PRIMITIVE IDEALS OF ℂ_{q}[G]

TIMOTHY J. HODGES AND THIERRY LEVASSEUR

Abstract. In [4] Joseph proved the classification of the primitive ideals of the quantum group ℂ_{q}[G] conjectured in [2]. We prove this result taking account of Joseph’s analysis in [4] and of the methods already developed in [3].

1. Introduction

Let G be a connected, simply connected semi-simple complex Lie group and let 𝔤 be its Lie algebra. Denote by 𝔥 a fixed Cartan subalgebra of 𝔤. The notation used for the weights and roots of 𝔤 with respect to 𝔥 is as in Bourbaki [1] (with the minor exception that the set of dominant weights will be denoted by P^{+}). Let q be a non-zero complex number which is not a root of unity and let Q be the subgroup of ℂ^{∗} generated by q. Following [5, 7], we define the quantised enveloping algebra U_{q}(𝔤) to be the ℂ-algebra generated by K_{i}^{±1}, X_{i}^{±}, 1 ≤ i ≤ n with relations

K_{i}^{−1}K_{i} = K_{i}K_{i}^{−1} = 1, K_{i}X_{j}^{±} = q^{±(α_{i},α_{j})}X_{j}^{±}K_{i}

[X_{i}^{+}, X_{j}^{−}] = δ_{ij} \frac{K_{i}^{2} − K_{i}^{−2}}{q^{2d_{i}} − q^{−2d_{i}}} , K_{i}K_{j} = K_{j}K_{i}

∑_{k=0}^{1−a_{ij}}(−1)^{k} [ \begin{matrix}1 − a_{ij} \\ k \end{matrix} ]_{q^{2d_{i}}} (X_{i}^{±})^{k}X_{j}^{±}(X_{i}^{±})^{1−a_{ij}−k} = 0, if i ≠ j

where [m]_{t} = (t−t^{−1}) … (t^{m}−t^{−m}) and [ \begin{matrix}m \\ k \end{matrix} ]_{t} = \frac{[m]_{t}}{[k]_{t}[m−k]_{t}} . Here [a_{ij}] is the Cartan matrix

and d_{i} = (α_{i}, ϖ_{i}) = (α_{i}, α_{i})/2. The algebra U_{q}(𝔤) is a Hopf algebra. The comultiplication Δ is defined by

Δ(X_{i}^{±}) = X_{i}^{±} ⊗ K_{i}^{−1} + K_{i} ⊗ X_{i}^{±}, Δ(K_{i}) = K_{i} ⊗ K_{i}

and the counit and antipode by

ε(X_{i}^{±}) = 0, ε(K_{i}) = 1, S(K_{i}) = K_{i}^{−1}, S(X_{i}^{±}) = −q^{∓2d_{i}}X_{i}^{±}.

We also set

E_{i} = K_{i}X_{i}^{+}, F_{i} = X_{i}^{−}K_{i}^{−1}, V^{+} = ℂ[E_{i}, 1 ≤ i ≤ n], V^{−} = ℂ[F_{i}, 1 ≤ i ≤ n]

and V_{+}^{±} = V^{±} ∩ ker ε. Notice that Δ(E_{i}) = E_{i} ⊗ 1 + K_{i}^{2} ⊗ E_{i}, S(E_{i}) = −K_{i}^{−2}E_{i}, and [E_{i}, F_{j}] = δ_{ij}q̂_{i}^{−1}(K_{i}^{2} − K_{i}^{−2}), where q̂_{i} = q^{2d_{i}} − q^{−2d_{i}}.

An element m of a U_{q}(𝔤)-module M is said to have weight μ ∈ P if K_{i} m = q^{(α_{i},μ)}m for all i = 1, …, n. For each dominant weight Λ ∈ P^{+} there exists a finite dimensional simple module L(Λ) of highest weight Λ. Denote by C the subcategory of U_{q}(𝔤)-modules consisting of finite direct sums of such modules. We consider the dual of a U_{q}(𝔤)-module M as a left module by the following action: uf(x) = f(S(u)x) for all x ∈ M, f ∈ M^{∗}, u ∈ U_{q}(𝔤). The category C is closed under tensor products and the formation of duals. Hence the corresponding restricted dual (the algebra of coordinate functions associated to objects of C) is a Hopf algebra. This Hopf algebra is denoted by ℂ_{q}[G] and is known as the standard quantization of ℂ[G].

Date: March, 1993.

The first author is partially supported by a grant from the N.S.A..

In [2], the authors conjectured that the primitive ideals of ℂ_{q}[G] should be in Hequivariant 1-1 correspondence with the symplectic leaves of G equipped with the standard Poisson group structure; that is, the H-orbits inside P rim ℂ_{q}[G] are parameterized by the double Weyl group W × W. This result was proved by the authors for G = SL(n) in [3] and in the general case by Joseph in [4]. As shown in [2, 3], the key to the main theorem is to understand the adjoint action of ℂ_{q}[G] on some algebras denoted by C_{w} (see section 3 for the definition). In these papers we were able to explicit the calculation of the adjoint action in order to get a complete description of C_{w} when G = SL(n). In the general case the key idea in the proof of [4, Theorem 9.2] is to use the “self-duality of U_{q}(𝔟^{+})”. We shall deduce Joseph’s theorem by combining this idea and the approach of [3]. In [4] Joseph also proves various other results that the present paper does not recover. Although no explicit references are made to the geometry, the proof follows closely the ideas involved in the description of the symplectic leaves given in the appendix of [2]. The original conjectures were inspired by work of Soibelman [10] on the irreducible unitary representations of the corresponding Hopf^{⋆}-algebra. For further details and background to these results the reader is referred to the introductions of [2] and [6].

2. Parameterization of the Prime Spectrum

Let M ∈ C and let f ∈ M^{∗}, v ∈ M. Define c_{f,v} to be the coordinate function given by:

∀u ∈ U_{q}(𝔤), c_{f,v}(u) = f(uv).

Then ℂ_{q}[G] = ⊕_{Λ∈P^{+}}C(Λ) where C(Λ) is the vector space spanned by the c_{f,v} where f ∈ L(Λ)^{∗} and v ∈ L(Λ). Denote by Δ, ε, S the comultiplication, counit and antipode in ℂ_{q}[G] (strictly speaking, these should be denoted Δ^{∗}, ε^{∗}, S^{∗}, but no ambiguity arises). Then if {v_{1}, …, v_{s}; f_{1}, …, f_{s}} is a dual basis for M,

Δ(c_{f,v}) = ∑_{i} c_{f,v_{i}} ⊗ c_{f_{i},v}, ε(c_{f,v}) = f(v), S(c_{f,v}) = c_{v,f}.

For each Λ ∈ P^{+}, choose v_{Λ} ∈ L(Λ)_{Λ}, v_{w_{0}Λ} ∈ L(Λ)_{w_{0}Λ} and f_{−Λ} ∈ L(Λ)_{−Λ}^{∗}, f_{−w_{0}Λ} ∈ L(Λ)_{−w_{0}Λ}^{∗} such that f_{−Λ}(v_{Λ}) = 1 and f_{−w_{0}Λ}(v_{w_{0}Λ}) = 1. Set

A^{+} = ∑_{μ∈P^{+}} ∑_{f∈L(μ)^{∗}} ℂc_{f,v_{μ}}, A^{−} = ∑_{μ∈P^{+}} ∑_{f∈L(μ)^{∗}} ℂc_{f,v_{w_{0}μ}}.

The multiplication map A^{+} ⊗ A^{−} → ℂ_{q}[G] is surjective, [10, 3.1]. A proof of this fact is given in [4, 3.7].

Consider the algebras U_{q^{−1}}(𝔤) and ℂ_{q^{−1}}[G] and use ˆ to distinguish elements, subalgebras, etc. of U_{q^{−1}}(𝔤) and ℂ_{q^{−1}}[G]. It is easily verified that the map σ : U_{q}(𝔤) → U_{q^{−1}}(𝔤) given by σ(X_{i}^{±}) = X̂_{i}^{∓} and σ(K_{i}) = K̂_{i} is an isomorphism of Hopf algebras. For each Λ ∈ P^{+}, σ gives a bijection σ : L(−w_{0}Λ) → L̂(Λ) which sends v ∈ L(−w_{0}Λ)_{μ} onto v̂ ∈ L̂(Λ)_{−μ}. Therefore σ induces an isomorphism σ : ℂ_{q^{−1}}[G] → ℂ_{q}[G] such that

∀f ∈ L(−w_{0}Λ)_{−λ}^{∗}, ∀v ∈ L(−w_{0}Λ)_{μ}, σ(ĉ_{f̂,v̂}) = c_{f,v}.

In particular we have σ(Â^{±}) = A^{∓}. This observation allows one to deduce results for A^{−} from analogous results for A^{+}.

The isomorphisms L(Λ) ⊗ L(Λ′) ≅ L(Λ′) ⊗ L(Λ) given by the universal R-matrix yield the following commutation relations.

Lemma 2.1. Let Λ, Λ′ ∈ P^{+} and let g ∈ L(Λ′)_{−η}^{∗} and f ∈ L(Λ)_{−μ}^{∗}. Then there exists a finite collection of triples (a_{ν}, f_{ν}, g_{ν})_{ν∈Q^{+}} where a_{ν} ∈ ℂ, f_{ν} ∈ (U_{q}(𝔟^{+})f)_{−μ+ν} and g_{ν} ∈ (U_{q}(𝔟^{−})g)_{−η−ν} such that for any v ∈ L(Λ)_{γ},

c_{g,v}c_{f,v_{Λ}} = q^{2[(Λ,γ)−(μ,η)]}c_{f,v_{Λ}}c_{g,v} + q^{2(Λ,γ)} ∑_{ν∈Q^{+}} a_{ν}c_{f_{ν},v_{Λ}}c_{g_{ν},v}

Define A_{λ,μ} = {c_{f,v} | f ∈ M_{λ}^{∗}, v ∈ M_{μ}, M ∈ C}. It is easily verified that this defines a P × P grading on ℂ_{q}[G]. Denote by R(ℂ_{q}[G]) the set of one dimensional representations of ℂ_{q}[G] with the usual group structure. Let H be the maximal torus of G with Lie algebra 𝔥. For each h ∈ H define χ_{h} ∈ R(ℂ_{q}[G]) by

∀c_{f,v} ∈ A_{λ,μ}, χ_{h}(c_{f,v}) = μ(h)ε(c_{f,v}).

The map h ↦ χ_{h} is easily seen to be an injective group homomorphism. The usual map r : R(ℂ_{q}[G]) → Aut ℂ_{q}[G] corresponding to right translation, r_{χ}(c) = ∑ c_{(1)}χ(c_{(2)}), induces an action of H on ℂ_{q}[G] given by

∀c_{f,v} ∈ A_{λ,μ}, h ∈ H, h.c_{f,v} = μ(h)c_{f,v}.

For each y ∈ W define

I_{y}^{+} = ⟨c_{f,v_{Λ}} | f ∈ (U_{q}(𝔟^{+})L(Λ)_{yΛ})^{⊥}⟩, I_{y}^{−} = ⟨c_{f,v_{w_{0}Λ}} | f ∈ (U_{q}(𝔟^{−})L(Λ)_{yw_{0}Λ})^{⊥}⟩ = σ(Î_{y}^{+}).

For w = (w_{+}, w_{−}) ∈ W × W set I_{w} = I_{w_{+}}^{+} + I_{w_{−}}^{−}. For a given w we denote by c_{wΛ} and c̃_{wΛ} the images of c_{f_{−w_{+}Λ},v_{Λ}} and c_{v_{w_{−}Λ},f_{−Λ}} respectively in ℂ_{q}[G]/I_{w}. It follows from Lemma 2.1 that c_{g,v}c_{wΛ} ∈ Qc_{wΛ}c_{g,v} and c_{g,v}c̃_{wΛ} ∈ Qc̃_{wΛ}c_{g,v} modulo I_{w_{+}}^{+} and I_{w_{−}}^{−} respectively. Therefore the sets E_{w_{+}} = {αc_{wΛ} | α ∈ ℂ^{∗}}, E_{w_{−}} = {αc̃_{wΛ} | α ∈ ℂ^{∗}} and E_{w} = E_{w_{+}}E_{w_{−}} are multiplicatively closed sets of normal elements. Thus E_{w} is an Ore set in ℂ_{q}[G]/I_{w}. Define

A_{w} = (ℂ_{q}[G]/I_{w})_{E_{w}}.

Notice that the map σ extends to an isomorphism σ : Âŵ → A_{w} where ŵ = (w_{−}, w_{+}).

Theorem 2.2. For all w ∈ W × W, A_{w} ≠ 0.

Proof. The proof is as in [2, 2.4]. □

The following lemma is well known.

Lemma 2.3. Let w_{Λ} ∈ W for all Λ ∈ P^{+}. Then the following are equivalent: a) ∀Λ, Λ′ ∈ P^{+}, (w_{Λ}Λ, w_{Λ′}Λ′) = (Λ, Λ′);

b) There exists a unique w ∈ W such that wΛ = w_{Λ}Λ for all Λ ∈ P^{+}.

The proof of the following result was found independently by the authors in [3, 1.2] and Joseph in [4, 6.2].

Theorem 2.4. Let P ∈ Spec ℂ_{q}[G]. There exists a unique w ∈ W × W such that P ⊃ I_{w} and (P/I_{w}) ∩ E_{w} = ∅.

Proof. Fix a dominant weight Λ. Define an ordering on the weight vectors of L(Λ)^{∗} by f ≤ f′ if f′ ∈ U_{q}(𝔟^{+})f. This is a preordering which induces a partial ordering on the set of one dimensional weight spaces. Consider the set:

D(Λ) = {f ∈ L(Λ)_{μ}^{∗} | c_{f,v_{Λ}} ∉ P }.

Let f be an element of D(Λ) which is maximal for the above ordering. Suppose that f′ has the same property and that f and f′ have weights μ and μ′ respectively. By Lemma 2.1 the two elements c_{f,v_{Λ}} and c_{f′,v_{Λ}} are normal modulo P. Therefore we have, modulo P,

c_{f,v_{Λ}}c_{f′,v_{Λ}} = (q^{2})^{(Λ,Λ)−(μ,μ′)}c_{f′,v_{Λ}}c_{f,v_{Λ}} = (q^{4})^{(Λ,Λ)−(μ′,μ)}c_{f,v_{Λ}}c_{f′,v_{Λ}}.

Since P is prime and q is not a root of unity we can deduce that (Λ, Λ) = (μ, μ′). This forces μ = μ′ ∈ W (−Λ). In conclusion, we have shown that for all dominant Λ there exists a unique (up to scalar multiplication) maximal element g_{Λ} ∈ D(Λ) with weight −w_{Λ}Λ, w_{Λ} ∈ W. Applying the argument above to a pair of such elements, c_{g_{Λ},v_{Λ}} and c_{g_{Λ′},v_{Λ′}} yields that (w_{Λ}Λ, w_{Λ′}Λ′) = (Λ, Λ′). Lemma 2.3 then furnishes a unique element w_{+} ∈ W such that w_{+}Λ = w_{Λ}Λ for all Λ ∈ P^{+} . Thus for each Λ ∈ P^{+},

c_{g,v_{Λ}} ∈ P ⇔ g ≰ f_{−w_{+}Λ}.

Hence P ⊇ I_{w_{+}}^{+} and P ∩ E_{w_{+}} = ∅. It is easily checked that such a w_{+} must be unique. Using σ one deduces the existence and uniqueness of w_{−}. □

A prime ideal P such that P ⊃ I_{w} and (P/I_{w}) ∩ E_{w} = ∅ will be called a prime ideal of type w. Denote by Spec_{w}ℂ_{q}[G] the subset of Spec ℂ_{q}[G] consisting of prime ideals of type w. Clearly Spec_{w} ℂ_{q}[G] ≅ Spec A_{w} and σ(Specŵ ℂ_{q^{−1}}[G]) = Spec_{w} ℂ_{q}[G].

Corollary 2.5. Spec ℂ_{q}[G] = ⨆_{w∈W ×W} Spec_{w}ℂ_{q}[G].

Henceforth we fix w and work inside A_{w}. For each Λ ∈ P^{+}, set d_{Λ} = (c̃_{wΛ}c_{wΛ})^{−1} and t_{Λ} = c̃_{wΛ}(c_{wΛ})^{−1}. For f ∈ L(Λ)^{∗} and v ∈ L(Λ) set

z_{f}^{+} = c_{wΛ}^{−1}(c_{f,v_{Λ}}) z_{v}^{−} = c̃_{wΛ}^{−1}(c_{v,f_{−Λ}})

Since the generators of I_{w} and the elements of E_{w} are eigenvectors for H, the action of H extends to an action on A_{w}. Let Γ = {h ∈ H | h^{2} = 1} and let B_{w} = A_{w}^{Γ}. Let C_{w} be the subalgebra generated by the elements of the form z_{f}^{+}, z_{v}^{−}, t_{Λ}^{±1}. Then it is clear that C_{w} ⊂ B_{w}. It is not difficult to show (e.g. using Lemma 3.4 below and its R_{q}^{−} analog) that the subalgebra generated by the q-commuting elements t_{Λ}^{±1}, Λ ∈ P^{+}, is equal to ℂ[t_{ϖ_{i}}^{±1} | 1 ≤ i ≤ n]. If λ = ∑_{i=1}^{n} m_{i}ϖ_{i} is an element of P we set t_{λ} = ∏_{i=1}^{n} t_{ϖ_{i}}^{m_{i}}, so that

ℂ[t_{Λ}^{±1} | Λ ∈ P^{+}] = ℂ[t_{λ} | λ ∈ P].

Theorem 2.6. 1. C_{w}^{H} = ℂ[z_{f}^{+}, z_{v}^{−} | f ∈ L(Λ)^{∗}, v ∈ L(Λ), Λ ∈ P^{+}].

2. The set D = {d_{Λ} | Λ ∈ P^{+}} is an Ore subset of C_{w}^{H} and C_{w}. Furthermore B_{w} = (C_{w})_{D} and A_{w}^{H} = B_{w}^{H} = (C_{w}^{H})_{D}.

3. The set {t_{λ} | λ ∈ P} forms a basis for C_{w} as a left or right C_{w}^{H}-module and for B_{w} as a left or right B_{w}^{H}-module.

Proof. Notice that for all h ∈ H, h.t_{Λ} = Λ(h^{−2})t_{Λ} and h.z_{v}^{±} = z_{v}^{±}. This proves part 1 and the first assertion of part 3. Let {v_{1}, …, v_{s}; f_{1}, …, f_{s}} be a dual basis for L(Λ). Then

1 = ε(c_{f_{−Λ},v_{Λ}}) = ∑_{i} S(c_{f_{−Λ},v_{i}})c_{f_{i},v_{Λ}} = ∑_{i} c_{v_{i},f_{−Λ}}c_{f_{i},v_{Λ}}.

Multiplying both sides of the equation by d_{Λ} and using the normality of c_{wΛ} and c̃_{wΛ} yields d_{Λ} = ∑_{i} a_{i}z_{v_{i}}^{−}z_{f_{i}}^{+} for some a_{i} ∈ ℂ. Thus D ⊂ C_{w}^{H} and hence C_{w} contains all elements of the form c_{wΛ_{1}}^{−1}c̃_{wΛ_{2}}^{−1} where Λ_{1} − Λ_{2} ∈ 2P. Now it follows from [10, 3.1] that A_{w} is spanned by elements of the form c_{f,v_{1}}c_{v,f_{2}}d_{Λ} where v_{1} = v_{Λ_{1}}, f_{2} = f_{Λ_{2}} and Λ_{1}, Λ_{2}, Λ ∈ P^{+}. So B_{w} is spanned by elements of the form c_{f,v_{1}}c_{v,f_{2}}d_{Λ} where Λ_{1} − Λ_{2} ∈ 2P. If further Λ − Λ_{1} − Λ_{2} ∈ P^{+}, then, up to a scalar, c_{f,v_{1}}c_{v,f_{2}}d_{Λ} = z_{f}^{+}z_{v}^{−}c_{wΛ_{2}}^{−1}c̃_{wΛ_{1}}^{−1}d_{Λ−Λ_{1}−Λ_{2}} ∈ C_{w}. It is then clear that B_{w} = (C_{w})_{D}. The remaining assertions then follow easily. □

3. The Adjoint Action

We now consider the adjoint action of ℂ_{q}[G] on A_{w}. Notice that B_{w} and B_{w}^{H} are ad-submodules of A_{w}.

Proposition 3.1. Let g ∈ L(Λ′)_{−η}^{∗}, v ∈ L(Λ′)_{γ}, f ∈ L(Λ)_{−μ}^{∗}. Then

ad c_{g,v}.z_{f}^{+} = ε(c_{g,v})q^{2(w_{+}Λ−μ,η)}z_{f}^{+} + ∑_{ν} q^{2(w_{+}Λ,η)}ε(c_{g_{ν},v})a_{ν}z_{f_{ν}}^{+}

where a_{ν} ∈ ℂ, f_{ν} ∈ (U_{q}(𝔟^{+})f)_{−μ+ν} and g_{ν} ∈ (U_{q}(𝔟^{−})g)_{−η−ν}.

Proof. Let {v_{1}, …, v_{s}; f_{1}, …, f_{s}} be a dual basis for L(Λ′) and suppose that v_{i} ∈ L(Λ)_{γ_{i}}. It follows from Lemma 2.1 that

ad c_{g,v}.z_{f}^{+} = ∑_{i} c_{g,v_{i}}c_{wΛ}^{−1}c_{f,v_{Λ}}S(c_{f_{i},v}) = ∑_{i} q^{−2[(Λ,γ_{i})−(w_{+}Λ,η)]}c_{wΛ}^{−1}c_{g,v_{i}}c_{f,v_{Λ}}S(c_{f_{i},v})

= ∑_{i} q^{−2[(Λ,γ_{i})−(w_{+}Λ,η)]}c_{wΛ}^{−1} (q^{2[(Λ,γ_{i})−(μ,η)]}c_{f,v_{Λ}}c_{g,v_{i}} + q^{2(Λ,γ_{i})} ∑_{ν} a_{ν}c_{f_{ν},v_{Λ}}c_{g_{ν},v_{i}}) S(c_{f_{i},v})

= q^{(w_{+}Λ,η)}c_{wΛ}^{−1} (q^{−2(μ,η)}c_{f,v_{Λ}} ∑_{i} c_{g,v_{i}}S(c_{f_{i},v}) + ∑_{ν} a_{ν}c_{f_{ν},v_{Λ}} ∑_{i} c_{g_{ν},v_{i}}S(c_{f_{i},v}))

= ε(c_{g,v})q^{2(w_{+}Λ−μ,η)}z_{f}^{+} + ∑_{ν} q^{2(w_{+}Λ,η)}ε(c_{g_{ν},v})a_{ν}z_{f_{ν}}^{+}

□

Set C_{w}^{+} = ℂ[z_{f}^{+} | f ∈ L(Λ)^{∗}, Λ ∈ P^{+}] and C_{w}^{−} = ℂ[z_{v}^{−} | v ∈ L(Λ), Λ ∈ P^{+}]. Set J^{+} = I_{(w_{0},e)} and J^{−} = I_{(e,w_{0})}. Recall that I_{w_{0}}^{+} = (0), so that J^{+} is the ideal generated by the elements of the form c_{f,v_{w_{0}μ}} where μ ∈ P^{+}, f ∈ L(μ)_{η}^{∗}, η ≠ −w_{0}μ. Similarly J^{−} is the ideal generated by the c_{f,v_{Λ}}, Λ ∈ P^{+}, f ∈ L(Λ)_{γ}^{∗}, γ ≠ −Λ.

Corollary 3.2. C_{w}^{H} is a locally finite ad-ℂ_{q}[G] module. Furthermore, Ann_{ad}C_{w}^{±} ⊃ J^{±}.

Proof. This follows from Proposition 3.1 together with a similar result for C_{w}^{−}. □

Set ℂ_{q}[B^{±}] = ℂ_{q}[G]/J^{∓} and define ℂ[B^{+}/N^{+}] to be the subalgebra of ℂ_{q}[B^{+}] generated by the images of the elements c_{Λ} := c_{f_{−Λ},v_{Λ}}, Λ ∈ P^{+}.

Theorem 3.3. 1. ℂ_{q}[G]/I_{e} ≅ ℂ[H].

2. All finite dimensional ℂ_{q}[G]-modules are one dimensional and are annihilated by I_{e}.

3. The map χ : H ↦ R(ℂ_{q}[G]) is an isomorphism of groups.

Proof. For each c ∈ ℂ_{q}[G] define φ_{c} : H → ℂ by φ_{c}(h) = χ_{h}(c). It is easily verified that φ defines an algebra map from ℂ_{q}[G] to ℂ[H] and that ker φ ⊃ I_{e}. From [10, 3.1] it follows that the kernel is exactly I_{e}.

Let P be the annihilator of a simple finite dimensional ℂ_{q}[G]-module. Suppose that P is of type w. Then ℂ_{q}[G]/P is a homomorphic image of A_{w}. Notice from the proof of 2.4, that z_{g}^{+} ≠ 0 in ℂ_{q}[G]/P if g ≤ f_{−w_{+}Λ}. By Proposition 3.1 the adjoint action of ℂ[B^{+}/N^{+}] on such elements is diagonalisable. Moreover, unless w_{+} = e, there are infinitely many distinct weight vectors of this form, contradicting the fact that ℂ_{q}[G]/P is finite dimensional. Thus w_{+} = e. Using σ one deduces that w_{−} = e also. The remaining assertions are then clear. □

As already explained in the introduction, the key to the main theorem is to understand the adjoint action of ℂ_{q}[G] on C_{w}^{+}. To do this we use the “self-duality of U_{q}(𝔟^{+})”. Since U_{q}(𝔟^{+}) is not exactly self-dual we are obliged to work with the “simply-connected” version of U_{q}(𝔤). What follows is a distillation of results in [4]. Define Ǔ_{q}(𝔤) to be the algebra generated by X_{i}^{±}, i = 1, … n and τ(λ), λ ∈ P subject to the relations:

∀λ, μ ∈ P, τ(λ)τ(μ) = τ(λ + μ), τ(λ)X_{i}^{±} = q^{±(λ,α_{i})}X_{i}^{±}τ(λ)

[X_{i}^{+}, X_{j}^{−}] = δ_{ij} \frac{τ(α_{i})^{2} − τ(α_{i})^{−2}}{q^{2d_{i}} − q^{−2d_{i}}}

and the usual Serre relations for the X_{i}^{±} described in the introduction. Setting K_{i} = τ(α_{i}) then identifies U_{q}(𝔤) with a subalgebra of Ǔ_{q}(𝔤). The Hopf algebra structure can be extended to Ǔ_{q}(𝔤) by defining

Δ(τ(λ)) = λ ⊗ λ, ε(τ(λ)) = 1, S(τ(λ)) = τ(−λ).

Set Ǔ_{q}(𝔟^{±}) = ℂ[X_{i}^{±}, τ(λ) | i = 1, …, n, λ ∈ P]. The action of U_{q}(𝔤) on the category C described in the introduction extends to an action of Ǔ_{q}(𝔤). Moreover ℂ_{q}[G] may equally well be considered as the restricted dual of Ǔ_{q}(𝔤) with respect to C. See [5] for further details.

Let R_{q}^{+} = ⨁_{Λ∈P^{+}} L(Λ)^{∗} considered as an algebra via the multiplication maps defined by L(Λ_{1})^{∗} ⊗L(Λ_{2})^{∗} → L(Λ_{1} +Λ_{2})^{∗}. These maps are obtained as follows. Recall that there is an embedding of Ǔ_{q}(𝔤)-modules L(Λ_{1} + Λ_{2}) → L(Λ_{2}) ⊗ L(Λ_{1}), mapping v_{y(Λ_{1}+Λ_{2})} onto v_{yΛ_{2}} ⊗ v_{yΛ_{1}} for all y ∈ W. Recall also that we have a canonical isomorphism of left Ǔ_{q}(𝔤)- modules (L(Λ_{2}) ⊗ L(Λ_{1}))^{∗} ≅ L(Λ_{1})^{∗} ⊗ L(Λ_{2})^{∗}, such that (f_{1} ⊗ f_{2})(v_{2} ⊗ v_{1}) = f_{1}(v_{1})f_{2}(v_{2}) when f_{i} ∈ L(Λ_{i})^{∗}, v_{i} ∈ L(Λ_{i}) for i = 1, 2. The multiplication in R_{q}^{+} is deduced from these two maps of left Ǔ_{q}(𝔤)-modules. It is then easily seen that R_{q}^{+} is a P-graded Ǔ_{q}(𝔤)-module algebra and that if x = ∑ x_{(2)} ⊗ x_{(1)} ∈ L(Λ_{1} + Λ_{2}) ⊂ L(Λ_{2}) ⊗ L(Λ_{1}), we have

(1)  ∀u ∈ Ǔ_{q}(𝔤), (f_{1}f_{2})(ux) = f_{1}(u_{(2)}x_{(1)})f_{2}(u_{(1)}x_{(2)}).

The proofs of the next three results are taken from [4].

Lemma 3.4. The map ψ : R_{q}^{+} → ℂ_{q}[G] given by ψ(f) = c_{f,v_{Λ}} for all f ∈ L(Λ)^{∗}, is an injective anti-algebra map with image A^{+}.

Proof. In ℂ_{q}[G] we have c_{f_{2},v_{Λ_{2}}}c_{f_{1},v_{Λ_{1}}} = c_{f_{2}⊗f_{1},v_{Λ_{2}}⊗v_{Λ_{1}}} = ψ(f_{1}f_{2}), since f_{1}f_{2} is the image of f_{2}⊗f_{1} by the multiplication map previously described. The other assertions are clear. □

For each y ∈ W, define F (yΛ) = Ǔ_{q}(𝔟^{+})v_{yΛ} and set

Q_{y} = ⨁_{Λ∈P^{+}} F (yΛ)^{⊥}.

Proposition 3.5. Q_{y} is a graded Ǔ_{q}(𝔟^{+})-invariant two-sided ideal of R_{q}^{+}. Furthermore 1. Soc_{Ǔ_{q}(𝔟^{+})}R_{q}^{+}/Q_{y} = ⨁_{Λ∈P^{+}} ℂf_{−yΛ}.

2. The set F = {af_{−yΛ} | a ∈ ℂ^{∗}, Λ ∈ P^{+}} is a multiplicatively closed Ǔ_{q}(𝔟^{+})-invariant set of homogeneous regular normal elements of R_{q}^{+}/Q_{y}.

Proof. It is clear that Q_{y} is Ǔ_{q}(𝔟^{+})-invariant. Let f ∈ F (yΛ)^{⊥}, g ∈ L(Λ′)^{∗}. Then if u ∈ Ǔ_{q}(𝔟^{+}), we have by (1): fg(uv_{y(Λ+Λ′)}) = f(u_{(2)}v_{yΛ})g(u_{(1)}v_{yΛ′}) = 0. Thus fg ∈ F (y(Λ + Λ′))^{⊥}. Hence Q_{y} is a right ideal. A similar argument shows that Q_{y} is a left ideal.

It is easily seen that Soc_{Ǔ_{q}(𝔟^{+})}R_{q}^{+}/Q_{y} ⊃ ⨁_{Λ∈P^{+}} ℂf_{−yΛ}. To check the converse notice that the natural map: L(Λ)^{∗}/F (yΛ)^{⊥} → F (yΛ)^{∗} is an isomorphism of Ǔ_{q}(𝔟^{+})- modules. Furthermore, F (yΛ) = ℂv_{yΛ} ⊕ ∑_{i} X_{i}^{+}F (yΛ). Now all finite dimensional simple Ǔ_{q}(𝔟^{+})-modules are one dimensional and are annihilated by all the X_{i}^{+}. Thus if f ∈ Soc_{Ǔ_{q}(𝔟^{+})}F (yΛ)^{∗}, then X_{i}^{+}f = 0 for all i. Hence f(∑ X_{i}^{+}F (yΛ)) = 0. Therefore Soc_{Ǔ_{q}(𝔟^{+})}F (yΛ)^{∗} = ℂf_{−yΛ}.

Notice that ψ(Q_{y}) ⊂ im ψ is the ideal generated by the c_{f,v_{Λ}}, f ∈ F (yΛ)^{⊥}. By Lemma 2.1, ψ(f_{−yΛ}) ∈ im ψ is normal modulo ψ(Q_{y}). Thus f_{−yΛ} is normal modulo Q_{y}.

Suppose that g ∈ L(Λ′)^{∗} is such that gf_{−yΛ} ∈ Q_{y}. From the above there exists a χ ∈ R(Ǔ_{q}(𝔟^{+})) such that uf_{−yΛ} = χ(u)f_{−yΛ} for all u ∈ Ǔ_{q}(𝔟^{+}). Then we obtain from (1)

0 = gf_{−yΛ}(S(u)v_{y(Λ′+Λ)}) = g(S(u_{(1)})v_{yΛ′})f_{−yΛ}(S(u_{(2)})v_{yΛ})
= g(χ(u_{(2)})S(u_{(1)})v_{yΛ′})f_{−yΛ}(v_{yΛ}) = g(S(r_{χ}(u))v_{yΛ′})

Hence g ∈ Q_{y}. This proves the regularity of f_{−yΛ}. □

Theorem 3.6. The localization (R_{q}^{+}/Q_{y})_{F} is a P-graded Ǔ_{q}(𝔟^{+})-module algebra. The zero-th degree part T_{y} = ((R_{q}^{+}/Q_{y})_{F})_{0} is a submodule algebra and Soc_{Ǔ_{q}(𝔟^{+})}T_{y} = ℂ.

Proof. The first comment about localizations is a consequence of a general result concerning localizations of module algebras. The action is defined in the following way. Let f ∈ F such that uf = χ_{f}(u)f and let l_{f} be the automorphism l_{f}(u) = χ_{f}(u_{(1)})u_{(2)}. Then for any x ∈ R_{q}^{+}/Q_{y}, the action is given by u(f^{−1}x) = f^{−1}(l_{f}^{−1}(u)x). It is then clear that (R_{q}^{+}/Q_{y})_{F} is a P-graded Ǔ_{q}(𝔟^{+})-algebra and that T_{y} is a submodule algebra. It also follows immediately that Soc_{Ǔ_{q}(𝔟^{+})}(R_{q}^{+}/Q_{y})_{F} = F^{−1}(Soc_{Ǔ_{q}(𝔟^{+})}R_{q}^{+}/Q_{y}). Hence

Soc_{Ǔ_{q}(𝔟^{+})}T_{y} = [F^{−1}(Soc_{Ǔ_{q}(𝔟^{+})}R_{q}^{+}/Q_{y})]_{0} = ℂ.
□

We shall apply these results to the element y = w_{+}.

We aim to show that Soc_{ad} C_{w}^{+}, the socle under the adjoint action, reduces to ℂ. Following an idea of [4], we shall deduce this result from a suitable anti-isomorphism between a subalgebra of Ǔ_{q}(𝔟^{−}), denoted by U_{q}(𝔠^{+}), and ℂ_{q}[B^{−}]. Our approach to this fact differs from [4]. We shall obtain it as a consequence of the existence of a nondegenerate pairing between Ǔ_{q}(𝔟^{+}) and Ǔ_{q}(𝔟^{−}), studied in [9, 11]. The properties of this pairing enables us to relate the adjoint actions of U_{q}(𝔠^{+}) and ℂ_{q}[B^{−}], see Proposition 3.11. For convenience we follow the presentation given in [11]. The following theorem summarizes the results from [11, section 2] that we need. Recall that if Ǔ^{0} = ℂ[τ(λ) | λ ∈ P], there are triangular decompositions

Ǔ_{q}(𝔟^{±}) = Ǔ^{0}V^{±}, Ǔ_{q}(𝔤) = Ǔ_{q}(𝔟^{−}) ⊕ Ǔ_{q}(𝔤)V_{+}^{+}.

Theorem 3.7. 1. There exists a unique nondegenerate bilinear form

( / ) : Ǔ_{q}(𝔟^{+}) × Ǔ_{q}(𝔟^{−}) ⟶ ℂ

satisfying the following properties:

(i) (u^{+}/u_{1}^{−}u_{2}^{−}) = (Δ(u^{+})/u_{1}^{−} ⊗ u_{2}^{−}), (u_{1}^{+}u_{2}^{+}/u^{−}) = (u_{2}^{+} ⊗ u_{1}^{+}/Δ(u^{−})) for all u^{±}, u_{i}^{±} ∈ Ǔ_{q}(𝔟^{±}), i = 1, 2 ; (ii) ∀λ, μ ∈ P, (τ(λ)/τ(μ)) = q^{−(λ,μ)/2} ; (iii) ∀λ ∈ P, ∀i ∈ {1, …, n}, (τ(λ)/F_{i}) = (E_{i}/τ(λ)) = 0 ; (iv) ∀i, j ∈ {1, …, n}, (E_{i}/F_{j}) = −δ_{ij}q̂_{i}^{−1}.

2. For all u^{±} ∈ Ǔ_{q}(𝔟^{±}), a^{±} ∈ V^{±}, λ, μ ∈ P we have

(S(u^{+})/S(u^{−})) = (u^{+}/u^{−}), (a^{+}τ(λ)/a^{−}τ(μ)) = q^{−(λ,μ)/2}(a^{+}/a^{−}).

3. (V_{γ}^{+}/V_{−η}^{−}) = 0 for any γ ≠ η ∈ Q^{+}, where V_{±η}^{±} denotes the space of elements of weight ±η under the adjoint action of Ǔ^{0}.

4. For all x ∈ Ǔ_{q}(𝔟^{+}), y ∈ Ǔ_{q}(𝔟^{−}) one has

xy = ∑(x_{(1)}/y_{(1)})(x_{(3)}/S(y_{(3)}))y_{(2)}x_{(2)}

where as usual (I ⊗ Δ)Δ(x) = ∑ x_{(1)} ⊗ x_{(2)} ⊗ x_{(3)}.

We denote by U_{q}(𝔠^{+}) the following Hopf subalgebra of Ǔ_{q}(𝔟^{−}) :

U_{q}(𝔠^{+}) = ℂ[E_{i}, τ(2λ) | 1 ≤ i ≤ n, λ ∈ P].

The embedding Ǔ_{q}(𝔟^{−}) → Ǔ_{q}(𝔤) yields a Hopf algebra map ϕ : ℂ_{q}[G] → Ǔ_{q}(𝔟^{−})^{∘} where Ǔ_{q}(𝔟^{−})^{∘} denotes the cofinite dual. On the other hand the bilinear form ( / ) yields an injective map θ : Ǔ_{q}(𝔟^{+}) → Ǔ_{q}(𝔟^{−})^{∗}.

Theorem 3.8. 1. ker ϕ = J^{+}.

2. ϕ(ℂ_{q}[G]) = θ(U_{q}(𝔠^{+})).

3. The map β = ϕ^{−1}θ : U_{q}(𝔠^{+}) → ℂ_{q}[B^{−}] is an isomorphism of coalgebras and an anti-isomorphism of algebras.

4. Let f ∈ L(Λ)_{−λ}^{∗}. There exists a unique x_{λ} ∈ V_{Λ−λ}^{+} such that β(x_{λ}τ(−2Λ)) = ϕ(c_{f,v_{Λ}}) ∈ ℂ_{q}[B^{−}]. In particular β(τ(−2Λ)) = ϕ(c_{Λ}).

Proof. Assertion 1 is proved in [4, 3.10]. For completeness we give a proof using our notation. Notice that A^{±} = {c ∈ ℂ_{q}[G] | c(Ǔ_{q}(𝔤)V^{±}) = 0}. and recall that ℂ_{q}[G] = A^{+}A^{−}. It is clear that J^{+} ⊂ ker ϕ, ϕ(ℂ_{q}[G]) = ϕ(ℂ_{q}[G]/J^{+}) = ϕ(A^{+}[c_{f_{−w_{0}μ},v_{w_{0}μ}}, μ ∈ P^{+}]). Futhermore Ǔ_{q}(𝔟^{−})^{∗} can be identified with {φ ∈ Ǔ_{q}(𝔤)^{∗} | φ(Ǔ_{q}(𝔤)V_{+}^{+}) = 0}. It follows that ϕ is injective on A^{+}. Fix a dual basis {v_{γ}, f_{−γ}} of L(μ). Then

∑_{γ} c_{v_{γ},f_{−μ}}c_{f_{−γ},v_{μ}} = ∑_{γ} S(c_{f_{−μ},v_{γ}})c_{f_{−γ},v_{μ}} = ε(c_{f_{−μ},v_{μ}}) = 1.

Thus c̃_{μ}c_{μ} = 1 modulo J^{+}. Therefore ϕ(ℂ_{q}[G]) = A^{+}[ϕ(c_{μ}^{−1}) | μ ∈ P^{+}] and the claim follows easily.

We now prove part 4. The other parts will then follow from 1 and Theorem 3.7. Let f ∈ L(Λ)_{−λ}^{∗} and set c = ϕ(c_{f,v_{Λ}}). Notice first that c(V_{−η}^{−}) = 0 unless η = Λ − λ. Denote still by c the induced element on V_{λ−Λ}^{−}. By Theorem 3.7 there exists a unique x_{λ} ∈ V_{Λ−λ}^{+} such that c = θ(x_{λ}). Let ν ∈ P, u ∈ V^{−}; observe that θ(x_{λ}τ(−2Λ))(uτ(ν)) = 0 unless u ∈ V_{λ−Λ}^{−} and that in this case we have

c(uτ(ν)) = f(uτ(ν)v_{Λ}) = q^{(Λ,ν)}c(u) = q^{(Λ,ν)}θ(x_{λ})(u) = θ(x_{λ}τ(−2Λ))(uτ(ν)).

Thus c = θ(x_{λ}τ(−2Λ)). The above calculation also shows that ϕ(c_{Λ}) = θ(τ(−2Λ)). Hence the result. □

Under the hypothesis in 4 of Theorem 3.8 we shall now simply write β(x_{λ}τ(−2Λ)) = c_{f,v_{Λ}} ∈ ℂ_{q}[B^{−}]. We then have c_{f,v_{Λ}} = c_{Λ}β(x_{λ}), x_{λ} ∈ V_{Λ−λ}^{+}.

Since I_{w} ⊂ I_{w_{+}}^{+} + J^{+} = I_{w_{+}}^{+} + I_{e}^{−}, we have a surjective map

(ℂ_{q}[G]/I_{w})_{E_{w_{+}}} ⟶ (ℂ_{q}[G]/(J^{+} + I_{w_{+}}^{+}))_{E_{w_{+}}} = (ℂ_{q}[B^{−}]/(J^{+} ∩ I_{w_{+}}^{+}))_{E_{w_{+}}}.

We denote by C̄_{w}^{+} the image of C_{w}^{+} under this map. We continue to denote by z_{f}^{+} the image of z_{f}^{+} ∈ C_{w}^{+}. This will be justified by the proof of Theorem 3.12 below.

Lemma 3.9. Let f ∈ L(Λ)_{−λ}^{∗}. Set c_{f,v_{Λ}} = β(x_{λ}τ(−2Λ)), x_{λ} ∈ V_{Λ−λ}^{+}. Then in ℂ_{q}[B^{−}] we have, for all i ∈ {1, …, n}

c_{E_{i}f,v_{Λ}} = c_{Λ}β(ad E_{i}.x_{λ} + q^{−(2α_{i},λ)}q_{i,Λ}x_{λ}E_{i})

where q_{i,Λ} = q^{(2α_{i},Λ)} − q^{−(2α_{i},Λ)}.

Proof. An easy computation yields

(ad E_{i}.x_{λ} + q^{−(2α_{i},λ)}q_{i,Λ}x_{λ}E_{i})τ(−2Λ) = E_{i}x_{λ}τ(−2Λ) + q^{−(2α_{i},Λ)}K_{i}^{2}x_{λ}τ(−2Λ)S(E_{i}).

Therefore for all y ∈ Ǔ_{q}(𝔟^{−}) we have

c_{Λ}β(ad E_{i}.x_{λ} + q^{−(2α_{i},λ)}q_{i,Λ}x_{λ}S(E_{i}))(y) = (E_{i}x_{λ}τ(−2Λ) + q^{−(2α_{i},Λ)}K_{i}^{2}x_{λ}τ(−2Λ)S(E_{i})/y).

But by 1 of Theorem 3.7 this is also

(1/y_{(1)})(x_{λ}τ(−2Λ)/y_{(2)})(E_{i}/y_{(3)}) + q^{−(2α_{i},Λ)}(S(E_{i})/y_{(1)})(x_{λ}τ(−2Λ)/y_{(2)})(τ(2α_{i})/y_{(3)}).

By 4 of Theorem 3.7 we know that S(E_{i})y is equal to

(1/y_{(1)})(E_{i}/y_{(3)})y_{(2)}+(1/y_{(1)})(τ(2α_{i})/y_{(3)})y_{(2)}S(E_{i})+(S(E_{i})/y_{(1)})(τ(2α_{i})/y_{(3)})y_{(2)}τ(−2α_{i}).

The result then follows from the above and the following facts: c_{E_{i}f,v_{Λ}}(y) = c_{f,v_{Λ}}(S(E_{i})y), f(y_{(2)}v_{Λ}) = c_{f,v_{Λ}}(y_{(2)}) = (x_{λ}τ(−2Λ)/y_{(2)}), S(E_{i})v_{Λ} = 0, τ(−2α_{i})v_{Λ} = q^{−(2α_{i},Λ)}v_{Λ}. □

Recall that z_{f}^{+} = (c_{wΛ})^{−1}c_{f,v_{Λ}}. Write c_{wΛ} = β(x_{w_{+}Λ}τ(−2Λ))), with x_{w_{+}Λ} ∈ V_{Λ−w_{+}Λ}^{+}. Then

β(x_{w_{+}Λ})^{−1} ∈ (ℂ_{q}[B^{−}]/J^{+} ∩ I_{w_{+}}^{+})_{E_{w_{+}}}

and we may write c_{wΛ}^{−1} = β(x_{w_{+}Λ})^{−1}c_{Λ}^{−1}.

From Lemma 3.9 one deduces the following result.

Corollary 3.10. Under the notation of Lemma 3.9 we have

z_{E_{i}f}^{+} = β(x_{w_{+}Λ})^{−1}β(ad E_{i}.x_{λ} + q^{−(2α_{i},λ)}q_{i,Λ}x_{λ}E_{i}).

The next result computes the adjoint action of ℂ_{q}[B^{−}] on C̄_{w}^{+}.

Proposition 3.11. 1. ∀u, x ∈ U_{q}(𝔠^{+}), ad β(S(u)).β(x) = β(ad u.x).

2. ∀i ∈ {1, …, n}, ad β(S(E_{i})).z_{f}^{+} = z_{E_{i}f}^{+}.

3. ∀ν ∈ P, ad β(S(τ(ν))).z_{f}^{+} = q^{(ν,w_{+}Λ−λ)}z_{f}^{+}, if f ∈ L(Λ)_{−λ}^{∗}.

Proof. 1. Recall that β is an anti-homomorphism of algebras and an homomorphism of coalgebras. Futhermore by 2 of Theorem 3.7, S^{∗}(β(a)) = β(S^{−1}(a)) if S^{∗} denotes the antipode in ℂ_{q}[B^{−}] and a ∈ Ǔ_{q}(𝔟^{+}). Hence we have

ad β(S(u)).β(x) = β(S(u_{(2)}))β(x)S^{∗}(β(S(u_{(1)}))) = β(u_{(1)}xS(u_{(2)})) = β(ad u.x).

This proves part 1.

Let f ∈ L(Λ)_{−λ}^{∗} and set c_{f,v_{λ}} = β(x_{λ}τ(−2Λ)), x_{λ} ∈ V_{Λ−λ}^{+}, c_{wΛ} = β(τ(−2Λ))β(x_{w_{+}Λ}) as above. It follows from 1 and the properties of the adjoint action that for all ν ∈ P, ad β(S(τ(ν))).β(x_{λ}) = q^{(ν,Λ−λ)}β(x_{λ}) and

(2) ad β(S(E_{i})).z_{f}^{+} = ad β(S(E_{i})).[β(x_{w_{+}Λ})^{−1}β(x_{λ})]

= β(x_{w_{+}Λ})^{−1}[ad β(S(E_{i})).β(x_{λ})] + q^{(2α_{i},Λ−λ)}[ad β(S(E_{i})).β(x_{w_{+}Λ})^{−1}]β(x_{λ}).

2. Since c_{E_{i}f_{−w_{+}Λ},v_{Λ}} = 0 modulo I_{w_{+}}^{+}, we obtain from Lemma 3.9 that β(ad E_{i}.x_{w_{+}Λ}) = −q^{−(2α_{i},w_{+}Λ)}q_{i,Λ}β(x_{w_{+}Λ}E_{i}). From ad β(S(E_{i})).1 = 0 = ad β(S(E_{i})).[β(x_{w_{+}Λ})^{−1}β(x_{w_{+}Λ})] we deduce that

ad β(S(E_{i})).β(x_{w_{+}Λ})^{−1} = q^{−(2α_{i},Λ)}q_{i,Λ}β(x_{w_{+}Λ})^{−1}β(E_{i}).

Therefore we obtain from (2) that

ad β(S(E_{i})).z_{f}^{+} = β(x_{w_{+}Λ})^{−1}β(ad E_{i}.x_{λ} + q^{−(2α_{i},λ)}q_{i,Λ}x_{λ}E_{i}).

Hence the result by Corollary 3.10.

3. Using 1, we obtain

ad β(S(τ(ν)).z_{f}^{+} = [ad β(S(τ(ν)).c_{wΛ}^{−1}][ad β(S(τ(ν)).c_{f,v_{Λ}}] = q^{(ν,Λ−λ+w_{+}Λ−Λ)}z_{f}^{+},

hence the result. □

We can now prove the main theorem of this section (cf. [4, Proposition 7.5]).

Theorem 3.12. The anti-algebra map ψ : R_{q}^{+} → ℂ_{q}[G] extends to an anti-isomorphism of algebras ψ : T_{w_{+}} → C_{w}^{+} such that

∀t ∈ T_{w_{+}}, ∀u ∈ U_{q}(𝔠^{+}), ψ(ut) = ad β(S(u)).ψ(t).

Hence in particular, Soc_{ad}C_{w}^{+} = ℂ.

Proof. We already noticed that ψ(Q_{w_{+}}) ⊂ I_{w_{+}}^{+}, ψ(f_{−w_{+}Λ}) = c_{wΛ}. Therefore we can extend ψ to a surjective anti-algebra map

ψ : T_{w_{+}} ⟶ C_{w}^{+}, ψ(f(f_{−w_{+}Λ})^{−1}) = z_{f}^{+} = c_{wΛ}^{−1}c_{f,v_{Λ}}.

As seen in Theorem 3.6, T_{w_{+}} is a Ǔ_{q}(𝔟^{+})-module algebra and we have Soc_{U_{q}(𝔠^{+})}T_{w_{+}} = Soc_{Ǔ_{q}(𝔟^{+})}T_{w_{+}} = ℂ. We can compose the map ψ with the projection C_{w}^{+} → C̄_{w}^{+} to obtain a surjective anti-algebra map, ψ̄ : T_{w_{+}} → C̄_{w}^{+}, mapping f(f_{−w_{+}Λ})^{−1} onto z_{f}^{+} ∈ C̄_{w}^{+}. Now observe that for all f ∈ L(Λ)_{−λ}^{∗} we have in T_{w_{+}},

E_{i}(f(f_{−w_{+}Λ})^{−1}) = (E_{i}f)(f_{−w_{+}Λ})^{−1} + (K_{i}^{2}f)(E_{i}(f_{−w_{+}Λ})^{−1}) = (E_{i}f)(f_{−w_{+}Λ})^{−1}

and

τ(ν)(f(f_{−w_{+}Λ})^{−1}) = (τ(ν)f)(τ(ν)(f_{−w_{+}Λ})^{−1}) = q^{(ν,w_{+}Λ−λ)}(f(f_{−w_{+}Λ})^{−1}) for all ν ∈ P.

Notice that C̄_{w}^{+} is a U_{q}(𝔠^{+})-module algebra by a.z = ad β(S(a)).z for all a ∈ U_{q}(𝔠^{+}) and z ∈ C̄_{w}^{+}. From Proposition 3.11 it follows that ψ̄ is a U_{q}(𝔠^{+})-module map. Hence ker ψ̄ is a U_{q}(𝔠^{+})-submodule. Since ψ̄(1) = 1 and Soc_{U_{q}(𝔠^{+})}T_{w_{+}} = ℂ, we have ker ψ̄ = 0. Therefore both ψ and ψ̄ are isomorphisms of U_{q}(𝔠^{+})-modules. This proves the theorem. □

4. The main theorem

In this section we shall prove that Spec_{w}ℂ_{q}[G] is an H-orbit. We first begin with a consequence of Theorem 3.12.

Theorem 4.1. The socle of C_{w}^{H} considered as a ℂ_{q}[G]-module via the adjoint action is ℂ.

Proof. By Theorem 3.12 we have that Soc_{ad}C_{w}^{+} = ℂ. Now the map σ : ℂ_{q^{−1}}[G] → ℂ_{q}[G] induces a map σ : Ĉ+ ; ŵ → C_{w}^{−} such that σ(ad ĉ.d) = ad σ(ĉ).σ(d). Hence it follows immediately that Soc_{ad}C_{w}^{−} = ℂ also. Since the map C_{w}^{+}⊗C_{w}^{−} → C_{w}^{+}C_{w}^{−} = C_{w}^{H} is a module map, it suffices to show that Soc_{ad} (C_{w}^{+} ⊗ C_{w}^{−}) = ℂ. Recall that C_{w}^{−} is a module over ℂ_{q}[B^{+}] and is diagonalisable as a module over the subalgebra ℂ[B^{+}/N^{+}]. An element of C_{w}^{+} ⊗ C_{w}^{−} can therefore be written in the form ∑ a_{i} ⊗ b_{i} where the b_{i} are linearly independent ℂ_{q}[B^{+}/N^{+}]-weight vectors. Suppose that ∑ a_{i} ⊗ b_{i} ∈ Soc_{ad} (C_{w}^{+} ⊗ C_{w}^{−}) and let c_{f,v_{Λ}} be a generator of J^{−} = I_{e}^{+}. Then

\begin{aligned}c_{f,v_{Λ}}(∑_{i} a_{i} ⊗ b_{i}) & = ∑_{i,j} c_{f,v_{j}}(a_{i}) ⊗ c_{f_{j},v_{Λ}}(b_{i}) = ∑_{i} c_{f,v_{Λ}}(a_{i}) ⊗ c_{f_{−Λ},v_{Λ}}(b_{i}) \\ & = ∑_{i} c_{f,v_{Λ}}(a_{i}) ⊗ α_{i}b_{i} \end{aligned}

for some α_{i} ∈ ℂ^{∗}. Thus c_{f,v_{Λ}}(a_{i}) = 0 for all i. Now modulo J^{+} = I_{e}^{−} it is easily seen that the left ideal generated by the elements c_{f,v_{Λ}} is two-sided. Since Ann_{ad}C_{w}^{+} ⊃ J^{+}, each a_{i} is annihilated by I_{e}. It follows that a_{i} ∈ Soc_{ad}C_{w}^{+} = ℂ. Thus ∑ a_{i} ⊗ b_{i} ∈ Soc_{ad} (ℂ ⊗ C_{w}^{−}) = ℂ ⊗ ℂ. □

Proposition 4.2. Let c ∈ A_{γ,μ}. Then, if λ ∈ P, we have ad c.t_{λ} = ε(c)q^{2((w_{+}−w_{−})λ,μ)}t_{λ}.

Proof. This follows easily from Lemma 2.1. □

Denote by Ψ : P → P the map given by Ψ(λ) = (w_{+} − w_{−})λ.

Corollary 4.3. The algebra of ad-invariant elements of C_{w} is given by

C_{w}^{ad} = ℂ[t_{λ} | λ ∈ ker Ψ].

Proof. This is consequence of Theorems 2.6, 4.1 and the proposition above. □

Recall the following notation from [2, Appendix A]: for each w = (w_{+}, w_{−}) ∈ W × W we set s(w) = codim_{𝔥^{∗}} ker(w_{+}w_{−}^{−1} − I). Denote by P rim ℂ_{q}[G] the primitive spectrum of ℂ_{q}[G] and set P rim_{w} ℂ_{q}[G] = P rim ℂ_{q}[G] ∩ Spec_{w} ℂ_{q}[G]. We are now able to deduce Joseph’s theorem [4, Theorem 9.2].

Theorem 4.4. 1. Prim ℂ_{q}[G] ≅ ⨆_{w∈W ×W} Prim_{w}ℂ_{q}[G].

2. For each w ∈ W × W, Prim_{w}ℂ_{q}[G] is a non-empty H-orbit. If P ∈ Prim_{w}ℂ_{q}[G], then H/Stab_{H}P is a torus of rank equal to rank G − s(w).

Proof. Part 1 is an immediate consequence of Corollary 2.5. Fiw w ∈ W × W and let P be a primitive ideal of type w. Then P A_{w} is a primitive ideal of A_{w}. Let K be a subgroup of P such that P = ker Ψ ⊕ K. Set Cẇ = C_{w}^{H}[t_{λ} | λ ∈ K]. Then clearly C_{w} ≅ Cẇ ⊗ C_{w}^{ad} as ℂ_{q}[G]-module algebras. Notice that the socle of Cẇ is ⨁_{λ∈K} ℂt_{λ}. Therefore any ad-invariant subspace of Cẇ contains some t_{λ} and hence Cẇ contains no nontrivial ad-invariant ideals. If I is a maximal ideal of C_{w}^{ad}, then C_{w}/IC_{w} ≅ Cẇ as ℂ_{q}[G]-module algebras. Thus IC_{w} is a maximal ad-invariant ideal of C_{w}. Similarly IB_{w} is a maximal ad-invariant ideal of B_{w}.

Since C_{w}^{ad} is contained in the center of A_{w}, it follows from the nullstellensatz [8, 9.1.7] that I = P A_{w} ∩ C_{w}^{ad} is a maximal ideal of C_{w}^{ad}. Thus P A_{w} ∩ B_{w} = IB_{w}. Let Q be another such primitive ideal of type w. Then QA_{w} ∩ B_{w} = JB_{w} for some maximal ideal J of C_{w}^{ad}. It is clear from the action of H on the t_{λ} that there exists an h ∈ H such that I = J^{h}. This implies that Q^{h}A_{w} ∩ B_{w} = P A_{w} ∩ B_{w}. Since B_{w} = A_{w}^{Γ}, there must exist a g ∈ Γ such that P A_{w} = (Q^{h}A_{w})^{g} = Q^{hg}A_{w}. Hence Q^{hg} = P. Thus the primitive ideals of type w form an H-orbit.

Since the action of H is algebraic and Γ is finite, H/Stab_{H}P must be a torus of the same dimension as H/Stab_{H}I = dim C_{w}^{ad} = dim H − s(w). □

References

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[6] S. Levendorskii and Ya. S. Soibelman, Algebras of functions on compact quantum groups, Schubert cells and quantum tori, Commun. Math. Phys., 139 (1991), 141–170.

[7] G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Advances in Math., 70 (1988), 237-249.

[8] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley Interscience, New York 1987.

[9] M. Rosso, Analogues de la forme de Killing et du théorème de Harish-Chandra pour les groupes quantiques, Ann. Sci. Ec. Norm. Sup., 23 (1990), 445-467.

[10] Ya. S. Soibelman, The algebra of functions on a compact quantum group, and its representations, Leningrad Math. J. 2 (1991), 161-178.

[11] T. Tanisaki, Killing forms, Harish-Chandra isomorphisms, and universal R-Matrices for Quantum Algebras, preprint (1990).

University of Cincinnati, Cincinnati, OH 45221-0025, U.S.A.

Université de Poitiers, 86022 Poitiers Cedex, France