μ be a weight of M, and let E be a finite dimensional subspace of M_{μ}. There is s ∈ F_{Σ} such that s.E ⊂ L. If β is a weight of s, we have s.E ⊂ L_{μ+β}. As s acts injectively, we get dim E ≤ dim L_{μ+β}, i.e. dim M_{μ} ≤ dim L_{μ+β}. Hence ch(M) ≤ rch(L), and the claim is proved.

As previouly, M = ⊕_{ν∈T_{Q′}^{∗}} F_{Σ}^{ν}.M is a coherent family and its relative degree equals the relative degree of M, i.e. is the same as the relative degree of L. Q.E.D.

We want to compute the character of all simple weight modules M. In order to state the theorem, we need to introduce some data attached to M. By Theorem 1.1, there exists a parabolic subalgebra 𝔭 with unipotent radical 𝔲 and a cuspidal representation N of 𝔭/𝔲 such that M ≃ L(𝔭, N). Let 𝔞 be the Levi component of 𝔭, and Δ′, Q′ be the set of roots and the root lattice of 𝔞. Choose a basis B of Δ which is adapted to 𝔭, and let B′ the corresponding basis of Δ′ (i.e. B′ = B ∩ Δ′). It follows from Proposition 6.2 that the coherent 𝔞-family EX T (N) contains a submodule L_{B′}(λ) which is strictly infinite dimensional. Choose ν ∈ −λ + Supp(N).

THEOREM 13.4 We have ch(M) = e^{ν} rch(L_{B}(λ)).

Proof: By Lemma 13.1, M is contained in a relative coherent family M of same relative degree as M. Let μ be a weight of N, and let N = M[μ+K ⊗Q′]. Then N is a 𝔞-coherent family which contains N and of same degree. By Proposition 4.8, we have N^{ss} = EX T (N), hence L_{B′}(λ) is a suquotient of N [λ + Q′]. By assumption on the relative degree, we have Supp(M) = Supp(M) + K ⊗ Q′. However, we have Supp(M) ⊂ Supp(N) + C, where C is the cone of 𝔭. It follows that Supp(M) ⊂ Supp(N) + C. In particular, we have Supp(M[λ + Q]) ⊂ Supp(N [λ + Q′]) + C. Apply Lemma 1.5 to the 𝔤-module M[λ + Q] and to the 𝔭-module N [λ + Q′]. We get that L_{B}(λ) is a subquotient of M[λ + Q]. Hence we get ch(L_{B}(λ)) ≤ rdeg(M) = rdeg(M). It follows that rch(M) ≥ e^{−ν} ch(L_{B}(λ)).

Similarly, by Lemma 13.3, there is a coherent family M′ containing L_{B}(λ) and of same relative degree. By the same argument, we prove that M is a suquotient of M′, and thus ch(M) ≤ e^{ν} rch(L_{B}(λ)).

For any α ∈ Δ′, e_{α} acts bijectively on M. So we have ch(M) = rch(M), and the equality ch(M) = e^{ν} rch(L_{B}(λ) follows. Q.E.D.

Appendix: Generalized Enright functors. In the paper, we repetively used some localizations with respect with some non-integral powers. Indeed this tool can be used to define various functors between categories of weight modules, and these functors are very analogous to Enright functors. In this appendix, we will give one example of functor using a localization with respect to some non-integral power of some root vector. Proposition A.3 was known [S], but we provide here a very elementary proof.

Fix once for all a basis B of 𝔤, what allows to define the category O. Let χ be a non necessarily integral central character and let O^{(χ)} the category of all modules M ∈ O with generalized central character χ. For u ∈ T^{∗}, denote by O^{(χ)}[u] the category of all modules M in O whose supports are included in u.

LEMMA A.1 Let u, v ∈ T^{∗} be W -conjugated. Then the categories O^{(χ)}[u] and O^{(χ)}[v] are equivalent.

Proof: It is enough to prove the assertion when u = s_{α}.v for some simple root α. Set e = e_{α}, f = f_{α} and choose x ∈ K such that u = x.α + v. We can assume that u ≠ v, i.e. that x ≠ Z. For M ∈ O^{(χ)}[u] (respectively in O^{(χ)}[u]), let F (M) (respectively G(M)) be the space of e-locally nilpotent vectors in f^{x}.M_{f} (respectively in f^{−x}.M_{f}). For any central element u ∈ U, we have f^{x} u f^{−x} = u. It follows that F (M) ∈ O^{(χ)}[v] (respectively F (M) ∈ O^{(χ)}[u]), and by standard 𝔰𝔩(2)-computations it is easy to check that the two functors F and G are inverse each other. Q.E.D.

For a central character χ, let Λ^{+}(χ) the set of weights λ such that L(λ) belongs to O^{(χ)} and (λ + ρ)(h_{α}) ∉ Z_{<0} for all α ∈ Δ^{+}.

LEMMA A.2 Let χ be a central character. The blocks of O^{(χ)} are the subcategories O^{(χ)}[λ + Q], for λ ∈ Λ^{+}(χ), and they are all equivalent.

Proof: For λ ∈ Λ^{+}(χ), let W_{λ} be the subgroup of W generated by the reflexions s_{α} such that λ(h_{α}) ∈ Z. Indeed, W_{λ} is the stabilizer of the coset λ + Q. For any μ such that L(μ) ∈ O^{(χ)}, there is a unique λ ∈ Λ^{+}(χ) such that μ = w (λ + ρ) − ρ for some w ∈ W_{λ}. As L(μ) is a subquotient of the Verma module M(μ) (see e.g. [D], ch. 7), L(μ) and L(λ) are in the same block. Hence the blocks of O^{(χ)} are exactly the subcategories O^{(χ)}[λ + Q], and by Lemma A.1 they are all equivalent. Q.E.D.

PROPOSITION A.3 ( Soergel [S]) Assume that 𝔤 is simply laced. Then any block in O is equivalent to a block with integral central character of the category O of some Levi subalgebra.

Proof: Let B be a block of O, let χ be its central character. Let u be the Q-coset such that B = O^{(χ)}[u], and let Δ′ be the set of all roots α such that u(h_{α}) = 0 modulo Z. As 𝔤 is simply laced, for any α, β ∈ Δ such that α + β is a root, we have h_{α+β} = h_{α} + h_{β}. Hence 𝔤′ =𝔥⊕_{α∈Δ′} 𝔤_{α} is a reductive Lie subalgebra, and it is easy to see that 𝔤’ is a Levi subagebra. Let Δ′ be set of roots of 𝔤’ and let B′ be a basis of Δ′. There is a w ∈ W such that B^{′w} is contained in B. By Lemmas A.1 and A.2, we can assume that w = 1, i.e B′ ⊂ B.

Let λ ∈ Λ^{+}(χ) be the unique weight such that L(λ) ∈ B, set u′ = λ + Q′ where Q′ is the root lattice of 𝔤' and let W′ be the Weyl group of 𝔤'. For any highest weight module L(μ) ∈ B, there is some w ∈ W′ such that μ = w(λ + ρ) − ρ. In particular μ ∈ u′. Let χ′ be the central character of the 𝔤'-module L(λ)[u′]. It is easy to show that the functor M ∈ O^{(χ)}[u] ↦ M[u′] ∈ O^{′(χ′)}[u′] is an equivalence of categories and its inverse is the induction from 𝔭 to 𝔤, where 𝔭 is the parabolic subalgebra 𝔟+𝔤', where 𝔟 is the Borel subalgebra of 𝔤. By definition χ′ is an integral character of the Levi subalgebra 𝔤’, so the proposition is proved. Q.E.D.

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Author’s address: IRMA, Université Louis Pasteur, 7 rue René Descartes, 67000 Strasbourg (FRANCE).

mathieu@math.u-strasbg.fr