THE COMPOSITION SERIES OF MODULES INDUCED FROM WHITTAKER MODULES

DRAGAN MILIČIĆ AND WOLFGANG SOERGEL

Abstract. We study a category of representations over a semisimple Lie algebra, which contains category O as well as the so-called Whittaker modules, and prove a generalization of the Kazhdan-Lusztig conjectures in this context.

1. Introduction

Let 𝔤 ⊃ 𝔟 ⊃ 𝔫 be a semisimple complex Lie algebra, a Borel subalgebra, and its nilradical. Let U(𝔤) = U ⊃ Z be the enveloping algebra of 𝔤 and its center. Let χ ⊂ Z be a maximal ideal and f ∈ (𝔫/[𝔫, 𝔫])^{∗} =: ch𝔫 a character of 𝔫, giving rise to a one-dimensional 𝔫-module ℂ_{f}. By [McD85] the 𝔤-module

Y (χ, f) = U/χU ⊗_{U(𝔫)} ℂ_{f}

is of finite length. For any ring A let MaxA denote its set of maximal ideals. We are interested in the following

Problem 1.1. Compute the composition factors of Y (χ, f) with their multiplicities, for all χ ∈ MaxZ and f ∈ ch𝔫.

We will solve this problem completely for integral χ and partially for other χ as well^{1}. Let us first consider the two extreme cases. Let 𝔥 ⊂ 𝔟 be a Cartan subalgebra and 𝔥^{∗} ⊃ R ⊃ R^{+} ⊃ Δ its dual, the roots, the roots of 𝔟 and the simple roots. So 𝔤 = ⊕_{α∈R} 𝔤_{α} ⊕ 𝔥. Call f ∈ ch𝔫 regular if and only if f |_{𝔤_{α}}≠ 0 for all α ∈ Δ. For regular f our problem is solved completely by the following theorem of Kostant [Kos78], (see also [MS95] for a geometric proof).

Theorem 1.2. If f ∈ ch𝔫 is regular, then Y (χ, f) is irreducible for all χ ∈ MaxZ.

These irreducible Y (χ, f) are the so-called Whittaker modules. In the other extreme, i.e. for f = 0, we have (see 2.5)

1991 Mathematics Subject Classification. 17B10, 22E47.

1The general case was recently settled by E. Backelin [Bac97]

Proposition 1.3. If χ ∈ MaxZ is regular, then Y (χ, 0) is the direct sum of all Verma modules for 𝔤 ⊃ 𝔟 with central character χ. For χ singular Y (χ, 0) still has a Verma flag such that each Verma module with central character χ appears with the same multiplicity and the length of the flag is the cardinality of the Weyl group.

Thus for f = 0 our problem is solved by the Kazhdan-Lusztig conjectures, which describe the composition series of Verma modules. The general case will be a mixture of these two. We will partially solve it by reducing to the Kazhdan-Lusztig conjectures. To explain how this is done, let us put our problem in a different perspective.

Consider the full subcategory N = N (𝔤, 𝔟) ⊂ 𝔤-mod of all 𝔤-modules M which are (1) finitely generated over 𝔤, (2) locally finite over 𝔫 and (3) locally finite over Z. By [McD85] all objects of N have finite length. From a geometric perspective [MS95] this is evident, they just correspond to holonomic D-modules. The action of Z decomposes N into a direct sum N = ⊕_{χ}N (χ) where χ runs over MaxZ. The action of 𝔫 also decomposes N into a direct sum N = ⊕_{f}N (f) over all f ∈ ch𝔫, where

N (f) = {M ∈ N | X − f(X) acts locally nilpotently on M, ∀X ∈ 𝔫}.

In total, we have N = ⊕_{χ,f}N (χ, f) with N (χ, f) = N (χ) ∩ N (f) and all these categories are stable under subquotients and extensions in 𝔤mod. Certainly Y (χ, f) ∈ N (χ, f). To solve our problem, we have to study the categories N (χ, f).

Again the two extreme cases are more or less well understood. For regular f there is an equivalence of categories

N (f) ≅ {M ∈ Z-mod | dim_{ℂ}M < ∞}

as was shown by Kostant [Kos78]. For f = 0 our N (f) = N (0) consists just of all finite length 𝔤-modules with only highest weight modules as composition factors. For general f, the situation was investigated by McDowell [McD85]. In fact, McDowell’s results as well as Kostant’s results cited above admit very natural geometric proofs if one uses localization. We worked this out in our joint paper [MS95].

Let (W, S) be the Coxeter system of 𝔤 ⊂ 𝔟. There is a bijection

Δ →^{∼} S, α ↦ s_{α}. Put ρ = \frac{1}{2} ∑_{α∈R^{+}} α as usual and set w · λ = w(λ +

ρ) − ρ for all w ∈ W, λ ∈ 𝔥^{∗}. Let the Harish-Chandra homomorphism ξ^{♯} : Z → S = S(𝔥) be normalized by the condition ξ^{♯}(z) − z ∈ U𝔫. For the corresponding ξ : 𝔥^{∗} → MaxZ we have ξ(λ) = ξ(μ) ⇔ W·λ = W·μ. For f ∈ ch𝔫 put Δ_{f} = {α ∈ Δ | f|_{𝔤_{α}} ≠ 0}, S_{f} = {s_{α} | α ∈ Δ_{f}} and W_{f} = ⟨S_{f}⟩ ⊂ W.

Now for any λ ∈ 𝔥^{∗}, f ∈ ch𝔫 one constructs a “standard module” M(λ, f) ∈ N (ξ(λ), f). It has a unique simple quotient L(λ, f) and M(λ, f) = M(μ, f) if and only if W_{f} · λ = W_{f} · μ. Finally any simple object L ∈ N (f) has the form L ≅ L(λ, f) for a unique λ ∈ 𝔥^{∗}/(W_{f}·). All this is due to McDowell. The definitions are set up in such a way that M(λ, 0) is just the usual Verma module M(λ) = U ⊗_{U(𝔟)} ℂ_{λ}. For simplicity we state the analog of our first Proposition 1.3 only for regular χ. The general case is contained in Corollary 2.5.

Proposition 1.4. Suppose χ ∈ MaxZ is regular. Then Y (χ, f) ≅ ⨁_{λ} M(λ, f) where λ runs over ξ^{−1}(χ)/(W_{f}·).

In this way our original problem of computing the composition series of the Y (χ, f) reduces to the following

Problem 1.5. Compute the multiplicities [M(λ, f) : L(μ, f)] for all λ, μ ∈ 𝔥^{∗}.

Let us just explain how we solve this problem for regular integral central character χ. Let μ ∈ 𝔥^{∗} be integral dominant (i.e. ⟨μ+ρ, α^{∨}⟩ ∈ ℤ_{≥0} for all α ∈ R^{+}) and such that W_{f} = {x ∈ W | x · μ = μ}. Let λ ∈ 𝔥^{∗} be integral dominant such that χ = ξ(λ). At the end of Section 5 we establish an equivalence of categories

N (χ, f) ≅ N (ξ(μ), 0)

under which M(x · λ, f) corresponds to M(x^{−1} · μ). This reduces our problem to the Kazhdan-Lusztig conjectures, which by now are a theorem.

2. Standard modules and simple modules

Remember f ∈ ch𝔫 determines a subset Δ_{f} = {α ∈ Δ | f|_{𝔤_{α}} ≠ 0} of simple roots. Let 𝔭_{f} ⊂ 𝔤 be the corresponding parabolic subalgebra containing 𝔟 and 𝔭_{f} = 𝔤_{f} ⊕ 𝔫^{f} its ad𝔥-stable Levi decomposition. Remark that 𝔤_{f} is not semisimple in general, but only reductive. For example Δ_{0} = ∅, 𝔭_{0} = 𝔟, 𝔤_{0} = 𝔥 and 𝔫^{0} = 𝔫. Let U(𝔤_{f}) = U_{f} ⊃ Z_{f} be the enveloping algebra of 𝔤_{f} and its center. Put 𝔟_{f} = 𝔟 ∩ 𝔤_{f} and let 𝔫_{f} ⊂ 𝔟_{f} be its nilradical, so that 𝔫 = 𝔫_{f} ⊕ 𝔫^{f}. Let ξ_{f}^{♯} : Z_{f} → S be the Harish-Chandra homomorphism of 𝔤_{f}, normalized as before by the condition ξ_{f}^{♯}(z) − z ∈ U_{f}𝔫_{f}. It induces on the maximal ideals a map ξ_{f} : 𝔥^{∗} → MaxZ_{f}. For any ideal I ⊂ Z_{f} define the 𝔤_{f}-module

Y_{f}(I, f) = U_{f}/IU_{f} ⊗_{U(𝔫_{f})} ℂ_{f}.

If I is a maximal ideal, then this is an irreducible 𝔤_{f}-module, since the restriction of f to 𝔫_{f} is nondegenerate by definition of 𝔫_{f}. We define

M(λ, f) = U ⊗_{U(𝔭_{f})} Y_{f}(ξ_{f}(λ), f)

where we extend the 𝔤_{f}-action on Y_{f} to an action of 𝔭_{f} letting 𝔫^{f} act by zero.

Proposition 2.1. (1) We have M(λ, f) ≅ M(μ, f) if and only if

W_{f} · λ = W_{f} · μ.

(2) M(λ, f) has a unique simple quotient L(λ, f). We have L(λ, f) ≅ L(μ, f) if and only if W_{f} · λ = W_{f} · μ.

(3) Ann_{U}M(λ, f) = ξ(λ)U.

Proof. All this is in fact contained in [McD85] and, from a geometric point of view, in [MS95]. However for us it is not a great detour, so we will give complete arguments. We start with 3. This is easily reduced to the case of Verma modules by some general considerations: Let 𝔞 → 𝔟 be a morphism of Lie algebras. For any 𝔟-module M let res_{𝔟}^{𝔞}M denote the 𝔞-module obtained by restriction. For any 𝔞-module N let ind_{𝔞}^{𝔟}N denote the 𝔟-module ind_{𝔞}^{𝔟}N = U(𝔟) ⊗_{U(𝔞)} N obtained by induction. For any module M over a Lie algebra let AnnM denote its annihilator in the enveloping algebra.

Lemma 2.2. (1) Let M, M′ be 𝔟-modules. Then from the inclusion AnnM ⊂ AnnM′ follows the inclusion Ann(res_{𝔟}^{𝔞}M) ⊂ Ann(res_{𝔟}^{𝔞}M′).

(2) Let N, N′ be 𝔞-modules. Then from the inclusion AnnN ⊂ AnnN′ follows the inclusion Ann(ind_{𝔞}^{𝔟}N) ⊂ Ann(ind_{𝔞}^{𝔟}N′).

Proof. Omitted. □

From this we deduce part 3 of the proposition, as follows: Let M_{f}(λ) denote the Verma module with highest weight λ for 𝔤_{f}. Then

Ann_{U_{f}} Y_{f}(ξ_{f}(λ), f) = ξ_{f}(λ)U_{f} = Ann_{U_{f}} M_{f}(λ)

by a theorem of Kostant [Kos78] and of Duflo [Dix74] respectively. Now we apply part 1 of our lemma with the surjection 𝔭_{f} ↠ 𝔤_{f} and then part 2 with the inclusion 𝔭_{f} ↪ 𝔤 and deduce Ann_{U}M(λ, f) = Ann_{U}(U ⊗_{U(𝔭_{f})} M_{f}(λ)). But certainly U ⊗_{U(𝔭_{f})} M_{f}(λ) = M(λ), hence Ann_{U}M(λ, f) = Ann_{U}M(λ) = Uξ(λ) by the theorem of Duflo once again, and 3 is proved.

Next we prove part 1 of the proposition. Let us decompose 𝔥 = 𝔥^{f} ⊕ 𝔥_{f} with 𝔥_{f} = 𝔥 ∩ [𝔤_{f}, 𝔤_{f}] and 𝔥^{f} = ∩_{α∈Δ_{f}} kerα the centralizer of f alias the center of 𝔤_{f}. The action of W_{f} as well as the dot-action of W_{f} on 𝔥^{∗} respect this decomposition, and they are trivial on (𝔥^{f})^{∗}.

Remark that a priori there are two dot-actions of W_{f} on 𝔥^{∗}, fixing the halfsum of positive roots of 𝔤 and of 𝔤_{f} respectively, but one checks that they coincide. Now it is clear that W_{f} ·λ = W_{f} ·μ implies ξ_{f}(λ) = ξ_{f}(μ) and hence M(λ, f) = M(μ, f). To prove the reverse implication, denote for any λ ∈ 𝔥^{∗} by λ^{f} ∈ (𝔥^{f})^{∗} its restriction to 𝔥^{f}. Clearly 𝔥^{f} acts via λ^{f} on Y_{f}(ξ_{f}(λ), f). Hence M(λ, f) decomposes under 𝔥^{f} into weight spaces M(λ, f)_{μ} with μ ∈ λ^{f} − ∑_{α∈R^{+}} ℤ_{≥0}α^{f}. Clearly all the M(λ, f)_{μ} are 𝔤_{f}-submodules, and M(λ, f)_{λ^{f}} = Y_{f}(ξ_{f}(λ), f). Thus M(λ, f) ≅ M(μ, f) implies ξ_{f}(λ) = ξ_{f}(μ), hence W_{f} · λ = W_{f} · μ and we proved 1.

Finally we go for 2. We have to show that M(λ, f) has a unique maximal proper submodule. But any submodule N ⊂ M(λ, f) is the sum of its 𝔥^{f}-weight spaces N = ⊕N_{μ}, and these are 𝔤_{f}-submodules of the M(λ, f)_{μ}. Since M(λ, f)_{λ^{f}} is irreducible over 𝔤_{f} and generates M(λ, f) over 𝔤, any proper submodule has to be contained in ⊕_{μ≠λ^{f}} M(λ, f)_{μ}. Hence the sum of all proper submodules is itself a proper submodule, necessarily the biggest one. Thus M(λ, f) has a unique simple quotient L(λ, f). Again L(λ, f) ≅ L(μ, f) implies λ^{f} = μ^{f} and L(λ, f)_{λ^{f}} ≅ L(μ, f)_{λ^{f}}, hence ξ_{f}(λ) = ξ_{f}(μ) and finally W_{f} · λ = W_{f} · μ. □

Next we establish Proposition 1.4 from the introduction. Let us define the “relative Harish-Chandra homomorphism” θ^{♯} : Z → Z_{f} by the condition ξ_{f}^{♯}θ^{♯} = ξ^{♯} : Z → S.

Lemma 2.3. For all z ∈ Z we have θ^{♯}(z) − z ∈ U𝔫^{f}.

Proof. Consider U_{f} as a 𝔭_{f}-module with 𝔫^{f} acting by zero and form the induced module U ⊗_{U(𝔭_{f})} U_{f}. Then Ann_{U}(1 ⊗ 1) = U𝔫^{f} and 1 ⊗ U_{f} ⊂ U ⊗_{U(𝔭_{f})} U_{f} is just the space of invariants of 𝔥^{f}. This means that there is a map θ̃: Z → U_{f} such that z(1⊗1) = 1⊗θ̃(z) for all z ∈ Z, and we see easily that this defines an algebra homomorphism θ̃: Z → Z_{f} and that furthermore θ̃(z) − z ∈ U𝔫^{f} for all z ∈ Z. From there we find that ξ_{f}^{♯} θ̃: Z → S is an algebra homomorphism such that ξ_{f}^{♯} θ̃(z) − z ∈ U𝔫, thus θ_{f}^{♯} θ̃ = ξ^{♯}, thus θ^{♯} = θ̃ and the Lemma is proved. □

Proposition 2.4. Let I ⊂ Z be an ideal. We have an isomorphism

Y (I, f) ≅ U ⊗_{U(𝔭_{f})} Y_{f}(θ^{♯}(I)Z_{f}, f).

Proof. Recall that we defined Y (I, f) = U/IU ⊗_{U(𝔫)} ℂ_{f}. Certainly this object represents the functor

U/IU-mod →  ℂ-mod
M  ↦ Hom_{𝔫}(ℂ_{f}, M).

Since θ^{♯}(z) − z ∈ U𝔫^{f}, the right hand side of our future isomorphism is annihilated by I also. It follows furthermore that for any M ∈ U/IU-mod we have θ^{♯}(I)M^{𝔫^{f}} = 0. Thus for any M ∈ U/IU-mod we have

Hom_{𝔫}(ℂ_{f}, M) = Hom_{𝔫_{f}}(ℂ_{f}, M^{𝔫^{f}})
= Hom_{𝔤_{f}}(Y_{f}(θ^{♯}(I)Z_{f}, f), M^{𝔫^{f}})
= Hom_{𝔭_{f}}(Y_{f}(θ^{♯}(I)Z_{f}, f), M)
= Hom_{𝔤}(U ⊗_{U(𝔭_{f})} Y_{f}(θ^{♯}(I)Z_{f}, f), M).

Our isomorphism follows now from the fact that both sides share a universal property, i.e. represent the same functor. □

Corollary 2.5. (1) For all λ ∈ 𝔥^{∗} the module Y (ξ(λ), f) admits a filtration with subquotients M(x · λ, f), where x runs over W_{f} \ W.

(2) If λ is regular, then we have even a decomposition into a direct sum Y (ξ(λ), f) ≅ ⨁_{x} M(x · λ, f) with x running our W_{f} \ W.

Proof. We start with some generalities. For any commutative ring A let A-mod^{fl} be the category of finite length A-modules and [A-mod^{fl}] its Grothendieck group. Any finite flat ring extension j : A → B gives a homomorphism [j] : [A-mod^{fl}] → [B-mod^{fl}], [M] ↦ [B ⊗_{A} M]. Remark that in case M = A/I we have B ⊗_{A} M ≅ B/BI. We remark further that [A-mod^{fl}] can be identified with the free abelian group ℤMaxA over MaxA via 𝔪 ↦ [A/𝔪], so we can view [j] as a morphism [j] : ℤMaxA → ℤMaxB. Now all our ring extensions Z ⊂ Z_{f} ⊂ S given by θ^{♯}, ξ_{f}^{♯} and ξ^{♯} are finite and flat. Let us identify as usual 𝔥^{∗} →^{∼} MaxS via λ ↦ ⟨λ⟩. Now we know from invariant theory, say, that for every λ ∈ 𝔥^{∗} we have in ℤMaxS the equalities

[ξ^{♯}]ξ(λ) = ∑_{x∈W}⟨x · λ⟩

and

[ξ_{f}^{♯}]ξ_{f}(λ) = ∑_{x∈W_{f}}⟨x · λ⟩.

Since [ξ^{♯}] = [ξ_{f}^{♯}] ∘ [θ^{♯}] we deduce [θ^{♯}]ξ(λ) = ∑_{x∈W_{f} \W} ξ_{f}(x · λ). In other words, Z_{f}/θ^{♯}(ξ(λ))Z_{f} admits a filtration with subquotients Z_{f}/ξ_{f}(x·λ) where x runs over W_{f} \ W. Certainly this is in fact a direct sum decomposition when the ξ_{f}(x · λ) are pairwise different, e.g. for λ regular. Now U_{f} is a free Z_{f}-module, and we deduce that U_{f}/θ^{♯}(ξ(λ))U_{f} admits a filtration with subquotients U_{f}/ξ_{f}(x · λ)U_{f}, x ∈ W_{f} \ W, which splits for regular λ to give a direct sum decomposition. But U_{f} is known to be even a free right module over Z ⊗U(𝔫_{f}), thus we find that Y_{f}(θ^{♯}(ξ(λ))Z_{f}, f) admits a filtration with subquotients Y_{f}(ξ_{f}(x · λ), f), x ∈ W_{f} \ W, which splits for regular λ to give a direct sum decomposition. We now apply U⊗_{U(𝔭_{f})} and the Corollary follows from the Proposition. □

We now reprove McDowell’s results.

Theorem 2.6. (1) Any M ∈ N is of finite length.

(2) The L(λ, f) with λ ∈ 𝔥^{∗}/(W_{f}·) represent the isomorphism classes of simple objects in N (f).

Proof. We start with 1. Put I = Ann_{Z}M. By definition of N this is an ideal of finite codimension in Z. Without restriction of generality we can assume I ∈ MaxZ. Using the definition of N once more we find a finite dimensional 𝔫-stable subspace E ⊂ M that generates M as a 𝔤-module. Thus M is a quotient of U/UI ⊗_{U(𝔫)} E and we may restrict our attention to such M. We can filter E by 𝔫-submodules with onedimensional subquotients. This way we reduce our problem to showing that the Y (I, f) are of finite length. Using Corollary 2.5 we further reduce to showing that all M(λ, f) are of finite length. By [Kos78] we know that for any finite dimensional 𝔤_{f}-module E and η ∈ MaxZ_{f} the 𝔤_{f}-module E⊗Y_{f}(η, f) is of finite length and has its composition factors among the Y_{f}(η′, f) with η′ ∈ MaxZ_{f}. Let 𝔫̄^{f} ⊂ 𝔤 be the ad𝔥-stable complement of 𝔭_{f}. Certainly

M(λ, f) ≅ U(𝔫̄^{f}) ⊗_{ℂ} Y_{f}(ξ_{f}(λ), f)

as 𝔤_{f}-modules, and we deduce that all M(λ, f)_{μ} with μ ∈ (𝔥^{f})^{∗} are finite length modules over 𝔤_{f} with their composition factors among the Y_{f}(η′, f).

Now any simple subquotient L of M(λ, f) has to have a “highest” weight μ ∈ (𝔥^{f})^{∗} such that L_{μ} ≠ 0 and 𝔫^{f}L_{μ} = 0. We then find an η ∈ MaxZ_{f} such that Hom_{𝔤_{f}}(Y_{f}(η, f), L_{μ}) ≠ 0, thus Hom_{𝔤}(M(ν, f), L) ≠ 0 if ν ∈ 𝔥^{∗} is such that ξ_{f}(ν) = η, thus L ≅ L(ν, f) since L is simple. Modulo the things we saw already this proves 2.

All simple subquotients of M(λ, f) have the same central character, hence are among the L(x · λ, f), x ∈ W by Proposition 2.1. We deduce that the length of M(λ, f) is bounded by the sum of the lengths of the 𝔤_{f}-modules M(λ, f)_{(x·λ)^{f}} , x ∈ W. □

3. Equivalences between categories of Harish-Chandra bimodules and of representations

In this section we recall results of [BG80] in a form suitable for our purposes. Let for the moment 𝔤 be any complex Lie algebra and U = U(𝔤) its enveloping algebra. On any U-bimodule X ∈ U-mod-U we can define a third 𝔤-action ad : 𝔤 → End_{ℂ}X by the formula (adA)x = Ax − xA for any A ∈ 𝔤, x ∈ X. This is called the adjoint action on a bimodule. We get a functor U-mod-U → 𝔤-mod, M ↦ M^{ad} considering any U-bimodule as a representation of 𝔤 for the adjoint action. For M, N ∈ 𝔤-mod we make Hom_{ℂ}(M, N) into a U-bimodule in the obvious way and consider on M ⊗ N the standard 𝔤-module structure. For M ∈ 𝔤-mod, X ∈ U-mod-U we define M ⊗ X ∈ U-mod-U by the prescriptions A(m ⊗ x) = (Am) ⊗ x + m ⊗ (Ax), (m ⊗ x)A = m ⊗ (xA) for all A ∈ 𝔤, m ∈ M, x ∈ X. With these definitions we find that

(1) For any N, M, E ∈ 𝔤-mod the canonical isomorphism

Hom_{ℂ}(E, Hom_{ℂ}(M, N)) →^{∼} Hom_{ℂ}(E ⊗ M, N)

induces an isomorphism

Hom_{𝔤}(E, Hom_{ℂ}(M, N)^{ad}) →^{∼} Hom_{𝔤}(E ⊗ M, N).

(2) Consider U as an object of U-mod-U. For any E ∈ 𝔤-mod and X ∈ U-mod-U we obtain a canonical isomorphism

Hom_{U−U}(E ⊗ U, X) → Hom_{𝔤}(E, X^{ad})

by composing a morphism E ⊗ U → X with the obvious map E → E ⊗ U, e ↦ e ⊗ 1.

For any X ∈ U-mod-U let X_{adf} ⊂ X denote the subspace of ad𝔤finite vectors, X_{adf} = {x ∈ X | There exists a finite dimensional ad𝔤stable subspace of X containing x}. For M ∈ 𝔤-mod the subspace (End_{ℂ}M)_{adf} ⊂ End_{ℂ}M is actually a subring.

If 𝔤 is finite dimensional, X_{adf} ⊂ X is a sub-U-bimodule. Let us define the category

H = {X ∈ U-mod-U | X = X_{adf} and X is finitely generated}.

It is of no importance here whether X is supposed to be finitely generated as left module, right module, or bimodule: For bimodules consisting of ad𝔤-finite vectors all these properties are equivalent.

Let us return now to our semisimple Lie algebra 𝔤. Let us denote by F(𝔤) = F the category of all finite dimensional representations of 𝔤. To M ∈ 𝔤-mod we associate two full subcategories of 𝔤-mod:

(1) The category ⟨F ⊗ M⟩ consisting of all subquotients of objects of the form E ⊗ M with E ∈ F.

(2) The category coker(F ⊗ M) consisting of all N ∈ 𝔤-mod that admit a two-step resolution E ⊗M → F ⊗M ↠ N with E, F ∈ F.

On the other hand define for any subset I ⊂ U the category

H(I) = {X ∈ H | XI = 0}.

We are now ready to state the result of [BG80] in the form in which we need it.

Theorem 3.1. Let I ⊂ U be a twosided ideal, Ū = U/I the quotient and M ∈ 𝔤-mod a representation with IM = 0. Suppose that

(1) The multiplication U → End_{ℂ}M induces an isomorphism Ū →^{∼} (End_{ℂ}M)_{adf} and

(2) M is a projective object in ⟨F ⊗ M⟩. Then the functor ⊗_{U}M : U-mod-U → 𝔤-mod induces an equivalence of categories

H(I) →^{∼} coker(F ⊗ M).

Proof. From the preceding considerations it is clear that for all E ∈ F, X ∈ H(I) we have

Hom_{U−U}(E ⊗ Ū, X) ≅ Hom_{𝔤}(E, X^{ad}).

Whence the E ⊗ Ū with E ∈ F are projective in H(I) and generate H(I). Since ⊗_{U}M is right exact, we see that it induces indeed a functor from H(I) to coker(F ⊗ M).

Next we claim that for all E, F ∈ F the functor ⊗_{U}M induces a bijection

Hom_{U−U}(E ⊗ Ū, F ⊗ Ū) → Hom_{𝔤}(E ⊗ M, F ⊗ M).

To see this, remark first that for any three vectorspaces V, W, F with dimF < ∞ there is a canonical isomorphism Hom_{ℂ}(V, F ⊗ W) ≅ Hom_{ℂ}(F^{∗} ⊗ V, W). This is compatible with all our actions, so we need only prove our displayed bijection in case F = ℂ. But now we identified the left hand side with Hom_{𝔤}(E, Ū^{ad}) and the right hand side with Hom_{𝔤}(E, (End_{ℂ}M)^{ad}) and the claim follows from assumption 1.

Remark next that from assumption 2 actually follows that all E ⊗M with E ∈ F are projectives in ⟨F ⊗ M⟩. Indeed, Hom_{𝔤}(E ⊗ M, N) = Hom_{𝔤}(M, E^{∗} ⊗ N) is an exact functor in N ∈ ⟨F ⊗ M⟩. So we proved that our functor goes from H(I) to ⟨F ⊗ M⟩, is fully faithful on a system of projective generators of H(I) and maps those to projective objects in ⟨F ⊗ M⟩. The theorem now follows by standard arguments, see for example [BG80], 5.10. □

4. Action of the center

Let 𝔤 for this section be any reductive complex Lie algebra, 𝔥 ⊂ 𝔤 a Cartan subalgebra. Let S = S(𝔥) be the symmetric algebra over 𝔥 and Ŝ its completion at the maximal ideal generated by 𝔥. This is acted upon by the Weyl group W and we consider the invariants Ŝ^{W}.

Let M = M(𝔤) be the category of all representations of 𝔤 that are locally finite over the center Z of U = U(𝔤). If E ∈ 𝔤-mod is finite dimensional and M ∈ M, then E ⊗ M ∈ M. Let id_{M} : M → M be the identity functor. In this section we are going to define a ring homomorphism θ : Ŝ^{W} → End(id_{M}). This gives rise, for every M ∈ M, to a ring homomorphism ϑ_{M} : Ŝ^{W} → End_{𝔤}M and we will prove:

Theorem 4.1. Let E ∈ 𝔤-mod be semisimple and finite dimensional, M ∈ M arbitrary, s ∈ Ŝ^{W}. Then id_{E} ⊗ ϑ_{M}(s) = ϑ_{E⊗M}(s) as endomorphisms of E ⊗ M.

Remark 4.2. Since 𝔤 was only supposed reductive, there may be finite dimensional representations E that are not semisimple.

We first construct ϑ and then prove the theorem. Certainly M = ×_{χ∈MaxZ}M_{χ} where

M_{χ} = {M ∈ 𝔤-mod | ∀v ∈ M ∃n > 0 such that χ^{n}v = 0}.

The notation M = ×_{χ}M_{χ} can be spelled out as follows: For any M ∈ M let M_{χ} ⊂ M be the maximal submodule contained in M_{χ}. Then M = ⊕_{χ}M_{χ} and furthermore Hom_{𝔤}(M, M′) = 0 if M ∈ M_{χ}, M′ ∈ M_{χ′} and χ ≠ χ′. Certainly the completion Z_{χ}^{∧} of Z at χ acts on M_{χ}. Now we have our bijection 𝔥^{∗} → MaxS, λ ↦ ⟨λ⟩. We let S_{λ}^{∧} denote the completion of S at ⟨λ⟩, so that Ŝ = S_{0}^{∧}. The Harish-Chandra homomorphism ξ^{♯} : Z → S induces an inclusion ξ^{♯} : Z_{ξ(λ)}^{∧} ↪ S_{λ}^{∧} for every λ ∈ 𝔥^{∗}. Then translation by λ induces an isomorphism T_{λ} : S_{λ}^{∧} →^{∼} Ŝ. It is clear that Ŝ^{W} lies in the image of the composition T_{λ} ∘ ξ^{♯} : Z_{ξ(λ)}^{∧} ↪ Ŝ, thus we get an inclusion ϑ_{λ} : Ŝ^{W} ↪ Z_{ξ(λ)}^{∧}. It is clear as well that this inclusion depends only on ξ(λ), not on λ itself, so for any χ ∈ MaxZ we defined an inclusion

ϑ_{χ} : Ŝ^{W} ↪ Z_{χ}^{∧}.

Now for any M ∈ M_{χ} we define ϑ_{M} : Ŝ^{W} → End_{𝔤}M by the prescription that ϑ_{M}(s) should be multiplication with ϑ_{χ}(s), and then define ϑ_{M} for arbitrary M ∈ M = ×_{χ}M_{χ} in the obvious way. This completes the construction of ϑ. Remark that for more convenience in other parts of our paper we define the Harish-Chandra homomorphism ξ^{♯} : Z → S in such a way that it actually depends on the choice of a Borel subalgebra of 𝔤 containing 𝔥. However it is clear that ϑ does not depend on this choice.

Proof[Theorem. ] Any M ∈ M is a quotient of a (possibly infinite) direct sum of objects of the form U/χ^{n}U, χ ∈ MaxZ. Thus we only need to prove the theorem for all M = U/χ^{n}U. Now put M^{i}(λ) = U ⊗_{U(𝔟)} S/⟨λ⟩^{i} and ξ^{i}(λ) = Ann_{Z}M^{i}(λ). The system of all ξ^{i}(λ) is cofinal to the system of all (ξ(λ))^{n}. Thus we only need to prove the theorem for M = U/ξ^{i}(λ)U. In [Soe86] it is shown that U/ξ^{i}(λ)U acts faithfully on M^{i}(λ). Hence U/ξ^{i}(λ)U injects into an (infinite) direct product of copies of M^{i}(λ). Thus we only need to show the theorem for M = M^{i}(λ).

Consider more generally any M ∈ Õ = {M ∈ 𝔤-mod | M is finitely generated over 𝔤 and locally finite over 𝔟}. (This category Õ coincides with our N (0), but we won’t use this fact.) The “nilpotent part of the 𝔥-action” gives rise to an action of Ŝ on M to be denoted n = n_{M} : Ŝ → End_{𝔤}M. This is just the ϑ constructed above when we consider M as a module over the reductive Lie algebra 𝔥. More explicitely we decompose M into generalized weight spaces under the action of 𝔥, M = ⊕_{μ∈𝔥^{∗}}M_{μ}, and define n(H)(m) = (H − μ(H))m for any μ ∈ 𝔥^{∗}, m ∈ M_{μ}, H ∈ 𝔥. For all λ, i the diagram

Ŝ^{W} →^{ϑ} End_{𝔤}M^{i}(λ)
↓  ∥
Ŝ →^{n} End_{𝔤}M^{i}(λ)

commutes, as one sees by comparing the actions of n(s) and ϑ(s) on the highest weight space M^{i}(λ)_{λ}, for s ∈ Ŝ^{W}. On the other hand it is clear that for M ∈ Õ , s ∈ Ŝ we have id_{E} ⊗ n_{M}(s) = n_{E⊗M}(s) : E ⊗ M → E ⊗ M, since E is semisimple over 𝔤 or, equivalently, over 𝔥. Now for generic λ ∈ 𝔥^{∗} we have an isomorphism

E ⊗ M^{i}(λ) ≅ ⨁_{ν∈P (E)} M^{i}(λ + ν)

where P (E) ⊂ 𝔥^{∗} is the multiset of weights of E. Indeed the tensor identity gives us an isomorphism E ⊗ M^{i}(λ) ≅ U ⊗_{U(𝔟)} (E ⊗ S/⟨λ⟩^{i}) and then a filtration of E as a 𝔟-module with subquotients the weight spaces E_{ν} induces a filtration of the 𝔤-module E ⊗ M^{i}(λ) with subquotients the U ⊗_{U(𝔟)} (E_{ν} ⊗ S/⟨λ⟩^{i}) ≅ E_{ν} ⊗ M^{i}(λ + ν). Since λ is generic, these subquotients have pairwise distinct central character, whence the filtration splits step by step to give the desired direct sum decomposition. Hence for generic λ we have in End_{𝔤}(E ⊗ M^{i}(λ)) the equalities (abbreviating M^{i}(λ) = M)

id_{E} ⊗ ϑ_{M}(s) = id_{E} ⊗ n_{M}(s) = n_{E⊗M}(s) = ϑ_{E⊗M}(s)

for all s ∈ Ŝ^{W}. We extend this result by Zariski continuity to all λ. Namely we identify all S/⟨λ⟩^{i} by translation with S/(𝔥)^{i} and then identify all M^{i}(λ) with the vector space U(𝔫̄) ⊗ S/(𝔥)^{i}. So all the E ⊗ M^{i}(λ) get identified canonically with the vector space E⊗U(𝔫̄)⊗S/(𝔥)^{i}, and for s ∈ Ŝ^{W} the endomorphisms id_{E} ⊗ ϑ_{M}(s) and ϑ_{E⊗M}(s) of E ⊗ M^{i}(λ) get identified with certain endomorphisms

ϕ(λ), ψ(λ) of E ⊗ U(𝔫̄) ⊗ S/(𝔥)^{i}.

But one sees that ϕ(λ), ψ(λ) are algebraic in λ ∈ 𝔥^{∗}, and since they coincide for generic λ they have to coincide for all λ. □

For later use we record the following fact from folklore.

Lemma 4.3. Let E be a finite dimensional 𝔤-module, P (E) ⊂ 𝔥^{∗} its weights, λ ∈ 𝔥^{∗}, M ∈ M_{ξ(λ)}. Then E ⊗ M ∈ ⨁_{ν∈P (E)} M_{ξ(λ+ν)}.

Proof. With obvious arguments we reduce to the case ξ(λ)M = 0. Then Ann_{U}M ⊃ Ann_{U}M(λ) by Duflo’s theorem, hence Ann_{U}(E ⊗ M) ⊃ Ann_{U}(E ⊗ M(λ)) and the lemma follows. □

5. The main results

Let now again 𝔤 be our semisimple Lie algebra, H ⊂ U-mod-U its category of Harish-Chandra bimodules. For χ ∈ MaxZ put

H_{χ} = {X ∈ H | Xχ^{n} = 0 for n ≫ 0}.

On the other hand put N (P, f) = ⨁_{η} N (η, f) where η runs over all integral elements of MaxZ, i.e. over the image ξ(P) under ξ of the lattice of integral weights P ⊂ 𝔥^{∗}. We will establish an equivalence of categories H_{ξ(μ)} ≅ N (P, f) for all dominant μ ∈ P such that W_{f} = {w ∈ W | w · μ = μ}.

We even want to prove a more general statement and have to introduce a finer decomposition of our category N (f). Namely remark that N (f) ⊂ M(𝔤_{f}), hence any M ∈ N (f) decomposes into (generalized) eigenspaces under the action of Z_{f}, say M = ⊕_{χ}M_{χ} with χ running over MaxZ_{f}. For any coset Λ ∈ 𝔥^{∗}/P put N (Λ, f) = {M ∈ N (f) | M_{χ} ≠ 0 only for χ ∈ ξ_{f}(Λ)}. This category is stable under taking tensor products with finite dimensional 𝔤-modules, as follows from Lemma 4.3. For μ ∈ 𝔥^{∗} put W_{μ} = {w ∈ W | w · μ = μ}. We call μ dominant if and only if ⟨μ + ρ, α^{∨}⟩ ∉ {−1, −2, … } for all α ∈ R^{+}. For μ ∈ 𝔥^{∗} let us define M^{n}(μ, f) = U ⊗_{U(𝔭_{f})} Y_{f}(ξ_{f}(μ)^{n}, f). These form a projective system in an obvious way. We will prove:

Theorem 5.1. Suppose μ ∈ 𝔥^{∗} is dominant with W_{μ} = W_{f}. Then the functor X ↦ \underleftarrow{lim}_{n}X ⊗_{U} M^{n}(μ, f) determines an equivalence of categories T : H_{ξ(μ)} ≅ N (μ + P, f).

Remark 5.2. The case f = 0 is treated in [Soe86].

To prove this theorem, we reduce it to a special case of the main result from Section 3, stated below as Theorem 5.3. Certainly N ⊂ M(𝔤_{f}), hence applying the results of the previous section to 𝔤 = 𝔤_{f} we find for any M ∈ N a canonical morphism

ϑ = ϑ_{M} : Ŝ^{W_{f}} → End_{𝔤_{f}}M.

Now 𝔤 is a semisimple 𝔤_{f}-module, and the equality ϑ_{𝔤⊗M}(s) = id_{𝔤} ⊗ ϑ_{M}(s) from Theorem 4.1 along with naturality tells us that in fact we constructed a homomorphism

ϑ : Ŝ^{W_{f}} → End_{𝔤}M.

Let 𝔪 ⊂ Ŝ^{W_{f}} be the maximal ideal and define N (f)^{n} = {M ∈ N (f) | ϑ(𝔪^{n})M = 0} and N (Λ, f)^{n} = N (Λ, f) ∩ N (f)^{n}. Our first Theorem 5.1 will follow easily from

Theorem 5.3. Suppose μ ∈ 𝔥^{∗} is dominant with W_{μ} = W_{f}. Then the functor X ↦ X ⊗_{U} M^{n}(μ, f) determines an equivalence of categories H(ξ(μ)^{n}) ≅ N (μ + P, f)^{n}.

Remark 5.4. The case f = 0, n = 1 is treated in [BG80].

Proof. We will apply Theorem 3.1 with I = ξ(μ)^{n}, M = M^{n}(μ, f). For this we need

Proposition 5.5. Suppose μ ∈ 𝔥^{∗} is dominant with W_{μ} = W_{f}. Then (1) M^{n}(μ, f) is a projective object in N (f)^{n}.

(2) U/ξ(μ)^{n}U →^{∼} (End_{ℂ}M^{n}(μ, f))_{adf}.

Proof. Postponed. □

Now by Theorem 4.1 and Lemma 4.3 we know that N (μ + P, f)^{n} is stable under tensoring with finite dimensional 𝔤-modules. Hence part 1 of the proposition implies that ⟨F ⊗ M^{n}(μ, f)⟩ ⊂ N (μ + P, f)^{n} and that M^{n}(μ, f) is projective in ⟨F ⊗ M^{n}(μ, f)⟩. Using also part two of the proposition we can now apply Theorem 3.1 and deduce that

⊗_{U}M^{n}(μ, f) : H(ξ(μ)^{n}) → N (μ + P, f)^{n}

is a fully faithful functor. It only remains to show that it is essentially surjective. We will do this by counting indecomposable projectives.

In both our categories the indecomposable objects are precisely those with a local endomorphism ring. Our functor being fully faithful, it maps indecomposables to indecomposables and defines an injection of isomorphism classes of objects. Since H(ξ(μ)^{n}) has enough projectives, the (isomorphism classes of) indecomposable projectives and of simple objects correspond bijectively. Recalling the classification of simple objects from [BG80], we see that all indecomposable projectivs in H(ξ(μ)^{n}) are annihilated by a power of ξ(λ) for some λ ∈ μ+P, and the isomorphism classes of those are parametrized by the (W_{μ})-orbits in (W · λ) ∩ (μ + P).

Now recall that the projective objects in any H(I) are just the direct sums of objects of the form E ⊗ U/IU, E ∈ F. By the proposition M^{n}(μ, f) is projective in N (μ+P, f)^{n}, hence so are all the E⊗M^{n}(μ, f) with E ∈ F, hence our functor maps projective objects to projective objects. But now the simple objects in N (μ + P, f)^{n} annihilated by ξ(λ) are parametrized by the (W_{f}·)-orbits in (W ·λ)∩(μ+P). (Remark this space is (W_{f}·)-stable, since W_{f} = W_{μ}). Just counting we see that N (μ + P, f)^{n} has enough projectives and they are all in the image of our functor. Thus indeed our functor gives an equivalence of categories

H(ξ(μ)^{n}) →^{∼} N (μ + P, f)^{n}. □

We now prepare the proof of Proposition 5.5. We begin with some lemmas on invariant theory. Recall ϑ_{μ} from Section 4. We now use it for 𝔤 = 𝔤_{f}.

Lemma 5.6. Let μ ∈ 𝔥^{∗} be given with W_{f} · μ = μ. Then ϑ_{μ} : Ŝ^{W_{f}} → (Z_{f})_{ξ_{f} (μ)}^{∧} is an isomorphism.

Proof. This is clear from the definitions. □

Lemma 5.7. Suppose stronger W_{f} = W_{μ}. Consider Z as a subring of Z_{f} via the relative Harish-Chandra homomorphism θ^{♯}. Then ξ_{f}(μ)^{n} ∩ Z = ξ(μ)^{n}.

Proof. We have to show that θ : SpecZ_{f} → SpecZ is étale at ξ_{f}(μ). But this is clear from the condition on μ. □

Proposition 5.8. Let μ ∈ 𝔥^{∗} be dominant and suppose W_{f} · μ = μ. Then M^{n}(μ, f) is projective in N (f)^{n}.

Proof. We need just to show that it is projective in N (ξ(μ), f)^{n}. Choose N ∈ N (ξ(μ), f)^{n}. Then

Hom_{𝔤}(M^{n}(μ, f), N) = Hom_{𝔭_{f}}(Y_{f}(ξ_{f}(μ)^{n}, f), N)
= Hom_{𝔭_{f}}(Y_{f}(ξ_{f}(μ)^{n}, f), N_{μ^{f}})
= Hom_{𝔤_{f}}(Y_{f}(ξ_{f}(μ)^{n}, f), N_{μ^{f}}),

the first equality by definition of M^{n}(μ, f), the second since 𝔥^{f} acts via μ^{f} on Y_{f}, the third since μ is dominant, thus the weight μ^{f} is highest possible for N ∈ N (ξ(μ), f), hence N_{μ^{f}} is annihilated by 𝔫^{f}.

Let us put N_{f} = N (𝔤_{f}, 𝔟_{f}) and for η ∈ MaxZ_{f} define N_{f}(η), N_{f}(η, f) as subcategories of 𝔤_{f}-mod in the obvious way. Since μ is dominant, N_{μ^{f}} lies in N_{f}(ξ_{f}(μ), f). By 5.7 we see that N ∈ N (f)^{n} implies already ξ_{f}(μ)^{n}N_{μ^{f}} = 0. Now a theorem in [Kos78] tells us that for any η ∈ MaxZ_{f} we have an equivalence of categories

{M ∈ (Z_{f}/η^{n}) − mod | dim_{ℂ}M < ∞} → {H ∈ N_{f}(η, f) | η^{n}H = 0}

given by the functor M ↦ (U_{f} ⊗_{Z_{f}} M) ⊗_{U(𝔫_{f})} ℂ_{f}. Thus Y_{f}(η^{n}, f) is a projective object on the right hand side, and thus Hom_{𝔤}(M^{n}(μ, f), N) = Hom_{𝔤_{f}}(Y_{f}(ξ_{f}(μ)^{n}, f), N_{μ^{f}}) is an exact functor when restricted to N ∈ N (f)^{n}. Thus it only remains to be shown that M^{n}(μ, f) ∈ N (f)^{n}. But this is clear from Lemma 5.6. □

We now prove the second part of Proposition 5.5. We begin with

Lemma 5.9. Let I ⊂ Z_{f} be an ideal, N ∈ 𝔤_{f}-mod a representation such that Ann_{U_{f}} N = U_{f}I. Then Ann_{U}(U ⊗_{U(𝔭_{f})} N) = U(I ∩ Z) when we view Z as a subring of Z_{f} via θ^{♯}.

Remark 5.10. We extend here the 𝔤_{f}-action on N to a 𝔭_{f}-action via the surjection 𝔭_{f} → 𝔤_{f} with kernel 𝔫^{f}.

Proof. Using Lemma 2.2, we may just check on one N. Let us consider Z_{f} as a subring of S via ξ_{f}^{♯}. By Duflo’s theorem (more precisely, its generalization from [Soe86]) we may take N = U_{f} ⊗_{U(𝔟_{f})} (S/IS). Then U ⊗_{U(𝔭_{f})}N = U ⊗_{U(𝔟)}S/IS and by the generalized Duflo theorem again we conclude Ann_{U}(U ⊗_{U(𝔭_{f})} N) = U(Z ∩ IS). But S is faithfully flat over Z_{f}, hence Z_{f} ∩ IS = I, hence Z ∩ IS = Z ∩ I. □

We deduce

Lemma 5.11. Let μ ∈ 𝔥^{∗} be given with W_{μ} = W_{f}. Then we have Ann_{U}M^{n}(μ, f) = Uξ(μ)^{n}.

Proof. Apply the previous lemma to N = Y_{f}(ξ_{f}(μ)^{n}, f) and use Lemma 5.7 to see that ξ_{f}(μ)^{n} ∩ Z = ξ(μ)^{n}. □

So we already have an injection U/ξ(μ)^{n}U ↪ (End_{ℂ}M^{n}(μ, f))_{adf}. To prove that it is a surjection we compare multiplicities under the adjoint 𝔤-action on both sides. For this we study how our standard modules behave under translation. For E ∈ F let P (E) ⊂ 𝔥^{∗} be the multiset of weights of E (counted with their multiplicities).

Lemma 5.12. E ⊗ M(λ, f) has a filtration with subquotients M(λ + ν, f), ν ∈ P (E).

Proof. If f is regular, thus M(λ, f) = Y (ξ(λ), f), this was proved by Kostant [Kos78]. In general write

E ⊗ M(λ, f) = U ⊗_{U(𝔭_{f})} (E ⊗ Y_{f}(ξ_{f}(λ), f)).

Now filter E|_{𝔭_{f}} in such a way that 𝔫^{f} annihilates the subquotients, and then apply Kostant’s result to the Lie algebra 𝔤_{f}. □

Finally we can prove what we were after.

Proposition 5.13. Let μ ∈ 𝔥^{∗} be dominant with W_{μ} = W_{f}. Then the multiplication

U/ξ(μ)^{n}U → (End_{ℂ}M^{n}(μ, f))_{adf}

is an isomorphism.

Proof. This map is injective by Lemma 5.11. Let E be a finite dimensional simple 𝔤-module and E_{0} its zero weight space. We only have to check that E occurs with the same multiplicity on both sides, regarded as 𝔤-modules via the adjoint action. We have

dim_{ℂ}Hom_{𝔤}(E, (U/ξ(μ)^{n}U)^{ad}) = (dim_{ℂ}Z/ξ(μ)^{n}) · (dim_{ℂ}E_{0})

by Kostant’s theorem describing U^{ad}. On the other hand

dim_{ℂ}Hom_{𝔤}(E, (End_{ℂ}M^{n}(μ, f))^{ad}) =
= dim_{ℂ}Hom_{𝔤}(M^{n}(μ, f), E^{∗} ⊗ M^{n}(μ, f))
= [E^{∗} ⊗ M^{n}(μ, f) : L(μ, f)]

since M^{n}(μ, f) is the projective cover of L(μ, f) in N (f)^{n}. Now M^{n}(μ, f) has a filtration with dim_{ℂ}(Z_{f}/ξ_{f}(μ)^{n}) steps where all subquotients are copies of M(μ, f). Certainly dim_{ℂ}(Z_{f}/ξ_{f}(μ)^{n}) = dim_{ℂ}(Z/ξ(μ)^{n}). Thus we only have to check the equality

[E^{∗} ⊗ M(μ, f) : L(μ, f)] = dim_{ℂ}E_{0}.

This however is clear from Lemma 5.12 since μ is dominant and W_{f} ⊂ W_{μ}. □

The proof of Theorem 5.3 is now complete. To deduce Theorem 5.1 we just have to check

Lemma 5.14. Let μ ∈ 𝔥^{∗} be given with W_{f} = W_{μ}. For n > m the canonical surjection M^{n}(μ, f) → M^{m}(μ, f) has kernel ξ(μ)^{m}M^{n}(μ, f).

Proof. Omitted. □

So we finally get for any dominant μ ∈ 𝔥^{∗} with W_{μ} = W_{f} our equivalence of categories

T : H_{ξ(μ)} →^{∼} N (μ + P, f).

We now investigate the effect of our equivalence on standard modules and simple modules. Recall from [BG80, Jan83] the description of simple objects in H_{ξ(μ)}. For any λ, μ ∈ 𝔥^{∗} one forms the U-bimodule L(λ, μ) = Hom_{ℂ}(M(μ), M(λ))_{adf}. It can be shown that L(λ, μ) is actually finitely generated, i.e. it is an object of H. It is nonzero if and only if λ + P = μ + P. Assume now that μ is dominant. Then L(λ, μ) ≅ L(λ′, μ) if and only if W_{μ}·λ = W_{μ}·λ′, the L(λ, μ) have unique simple quotients L̄(λ, μ) for every λ ∈ μ + P, and the L̄(λ, μ) with λ running over the (W_{μ}·)-orbits in μ+P form a system of representatives for the simple isomorphism classes in H_{ξ(μ)}.

Proposition 5.15. Let μ ∈ 𝔥^{∗} be dominant with W_{μ} = W_{f}. For any λ ∈ μ + P we have T L(λ, μ) ≅ M(λ, f) and T L̄(λ, μ) ≅ L(λ, f).

Proof. Let for any abelian category A denote [A] its Grothendieck group. Any object M ∈ A determines an element [M] ∈ [A]. Any exact functor T : A → B to another abelian category gives rise to a group homomorphism T : [A] → [B]. In our situation we know already that T L(μ, μ) ≅ U/ξ(μ)U ⊗_{U} M(μ, f) ≅ M(μ, f). We know furthermore that our functor T commutes with all functors E⊗ for (E ∈ F) and with the (left) Z-action on our categories. Now E ⊗ L(μ, μ) has a filtration with subquotients L(μ + ν, μ), ν running over the multiset P (E) of weights of E. By 5.12 we know that E ⊗ M(μ, f) similarily has a filtration with subquotients M(μ + ν, f), ν ∈ P (E). Since this holds for all E, we deduce for all integral weights ν ∈ P the equality

T ∑_{w∈W}[L(wν + μ, μ)] = ∑_{w∈W}[M(wν + μ, f)].

If we split it up according to central character and use the isomorphisms L(v · λ, μ) ≅ L(λ, μ), M(v · λ, f) ≅ M(λ, f) for v ∈ W_{μ} = W_{f}, we deduce |W_{μ}| T [L(λ, μ)] = |W_{f}| [M(λ, f)] and thus T [L(λ, μ)] = [M(λ, f)] for all λ ∈ μ + P. Choose now representatives λ_{1}, …, λ_{n} of the (W_{μ}·)-orbits in (W · λ) ∩ (μ + P) such that λ_{i} ∈ λ_{j} − ℝ_{≥0}R^{+} implies i ≥ j. Then the multiplicity matrices [L(λ_{i}, μ) : L̄(λ_{j}, μ)] and [M(λ_{i}, f) : L(λ_{j}, f)] are upper triangular with ones on the diagonal, thus the equations T [L(λ_{i}, μ)] = [M(λ_{i}, f)] imply T [L̄(λ_{i}, μ)] = [L(λ_{i}, f)] and the effect of T on simples is as asserted.

Next we determine the effect of T on standard objects. We claim that for any N ∈ N (μ + P, f) annihilated by some power of ξ(λ) and such that [N : L(λ_{i}, f)] ≠ 0 and [N : L(λ_{j}, f)] = 0 for j < i there is a nonzero morphism M(λ_{i}, f) → N. Indeed the conditions on N imply that its 𝔥^{f}-weight space of weight λ_{i}^{f} is not zero, annihilated by 𝔫^{f} and isomorphic to Y_{f}(ξ_{f}(λ_{i}), f) as a 𝔤_{f}-module. We apply this to N = T L(λ_{i}, μ) and find a nonzero morphism φ : M(λ_{i}, μ) → T L(λ_{i}, μ). By construction this morphism φ has to induce a surjection onto the unique simple quotient L(λ_{i}, f) of T L(λ_{i}, μ), thus φ is a surjection itself. Since we know already [M(λ_{i}, f)] = [T L(λ_{i}, μ)], this surjection φ has even to be an isomorphism. □

Let us finally fulfill our promise from the introduction. Let us define for χ, η ∈ MaxZ the category

χH_{η} = {X ∈ H | χ^{n}X = 0, Xη^{n} = 0 for n ≫ 0}.

Remark that for an integral weight λ ∈ P actually

N (ξ(λ), f) ⊂ N (λ + P, f).

So if λ, μ are dominant integral weights with λ regular and W_{μ} = W_{f}, then we find equivalences of categories

\begin{aligned}_{ξ(μ)}H_{ξ(λ)} & ≅ & N (ξ(μ), 0) \\ _{ξ(λ)}H_{ξ(μ)} & ≅ & N (ξ(λ), f), \end{aligned}

and since the two categories of bimodules can be identified by interchanging the left and the right action via the Chevalley antiautomorphism of 𝔤, we finally find an equivalence

N (ξ(μ), 0) ≅ N (ξ(λ), f).

Using the proposition and [Jan83], 6.34 it can be checked that under this equivalence M(x · λ, f) corresponds to M(x^{−1} · μ).

References

[Bac97] Erik Backelin. Representation of the category ≀ in Whittaker categories. Duke IMRN, page to appear, 1997.

[BG80] Joseph N. Bernstein and Sergei I. Gelfand. Tensor products of finite and infinite representations of semisimple Lie algebras. Compositio Math., 41:245–285, 1980.

[Dix74] Jacques Dixmier. algèbres enveloppantes. Cahiers Scientifiques. Gauthier-Villars, 1974.

[Jan83] Jens C. Jantzen. Einhüllende Algebren halbeinfacher Lie-Algebren. Springer, 1983.

[Kos78] B. Kostant. On Whittaker vectors and representation theory. Inventiones, 48:101–184, 1978.

[McD85] E. McDowell. On modules induced from Whittaker modules. J. Algebra, 96:161–177, 1985.

[MS95] Dragan Miličić and Wolfgang Soergel. Twisted Harish-Chandra sheaves and Whittaker modules. Preprint, 1995.

[Soe86] Wolfgang Soergel. Équivalences de certaines catégories de 𝔤-modules. C. R. Acad. Sci. Paris, Sér. 1, 303(15):725–728, 1986.

University of Utah, Department of Mathematics, Salt Lake City, Utah 84112, USA

  E-mail address: milicic@math.utah.edu

Universität Freiburg, Mathematisches Institut, Eckerstraße 1, D-79104 Freiburg, Germany

  E-mail address: soergel@mathematik.uni-freiburg.de