ALGEBRAIC STRUCTURE OF MULTI-PARAMETER QUANTUM GROUPS

TIMOTHY J. HODGES, THIERRY LEVASSEUR, AND MARGARITA TORO

Introduction

Let G be a connected semi-simple complex Lie group. We define and study the multi-parameter quantum group ℂ_{q,p}[G] in the case where q is a complex parameter that is not a root of unity. Using a method of twisting bigraded Hopf algebras by a cocycle, [2], we develop a unified approach to the construction of ℂ_{q,p}[G] and of the multi-parameter Drinfeld double D_{q,p}. Using a general method of deforming bigraded pairs of Hopf algebras, we construct a Hopf pairing between these algebras from which we deduce a Peter-Weyl-type theorem for ℂ_{q,p}[G]. We then describe the prime and primitive spectra of ℂ_{q,p}[G], generalizing a result of Joseph. In the one-parameter case this description was conjectured, and established in the SL(n)-case, by the first and second authors in [15, 16]. It was proved in the general case by Joseph in [18, 19]. In particular the orbits in Prim ℂ_{q,p}[G] under the natural action of the maximal torus H are indexed, as in the one-parameter case by the elements of the double Weyl group W × W. Unlike the one-parameter case there is not in general a bijection between Symp G and Prim ℂ_{q,p}[G]. However in the case when the symplectic leaves are algebraic such a bijection does exist since the orbits corresponding to a given w ∈ W × W have the same dimension.

In the first section we discuss the Poisson structures on G defined by classical r-matrices of the form r = a − u where a = ∑_{α∈R_{+}} e_{α} ∧ e_{−α} ∈ ∧^{2}𝔤 and u ∈ ∧^{2}𝔥. Given such an r one constructs a Manin triple of Lie groups (G × G, G, G_{r}). Unlike the one-parameter case (where u = 0), the dual group G_{r} will generally not be an algebraic subgroup of G × G. In fact this happens if and only if u ∈ ∧^{2}𝔥_{ℚ}. Since the quantized universal enveloping algebra U_{q}(𝔤) is a deformation of the algebra of functions on the algebraic group G_{r} [11], this explains the difficulty in constructing multi-parameter versions of U_{q}(𝔤). From [22, 30], one has that the symplectic leaves are the connected components of G ∩ G_{r}xG_{r} where x ∈ G. Since r is H-invariant, the symplectic leaves are permuted by H with the orbits being contained in Bruhat cells in G × G indexed by W × W. In the case where G_{r} is algebraic, the symplectic leaves are also algebraic and an explicit formula is given for their dimension.

The philosophy of [15, 16] was that, as in the case of enveloping algebras of algebraic solvable Lie algebras, the primitive ideals of ℂ_{q}[G] should be in bijection with the symplectic leaves of G (in the case u = 0). Indeed, since the Lie bracket on 𝔤_{r} = Lie(G_{r}) is the linearization of the Poisson structure on G, Prim ℂ_{q,p}[G] should resemble Prim U(𝔤_{r}). The study of the multi-parameter versions ℂ_{q,p}[G] is similar to the case of enveloping algebras of general solvable Lie algebras. In the general case Prim U(𝔤_{r}) is in bijection with the co-adjoint orbits in 𝔤_{r}^{∗} under the action of the ‘adjoint algebraic group’ of 𝔤_{r}, [12]. It is therefore natural that, only in the case where the symplectic leaves are algebraic, does one expect and obtain a bijection between the symplectic leaves and the primitive ideals.

In section 2 we define the notion of an L-bigraded Hopf 𝕂-algebra, where L is an abelian group. When A is finitely generated such bigradings correspond bijectively to morphisms from the algebraic group L^{∨} to the (algebraic) group R(A) of one- dimensional representations of A. For any antisymmetric bicharacter p on L, the multiplication in A may be twisted to give a new Hopf algebra A_{p}. Moreover, given a pair of L-bigraded Hopf algebras A and U equipped with an L-compatible Hopf pairing A × U → 𝕂, one can deform the pairing to get a new Hopf pairing between A_{p^{−1}} and U_{p}. This deformation commutes with the formation of the Drinfeld double in the following sense. Suppose that A and U are bigraded Hopf algebras equipped with a compatible Hopf pairing A^{op} × U → 𝕂. Then the Drinfeld double A ⋊ ⋉ U inherits a bigrading such that (A ⋊ ⋉ U)_{p} ≅ A_{p} ⋊ ⋉ U_{p}.

Date: June, 1994.

The first author was partially supported by grants from the National Security Agency and the C. P. Taft Memorial Fund.

The third author was partially supported by a grant from Colciencias.

Let ℂ_{q}[G] denote the usual one-parameter quantum group (or quantum function algebra) and let U_{q}(𝔤) be the quantized enveloping algebra associated to the lattice L of weights of G. Let U_{q}(𝔟^{+}) and U_{q}(𝔟^{−}) be the usual sub-Hopf algebras of U_{q}(𝔤) corresponding to the Borel subalgebras 𝔟^{+} and 𝔟^{−} respectively. Let D_{q}(𝔤) = U_{q}(𝔟^{+}) ⋊ ⋉ U_{q}(𝔟^{−}) be the Drinfeld double. Since the groups of one-dimensional representations of U_{q}(𝔟^{+}), U_{q}(𝔟^{−}), D_{q}(𝔤) and ℂ_{q}[G] are all isomorphic to H = L^{∨}, these algebras are all equipped with L-bigradings. Moreover the Rosso-Tanisaki pairing is compatible with the bigradings on U_{q}(𝔟^{+}) and U_{q}(𝔟^{−}). For any anti-symmetric bicharacter p on L one may therefore twist simultaneously the Hopf algebras U_{q}(𝔟^{+}), U_{q}(𝔟^{−}) and D_{q}(𝔤) in such a way that D_{q,p}(𝔤) ≅ U_{q,p}(𝔟^{+}) ⋊ ⋉ U_{q,p}(𝔟^{−}). The algebra D_{q,p}(𝔤) is the ‘multi-parameter quantized universal enveloping algebra’ constructed by Okado and Yamane [25] and previously in special cases in [9, 32]. The canonical pairing between ℂ_{q}[G] and U_{q}(𝔤) induces a L-compatible pairing between ℂ_{q}[G] and D_{q}(𝔤). Thus there is an induced pairing between the multi-parameter quantum group ℂ_{q,p}[G] and the multi-parameter double D_{q,p^{−1}}(𝔤). Recall that the Hopf algebra ℂ_{q}[G] is defined as the restricted dual of U_{q}(𝔤) with respect to a certain category C of modules over U_{q}(𝔤). There is a natural deformation functor from this category to a category C_{p} of modules over D_{q,p^{−1}}(𝔤) and ℂ_{q,p}[G] turns out to be the restricted dual of D_{q,p^{−1}}(𝔤) with respect to this category. This Peter-Weyl theorem for ℂ_{q,p}[G] was also found by Andruskiewitsch and Enriquez in [1] using a different construction of the quantized universal enveloping algebra and in special cases in [5, 14].

The main theorem describing the primitive spectrum of ℂ_{q,p}[G] is proved in the final section. Since ℂ_{q,p}[G] inherits an L-bigrading, there is a natural action of H as automorphisms of ℂ_{q,p}[G]. For each w ∈ W ×W, we construct an algebra A_{w} = (ℂ_{q,p}[G]/I_{w})_{E_{w}} which is a localization of a quotient of ℂ_{q,p}[G]. For each prime P ∈ Spec ℂ_{q,p}[G] there is a unique w ∈ W × W such that P ⊃ I_{w} and P A_{w} is proper. Thus Spec ℂ_{q,p}[G] ≅ ⨆_{w∈W ×W} Spec_{w} ℂ_{q,p}[G] where Spec_{w} ℂ_{q,p}[G] ≅ Spec A_{w} is the set of primes of type w. The key results are then Theorems 4.14 and 4.15 which state that an ideal of A_{w} is generated by its intersection with the center and that H acts transitively on the maximal ideals of the center. From this it follows that the primitive ideals of ℂ_{q,p}[G] of type w form an orbit under the action of H.

An earlier version of our approach to the proof of Joseph’s theorem is contained in the unpublished article [17]. The approach presented here is a generalization of this proof to the multi-parameter case.

These results are algebraic analogs of results of Levendorskii [20, 21] on the irreducible representations of multi-parameter function algebras and compact quantum groups. The bijection between symplectic leaves of the compact Poisson group and irreducible ∗-representations of the compact quantum group found by Soibelman in the one-parameter case, breaks down in the multi-parameter case.

After this work was completed, the authors became aware of the work of Constantini and Varagnolo [7, 8] which has some overlap with the results in this paper.

1. Poisson Lie Groups

1.1. Notation. Let 𝔤 be a complex semi-simple Lie algebra associated to a Cartan matrix [a_{ij}]_{1⩽i,j⩽n}. Let {d_{i}}_{1⩽i⩽n} be relatively prime positive integers such that [d_{i}a_{ij}]_{1⩽i,j⩽n} is symmetric positive definite.

Let 𝔥 be a Cartan subalgebra of 𝔤, R the associated root system, B = {α_{1}, …, α_{n}} a basis of R, R_{+} the set of positive roots and W the Weyl group. We denote by P and Q the lattices of weights and roots respectively. The fundamental weights are denoted by ϖ_{1}, …, ϖ_{n} and the set of dominant integral weights by P^{+} = ∑_{i=1}^{n} ℕϖ_{i}. Let (−, −) be a non-degenerate 𝔤-invariant symmetric bilinear form on 𝔤; it will identify 𝔤, resp. 𝔥, with its dual 𝔤^{∗}, resp. 𝔥^{∗}. The form (−, −) can be chosen in order to induce a perfect pairing P × Q → ℤ such that

(ϖ_{i}, α_{j}) = δ_{ij}d_{i}, (α_{i}, α_{j}) = d_{i}a_{ij}.

Hence d_{i} = (α_{i}, α_{i})/2 and (α, α) ∈ 2ℤ for all α ∈ R. For each α_{j} we denote by h_{j} ∈ 𝔥 the corresponding coroot: ϖ_{i}(h_{j}) = δ_{ij}. We also set

𝔫^{±} = ⊕_{α∈R_{+}}𝔤_{±α}, 𝔟^{±} = 𝔥 ⊕ 𝔫^{±}, 𝔡 = 𝔤 × 𝔤, 𝔱 = 𝔥 × 𝔥, 𝔲^{±} = 𝔫^{±} × 𝔫^{∓}.

Let G be a connected complex semi-simple algebraic group such that Lie(G) = 𝔤 and set D = G × G. We identify G (and its subgroups) with the diagonal copy inside D. We denote by exp the exponential map from 𝔡 to D. We shall in general denote a Lie subalgebra of 𝔡 by a gothic symbol and the corresponding connected analytic subgroup of D by a capital letter.

1.2. Poisson Lie group structure on G. Let a = ∑_{α∈R_{+}} e_{α}∧e_{−α} ∈ ∧^{2}𝔤 where the e_{α} are root vectors such that (e_{α}, e_{β}) = δ_{α,−β}. Let u ∈ ∧^{2}𝔥 and set r = a − u. Then it is well known that r satisfies the modified Yang-Baxter equation [3, 20] and that therefore the tensor π(g) = (l_{g})_{∗}r − (r_{g})_{∗}r furnishes G with the structure of a Poisson Lie group, see [13, 22, 30] ((l_{g})_{∗} and (r_{g})_{∗} are the differentials of the left and right translation by g ∈ G).

We may write u = ∑_{1≤i,j≤n} u_{ij}h_{i} ⊗ h_{j} for a skew-symmetric n × n matrix [u_{ij}]. The element u can be considered either as an alternating form on 𝔥^{∗} or a linear map u ∈ End 𝔥 by the formula

∀ x ∈ 𝔥, u(x) = ∑_{i,j} u_{i,j}(x, h_{i})h_{j}.

The Manin triple associated to the Poisson Lie structure on G given by r is described as follows. Set u_{±} = u ± I ∈ End 𝔥 and define

ϑ : 𝔥 ↪ 𝔱, ϑ(x) = −(u_{−}(x), u_{+}(x)),
𝔞 = ϑ(𝔥), 𝔤_{r} = 𝔞 ⊕ 𝔲^{+}.

Following [30] one sees easily that the associated Manin triple is (𝔡, 𝔤, 𝔤_{r}) where 𝔤 is identified with the diagonal copy inside 𝔡. Then the corresponding triple of Lie groups is (D, G, G_{r}), where A = exp(𝔞) is an analytic torus and G_{r} = AU^{+}. Notice that 𝔤_{r} is a solvable, but not in general algebraic, Lie subalgebra of 𝔡.

The following is an easy consequence of the definition of 𝔞 and the identities u_{+}+u_{−} = 2u, u_{+}−u_{−} = 2I:

𝔞 = {(x, y) ∈ 𝔱 | x + y = u(y − x)} = {(x, y) ∈ 𝔱 | u_{+}(x) = u_{−}(y)}. (1.1)

Recall that exp : 𝔥 → H is surjective; let L_{H} be its kernel. We shall denote by X(K) the group of characters of an algebraic torus K. Any χ ∈ X(H) is given by χ(exp x) = exp dχ(x), x ∈ 𝔥, where dχ ∈ 𝔥^{∗} is the differential of χ. Then

X(H) ≅ L = L_{H}^{∘} := {ξ ∈ 𝔥^{∗} | ξ(L_{H}) ⊂ 2iπℤ}.

One can show that L has a basis consisting of dominant weights.

Recall that if G̃ is a connected simply connected algebraic group with Lie algebra 𝔤 and maximal torus H̃, we have

LH̃ = P^{∘} = ⊕_{j=1}^{n}2iπℤh_{j}, X(H̃) ≅ P,
Q ⊆ L ⊆ P, π_{1}(G) = L_{H}/P^{∘} ≅ P/L.

Notice that L_{H}/P^{∘} is a finite group and exp u(L_{H}) is a subgroup of H. We set

Γ_{0} = {(a, a) ∈ T | a^{2} = 1}, Δ = {(a, a) ∈ T | a^{2} ∈ exp u(L_{H})},
Γ = A ∩ H = {(a, a) ∈ T | a = exp x = exp y, x + y = u(y − x)}.

It is easily seen that Γ = G ∩ G_{r}.

Proposition 1.1. We have Δ = Γ.Γ_{0}.

Proof. We obviously have Γ_{0} ⊂ Δ. Let (exp h, exp h) ∈ Γ, h ∈ 𝔥. By definition there exist (x, y) ∈ 𝔞, ℓ_{1}, ℓ_{2} ∈ L_{H} such that

x = h + ℓ_{1}, y = h + ℓ_{2}, y + x = u(y − x).

Hence y+x = 2h+ℓ_{1} +ℓ_{2} = u(ℓ_{2} −ℓ_{1}) and (exp h)^{2} = exp 2h = exp u(ℓ_{2} −ℓ_{1}). This shows (exp h, exp h) ∈ Δ. Thus Γ.Γ_{0} ⊆ Δ.

Let (a, a) ∈ Δ, a = exp h, h ∈ 𝔥. From a^{2} ∈ exp u(L_{H}) we get ℓ, ℓ′ ∈ L_{H} such that 2h = u(ℓ′) + ℓ. Set x = h − ℓ/2 − ℓ′/2, y = h + ℓ′/2 − ℓ/2. Then y + x = u(y − x) and we have exp(−ℓ/2 − ℓ′/2) = exp(ℓ′/2 − ℓ/2), since ℓ′ ∈ L_{H}. If b = exp(−ℓ′/2 + ℓ/2) we obtain exp x = exp y = ab^{−1}, hence (a, a) = (exp x, exp y).(b, b) ∈ Γ.Γ_{0}. Therefore Γ.Γ_{0} = Δ.

Remark . When u is “generic” Γ_{0} is not contained in Γ. For example, take G to be SL(3, ℂ) and u = α(h_{1} ⊗ h_{2} − h_{2} ⊗ h_{1}) with α ∉ ℚ.

Considered as a Poisson variety, G decomposes as a disjoint union of symplectic leaves. Denote by Symp G the set of these symplectic leaves. Since r is H-invariant, translation by an element of H is a Poisson morphism and hence there is an induced action of H on Symp G. The key to classifying the symplectic leaves is the following result, cf. [22, 30].

Theorem 1.2. The symplectic leaves of G are exactly the connected components of G∩G_{r}xG_{r} for x ∈ G.

Remark that A, Γ and G_{r} are in general not closed subgroups of D. This has for consequence that the analysis of Symp G made in [15, Appendix A] in the case u = 0 does not apply in the general case.

Set Q = HG_{r} = T U^{+}. Then Q is a Borel subgroup of D and, recalling that the Weyl group associated to the pair (G, T) is W × W, the corresponding Bruhat decomposition yields D = ⊔_{w∈W ×W} QwQ = ⊔_{w∈W ×W} QwG_{r}. Therefore any symplectic leaf is contained in a Bruhat cell QwQ for some w ∈ W × W.

Definition . A leaf A is said to be of type w if A ⊂ QwQ. The set of leaves of type w is denoted by Symp_{w} G.

For each w ∈ W × W set w = (w_{+}, w_{−}), w_{±} ∈ W, and fix a representative ẇ in the normaliser of T. One shows as in [15, Appendix A] that G ∩ QẇG_{r} ≠ ∅, for all w ∈ W × W; hence Symp_{w} G ≠ ∅ and G ∩ G_{r}ẇG_{r} ≠ ∅, since QwQ = ∪_{h∈H}hG_{r}ẇG_{r}.

The adjoint action of D on itself is denoted by Ad. Set

U_{w}^{−} = Ad w(U) ∩ U^{+}, A′_{w} = {a ∈ A | aẇG_{r} = ẇG_{r}},
T′_{w} = {t ∈ T | tG_{r}ẇG_{r} = G_{r}ẇG_{r}}, H′_{w} = H ∩ T′_{w}.

Recall that U_{w}^{−} is isomorphic to ℂ^{l(w)} where l(w) = l(w_{+}) + l(w_{−}) is the length of w. We set s(w) = dim H′_{w}.

Lemma 1.3. (i) A′_{w} = Ad w(A) ∩ A and T′_{w} = A. Ad w(A) = AH′_{w}.

(ii) We have Lie(A′_{w}) = 𝔞′_{w} = {ϑ(x) | x ∈ Ker(u_{−}w_{−}^{−1}u_{+} − u_{+}w_{+}^{−1}u_{−})} and dim 𝔞′_{w} = n − s(w).

Proof. (i) The first equality is obvious and the second is an easy consequence of the bijection, induced by multiplication, between U_{w}^{−} × T × U^{+} and QwQ = QwG_{r}.

(ii) By definition we have 𝔞′_{w} = {ϑ(x) | x ∈ 𝔥, w^{−1}(ϑ(x)) ∈ 𝔞}. From (1.1) we deduce that ϑ(x) ∈ 𝔞′_{w} if and only if u_{+}w_{+}^{−1}(−u_{−}(x)) = u_{−}w_{−}^{−1}(−u_{+}(x)).

It follows from (i) that dim T′_{w} = n + dim H′_{w} = 2n − dim A′_{w}, hence dim 𝔞′_{w} = n − s(w).

Recall that u ∈ End 𝔥 is an alternating bilinear form on 𝔥^{∗}. It is easily seen that ∀ λ, μ ∈ 𝔥^{∗}, u(λ, μ) = −(^{t}u(λ), μ), where ^{t}u ∈ End 𝔥^{∗} is the transpose of u.

Notation . Set ^{t}u = −Φ, Φ_{±} = Φ ± I, σ(w) = Φ_{−}w_{−}Φ_{+} − Φ_{+}w_{+}Φ_{−}, where w_{±} ∈ W is considered as an element of End 𝔥^{∗}.

Observe that ^{t}u_{±} = −Φ_{∓} and that

u(λ, μ) = (Φλ, μ), for all λ, μ ∈ 𝔥^{∗}. (1.2)

Furthermore, since the transpose of w_{±} ∈ End 𝔥^{∗} is w_{±}^{−1} ∈ End 𝔥, we have^{t}σ(w) = u_{−}w_{−}^{−1}u_{+}−u_{+}w_{+}^{−1}u_{−}. Hence by Lemma 1.3

(1.3) s(w) = codim Ker_{𝔥^{∗}} σ(w), dim A′_{w} = dim Ker_{𝔥^{∗}} σ(w).

1.3. The algebraic case. As explained in 1.1 the Lie algebra 𝔤_{r} is in general not algebraic. We now describe its algebraic closure. Recall that a Lie subalgebra 𝔪 of 𝔡 is said to be algebraic if 𝔪 is the Lie algebra of a closed (connected) algebraic subgroup of D.

Definition . Let 𝔪 be a Lie subalgebra of 𝔡. The smallest algebraic Lie subalgebra of 𝔡 containing 𝔪 is called the algebraic closure of 𝔪 and will be denoted by 𝔪̃.

Recall that 𝔤_{r} = 𝔞 ⊕ 𝔲^{+}. Notice that 𝔲^{+} is an algebraic Lie subalgebra of 𝔡; hence it follows from [4, Corollary II.7.7] that 𝔤̃_{r} = 𝔞̃ ⊕ 𝔲^{+}. Thus we only need to describe 𝔞̃. Since 𝔱 is algebraic we have 𝔞̃ ⊆ 𝔱 and we are reduced to characterize the algebraic closure of a Lie subalgebra of 𝔱 = Lie(T).

The group T = H ×H is an algebraic torus (of rank 2n). The map χ ↦ dχ identifies X(T) with L×L. Let 𝔨 ⊂ 𝔱 be a subalgebra. We set

𝔨^{⊥} = {θ ∈ X(T) | 𝔨 ⊂ Ker_{𝔱} θ}.

The following proposition is well known. It can for instance be deduced from the results in [4, II. 8].

Proposition 1.4. Let 𝔨 be a subalgebra of 𝔱. Then 𝔨̃ = ∩_{θ∈𝔨^{⊥}} Ker_{𝔱} θ and 𝔨̃ is the Lie algebra of the closed connected algebraic subgroup K̃ = ∩_{θ∈𝔨^{⊥}} Ker_{T} θ.

Corollary 1.5. We have

𝔞^{⊥} = {(λ, μ) ∈ X(T) | Φ_{+}λ + Φ_{−}μ = 0},
𝔞̃ = ∩_{(λ,μ)∈𝔞^{⊥}} Ker_{𝔱}(λ, μ), Ã = ∩_{(λ,μ)∈𝔞^{⊥}} Ker_{T} (λ, μ).

Proof. From the definition of 𝔞 = ϑ(𝔥) we obtain

(λ, μ) ∈ 𝔞^{⊥} ⟺ ∀ x ∈ 𝔥, λ(−u_{−}(x)) + μ(−u_{+}(x)) = 0.

The first equality then follows from^{t}u_{±} = −Φ_{∓}. The remaining assertions are consequences of Proposition 1.4.

Set

𝔥_{ℚ} = ℚ ⊗_{ℤ} P^{∘} = ⊕_{i=1}^{n}ℚh_{i}, 𝔥_{ℚ}^{∗} = ℚ ⊗_{ℤ} P = ⊕_{i=1}^{n}ℚϖ_{i}
𝔞_{ℚ}^{⊥} = ℚ ⊗_{ℤ} 𝔞^{⊥} = {(λ, μ) ∈ 𝔥_{ℚ}^{∗} × 𝔥_{ℚ}^{∗} | Φ_{+}λ + Φ_{−}μ = 0}.

Observe that dim_{ℚ} 𝔞_{ℚ}^{⊥} = rk_{ℤ} 𝔞^{⊥} and that, by Corollary 1.5,

dim 𝔞̃ = 2n − dim_{ℚ} 𝔞_{ℚ}^{⊥}. (1.4)

Lemma 1.6. 𝔞_{ℚ}^{⊥} ≅ {ν ∈ 𝔥_{ℚ}^{∗} | Φν ∈ 𝔥_{ℚ}^{∗}}.

Proof. Define a ℚ-linear map

{ν ∈ 𝔥_{ℚ}^{∗} | Φν ∈ 𝔥_{ℚ}^{∗}} ⟶ 𝔞_{ℚ}^{⊥}, ν ↦ (−Φ_{−}ν, Φ_{+}ν).

It is easily seen that this provides the desired isomorphism.

Theorem 1.7. The following assertions are equivalent:

(i) 𝔤_{r} is an algebraic Lie subalgebra of 𝔡;

(ii) u(P × P) ⊂ ℚ;

(iii) ∃ m ∈ ℕ^{∗}, Φ(mP) ⊂ P;

(iv) Γ is a finite subgroup of T.

Proof. Recall that 𝔤_{r} is algebraic if and only if 𝔞 = 𝔞̃, i.e. n = dim 𝔞 = dim 𝔞̃. By (1.4) and Lemma 1.6 this is equivalent to Φ(P) ⊂ 𝔥_{ℚ}^{∗} = ℚ⊗_{ℤ} P. The equivalence of (i) to (iii) then follows from the definitions, (1.2) and the fact that ^{t}u = −Φ.

To prove the equivalence with (iv) we first observe that, by Proposition 1.1, Γ is finite if and only if exp u(L_{H}) is finite. Since L_{H}/P^{∘} is finite this is also equivalent to exp u(P^{∘}) being finite. This holds if and only if u(mP^{∘}) ⊂ P^{∘} for some m ∈ ℕ^{∗}. Hence the result.

When the equivalent assertions of Theorem 1.7 hold, we shall say that we are in the algebraic case or that u is algebraic. In this case all the subgroups previously introduced are closed algebraic subgroups of D and we may define the algebraic quotient varieties D/G_{r} and Ḡ = G/Γ. Let p be the projection G → Ḡ. Observe that Ḡ is open in in D/G_{r} and that the Poisson bracket of G passes to Ḡ. We set

Cẇ = G_{r}ẇG_{r}/G_{r}, C_{w} = QwG_{r}/G_{r} = ∪_{h∈H}hCẇ
Bẇ = Cẇ ∩ Ḡ, B_{w} = C_{w} ∩ Ḡ, A_{w} = p^{−1}(B_{w}).

The next theorem summarizes the description of the symplectic leaves in the algebraic case.

Theorem 1.8. 1. Symp_{w} G ≠ ∅ for all w ∈ W × W, Symp G = ⊔_{w∈W ×W} Symp_{w} G.

2. Each symplectic leaf of Ḡ, resp. G, is of the form hBẇ, resp. hAẇ, for some h ∈ H and w ∈ W ×W, where Aẇ denotes a fixed connected component of p^{−1}(Bẇ).

3. Cẇ ≅ A_{w} × U_{w}^{−} where A_{w} = A/A′_{w} is a torus of rank s(w). Hence dim Cẇ = dim Bẇ = dim Aẇ = l(w) + s(w) and H/ Stab_{H} Aẇ is a torus of rank n − s(w).

Proof. The proofs are similar to those given in [15, Appendix A] for the case u = 0.

2. Deformations of Bigraded Hopf Algebras

2.1. Bigraded Hopf Algebras and their deformations. Let L be an (additive) abelian group. We will say that a Hopf algebra (A, i, m, ϵ, Δ, S) over a field 𝕂 is an L-bigraded Hopf algebra if it is equipped with an L × L grading

A = ⨁_{(λ,μ)∈L×L} A_{λ,μ}

such that 1. 𝕂 ⊂ A_{0,0}, A_{λ,μ}A_{λ′,μ′} ⊂ A_{λ+λ′,μ+μ′} (i.e. A is a graded algebra)

2. Δ(A_{λ,μ}) ⊂ ∑_{ν∈L} A_{λ,ν} ⊗ A_{−ν,μ}

3. λ ≠ −μ implies ϵ(A_{λ,μ}) = 0

4. S(A_{λ,μ}) ⊂ A_{μ,λ}.

For sake of simplicity we shall often make the following abuse of notation: If a ∈ A_{λ,μ} we will write Δ(a) = ∑_{ν} a_{λ,ν} ⊗ a_{−ν,μ}, a_{λ,ν} ∈ A_{λ,ν}, a_{−ν,μ} ∈ A_{−ν,μ}.

Let p : L × L → 𝕂^{∗} be an antisymmetric bicharacter on L in the sense that p is multiplicative in both entries and that, for all λ, μ ∈ L,

(1) p(μ, μ) = 1 ; (2) p(λ, μ) = p(μ, −λ).

Then the map p̃: (L × L) × (L × L) → 𝕂^{∗} given by

p̃((λ, μ), (λ′, μ′)) = p(λ, λ′)p(μ, μ′)^{−1}

is a 2-cocycle on L × L such that p̃(0, 0) = 1.

One may then define a new multiplication, m_{p}, on A by

∀ a ∈ A_{λ,μ}, b ∈ A_{λ′,μ′}, a · b = p(λ, λ′)p(μ, μ′)^{−1}ab. (2.1)

Theorem 2.1. A_{p} := (A, i, m_{p}, ϵ, Δ, S) is an L-bigraded Hopf algebra.

Proof. The proof is a slight generalization of that given in [2]. It is well known that A_{p} = (A, i, m_{p}) is an associative algebra. Since Δ and ϵ are unchanged, (A, Δ, ϵ) is still a coalgebra. Thus it remains to check that ϵ, Δ are algebra morphisms and that S is an antipode.

Let x ∈ A_{λ,μ} and y ∈ A_{λ′,μ′}. Then

ϵ(x · y) = p(λ, λ′)p(μ, μ′)^{−1}ϵ(xy)
= p(λ, λ′)p(μ, μ′)^{−1}δ_{λ,−μ}δ_{λ′,−μ′}ϵ(x)ϵ(y)
= p(λ, λ′)p(−λ, −λ′)^{−1}ϵ(x)ϵ(y)
= ϵ(x)ϵ(y)

So ϵ is a homomorphism. Now suppose that Δ(x) = ∑ x_{λ,ν} ⊗ x_{−ν,μ} and Δ(y) = ∑ y_{λ′,ν′} ⊗ y_{−ν′,μ′}. Then

Δ(x) · Δ(y) = (∑ x_{λ,ν} ⊗ x_{−ν,μ}) · (∑ y_{λ′,ν′} ⊗ y_{−ν′,μ′})
= ∑ x_{λ,ν} · y_{λ′,ν′} ⊗ x_{−ν,μ} · y_{−ν′,μ′}
= p(λ, λ′)p(μ, μ′)^{−1} ∑ p(ν, ν′)^{−1}p(−ν, −ν′)x_{λ,ν}y_{λ′,ν′} ⊗ x_{−ν,μ}y_{−ν′,μ′}
= p(λ, λ′)p(μ, μ′)^{−1}Δ(xy)
= Δ(x · y)

So Δ is also a homomorphism. Finally notice that

∑ S(x_{(1)}) · x_{(2)} = ∑ S(x_{λ,ν}) · x_{−ν,μ}
= ∑ p(ν, −ν)p(λ, μ)^{−1}S(x_{λ,ν})x_{−ν,μ}
= p(λ, μ)^{−1} ∑ S(x_{λ,ν}) · x_{−ν,μ}
= p(λ, μ)^{−1}ϵ(x)
= ϵ(x)

A similar calculation shows that ∑ x_{(1)} · S(x_{(2)}) = ϵ(x). Hence S is indeed an antipode.

Remark . The isomorphism class of the algebra A_{p} depends only on the cohomology class [p̃] ∈ H^{2}(L × L, 𝕂^{∗}), [2, §3].

Remark . Theorem 2.1 is a particular case of the following more general construction. Let (A, i, m) be a 𝕂-algebra. Assume that F ∈ GL_{𝕂}(A ⊗ A) is given such that (with the usual notation)

(1) F (m ⊗ 1) = (m ⊗ 1)F_{23}F_{13} ; F (1 ⊗ m) = (1 ⊗ m)F_{12}F_{13}

(2) F (i ⊗ 1) = i ⊗ 1 ; F (1 ⊗ i) = 1 ⊗ i

(3) F_{12}F_{13}F_{23} = F_{23}F_{13}F_{12}, i.e. F satisfies the Quantum Yang-Baxter Equation. Set m_{F} = m ∘ F. Then (A, i, m_{F}) is a 𝕂-algebra. Assume furthermore that (A, i, m, ϵ, Δ, S) is a Hopf algebra and that

(4) F : A ⊗ A → A ⊗ A is morphism of coalgebras

(5) mF (S ⊗ 1)Δ = m(S ⊗ 1)Δ ; mF (1 ⊗ S)Δ = m(1 ⊗ S)Δ. Then A_{F} := (A, i, m_{F} , ϵ, Δ, S) is a Hopf algebra. The proofs are straightforward verifications and are left to the interested reader.

When A is an L-bigraded Hopf algebra and p is an antisymmetric bicharacter as above, we may define F ∈ GL_{𝕂}(A ⊗ A) by

∀a ∈ A_{λ,μ}, ∀b ∈ A_{λ′,μ′}, F (a ⊗ b) = p(λ, λ′)p(μ, μ′)^{−1}a ⊗ b.

It is easily checked that F satisfies the conditions (1) to (5) and that the Hopf algebras A_{F} and A_{p} coincide.

A related construction of the quantization of a monoidal category is given in [24].

2.2. Diagonalizable subgroups of R(A). In the case where L is a finitely generated group and A is a finitely generated algebra (which is the case for the multi-parameter quantum groups considered here), there is a simple geometric interpretation of L-bigradings. They correspond to algebraic group maps from the diagonalizable group L^{∨} to the group of one dimensional representations of A.

Assume that 𝕂 is algebraically closed. Let (A, i, m, ϵ, Δ, S) be a Hopf 𝕂-algebra. Denote by R(A) the multiplicative group of one dimensional representations of A, i.e. the character group of the algebra A. Notice that when A is a finitely generated 𝕂-algebra, R(A) has the structure of an affine algebraic group over 𝕂, with algebra of regular functions given by 𝕂[R(A)] = A/J where J is the semi-prime ideal ∩_{h∈R(A)} Ker h. Recall that there are two natural group homomorphisms l, r : R(A) → Aut_{𝕂}(A) given by

l_{h}(x) = ∑ h(S(x_{(1)}))x_{(2)} = ∑ h^{−1}(x_{(1)})x_{(2)}
r_{h}(x) = ∑ x_{(1)}h(x_{(2)}).

Theorem 2.2. Let A be a finitely generated Hopf algebra and let L be a finitely generated abelian group. Then there is a natural bijection between:

1. L-bigradings on A;

2. Hopf algebra maps A → 𝕂L (where 𝕂L denotes the group algebra);

3. morphisms of algebraic groups L^{∨} → R(A).

Proof. The bijection of the last two sets of maps is well-known. Given an L-bigrading on A, we may define a map ϕ : A → 𝕂L by ϕ(a_{λ,μ}) = ϵ(a)u_{λ}. It is easily verified that this is a Hopf algebra map. Conversely, given a map L^{∨} → R(A) we may construct an L bigrading using the following result.

Theorem 2.3. Let (A, i, m, ϵ, Δ, S) be a finitely generated Hopf algebra over 𝕂. Let H be a closed diagonalizable algebraic subgroup of R(A). Denote by L the (additive) group of characters of H and by ⟨−, −⟩ : L × H → 𝕂^{∗} the natural pairing. For (λ, μ) ∈ L × L set

A_{λ,μ} = {x ∈ A | ∀h ∈ H, l_{h}(x) = ⟨λ, h⟩x, r_{h}(x) = ⟨μ, h⟩x}.

Then (A, i, m, ϵ, Δ, S) is an L-bigraded Hopf algebra.

Proof. Recall that any element of A is contained in a finite dimensional subcoalgebra of A. Therefore the actions of H via r and l are locally finite. Since they commute and H is diagonalizable, A is L × L diagonalizable. Thus the decomposition A = ⨁_{(λ,μ)∈L×L} A_{λ,μ} is a grading.

Now let C be a finite dimensional subcoalgebra of A and let {c_{1}, …, c_{n}} be a basis of H × H weight vectors. Suppose that Δ(c_{i}) = ∑ t_{ij} ⊗ c_{j}. Then since c_{i} = ∑ t_{ij}ϵ(c_{j}), the t_{ij} span C and it is easily checked that Δ(t_{ij}) = ∑ t_{ik} ⊗t_{kj}. Since l_{h}(c_{i}) = ∑ h^{−1}(t_{ij})c_{j} for all h ∈ H and the c_{i} are weight vectors, we must have that h(t_{ij}) = 0 for i ≠ j. This implies that

l_{h}(t_{ij}) = h^{−1}(t_{ii})t_{ij}, r_{h}(t_{ij}) = h(t_{jj})t_{ij}

and that the map λ_{i}(h) = h(t_{ii}) is a character of H. Thus t_{ij} ∈ A_{−λ_{i},λ_{j}} and hence

Δ(t_{ij}) = ∑ t_{ik} ⊗ t_{kj} ∈ ∑ A_{−λ_{i},λ_{k}} ⊗ A_{−λ_{k},λ_{j}}.

This gives the required condition on Δ. If λ+μ ≠ 0 then there exists an h ∈ H such that ⟨−λ, h⟩ ≠ ⟨μ, h⟩. Let x ∈ A_{λ,μ}. Then

⟨μ, h⟩ϵ(x) = ϵ(r_{h}(x)) = h(x) = ϵ(l_{h^{−1}}(x)) = ⟨−λ, h⟩ϵ(x).

Hence ϵ(x) = 0. The assertion on S follows similarly.

Remark . In particular, if G is any algebraic group and H is a diagonalizable subgroup with character group L, then we may deform the Hopf algebra 𝕂[G] using an antisymmetric bicharacter on L. Such deformations are algebraic analogs of the deformations studied by Rieffel in [27].

2.3. Deformations of dual pairs. Let A and U be a dual pair of Hopf algebras. That is, there exists a bilinear pairing ⟨ | ⟩ : A × U → 𝕂 such that:

1. ⟨a | 1⟩ = ϵ(a) ; ⟨1 | u⟩ = ϵ(u)

2. ⟨a | u_{1}u_{2}⟩ = ∑⟨a_{(1)} | u_{1}⟩⟨a_{(2)} | u_{2}⟩

3. ⟨a_{1}a_{2} | u⟩ = ∑⟨a_{1} | u_{(1)}⟩⟨a_{2} | u_{(2)}⟩

4. ⟨S(a) | u⟩ = ⟨a | S(u)⟩.

Assume that A is bigraded by L, U is bigraded by an abelian group Q and that there is a homomorphism ˘ : Q → L such that

⟨A_{λ,μ} | U_{γ,δ}⟩ ≠ 0 only if λ + μ = γ̆ + δ̆. (2.2)

In this case we will call the pair {A, U} an L-bigraded dual pair. We shall be interested in §3 and §4 in the case where Q = L and ˘ = Id.

Remark . Suppose that the bigradings above are induced from subgroups H and H̆ of R(A) and R(U) respectively and that the map Q → L is induced from a map h ↦ h̆ from H to H̆. Then the condition on the pairing may be restated as the fact that the form is ad-invariant in the sense that for all a ∈ A, u ∈ U and h ∈ H,

⟨ad_{h} a | u⟩ = ⟨a | adh̆ u⟩

where ad_{h} a = r_{h}l_{h}(a).

Theorem 2.4. Let {A, U} be the bigraded dual pair as described above. Let p be an antisymmetric bicharacter on L and let p̆ be the induced bicharacter on Q. Define a bilinear form ⟨ | ⟩_{p} : A_{p^{−1}} × Up̆ → 𝕂 by:

⟨a_{λ,μ} | u_{γ,δ}⟩_{p} = p(λ, γ̆)^{−1}p(μ, δ̆)^{−1}⟨a_{λ,μ} | u_{γ,δ}⟩.

Then ⟨ | ⟩_{p} is a Hopf pairing and {A_{p^{−1}}, Up̆} is an L-bigraded dual pair.

Proof. Let a ∈ A_{λ,μ} and let u_{i} ∈ U_{γ_{i},δ_{i}}, i = 1, 2. Then

⟨a | u_{1}u_{2}⟩_{p} = p(γ̆_{1}, γ̆_{2})p(δ̆_{1}, δ̆_{2})^{−1}p(λ, γ̆_{1} + γ̆_{2})^{−1}p(μ, δ̆_{1} + δ̆_{2})^{−1}⟨a | u_{1}u_{2}⟩.

Suppose that Δ(a) = ∑_{ν} a_{λ,ν} ⊗ a_{−ν,μ}. Then by the assumption on the pairing, the only possible value of ν for which ⟨a_{λ,ν} | u_{1}⟩⟨a_{−ν,μ} | u_{2}⟩ is non-zero is ν = γ̆_{1} + δ̆_{1} − λ = μ − γ̆_{2} − δ̆_{2}. Therefore

⟨a_{(1)} | u_{1}⟩_{p}⟨a_{(2)} | u_{2}⟩_{p} = p(λ, γ̆_{1})^{−1}p(ν, δ̆_{1})^{−1}p(−ν, γ̆_{2})^{−1}p(μ, δ̆_{2})^{−1}⟨a_{(1)} | u_{1}⟩⟨a_{(2)} | u_{2}⟩

= p(λ, γ̆_{1})^{−1}p(μ − γ̆_{2} − δ̆_{2}, δ̆_{1})^{−1}p(λ − γ̆_{1} − δ̆_{1}, γ̆_{2})^{−1}p(μ, δ̆_{2})^{−1}⟨a_{(1)} | u_{1}⟩⟨a_{(2)} | u_{2}⟩

= p(γ̆_{1}, γ̆_{2})p(δ̆_{1}, δ̆_{2})^{−1}p(λ, γ̆_{1} + γ̆_{2})^{−1}p(μ, δ̆_{1} + δ̆_{2})^{−1}⟨a | u_{1}u_{2}⟩ = ⟨a | u_{1}u_{2}⟩_{p}.

This proves the first axiom. The others are verified similarly.

Corollary 2.5. Let {A, U, p} be as in Theorem 2.4. Let M be a right A-comodule with structure map ρ : M → M ⊗ A. Then M is naturally endowed with U and Up̆ left module structures, denoted by (u, x) ↦ ux and (u, x) ↦ u · x respectively. Assume that M = ⊕_{λ∈L}M_{λ} for some 𝕂-subspaces such that ρ(M_{λ}) ⊂ ∑_{ν} M_{−ν} ⊗ A_{ν,λ}. Then we have U_{γ,δ}M_{λ} ⊂ M_{λ−γ̆−δ̆} and the two structures are related by

∀u ∈ U_{γ,δ}, ∀x ∈ M_{λ}, u · x = p(λ, γ̆ − δ̆)p(γ̆, δ̆)ux.

Proof. Notice that the coalgebras A and A_{p^{−1}} are the same. Set ρ(x) = ∑ x_{(0)} ⊗ x_{(1)} for all x ∈ M. Then it is easily checked that the following formulas define the desired U and Up̆ module structures:

∀u ∈ U, ux = ∑ x_{(0)}⟨x_{(1)} | u⟩, u · x = ∑ x_{(0)}⟨x_{(1)} | u⟩_{p}.

When x ∈ M_{λ} and u ∈ U_{γ,δ} the additional condition yields

u · x = ∑ x_{(0)}p(ν, −γ̆)p(λ, −δ̆)⟨x_{(1)} | u⟩.

But ⟨x_{(1)} | u⟩ ≠ 0 forces −ν = λ − γ̆ − δ̆, hence u · x = p(λ, γ̆ − δ̆)p(γ̆, δ̆) ∑ x_{(0)}⟨x_{(1)} | u⟩ = p(λ, γ̆ − δ̆)p(γ̆, δ̆)ux.

Denote by A^{op} the opposite algebra of the 𝕂-algebra A. Let {A^{op}, U, ⟨ | ⟩} be a dual pair of Hopf algebras. The double A ⋊ ⋉ U is defined as follows, [10, 3.3]. Let I be the ideal of the tensor algebra T (A ⊗ U) generated by elements of type

1 − 1_{A}, 1 − 1_{U} (a)

xx′ − x ⊗ x′, x, x′ ∈ A, yy′ − y ⊗ y′, y, y′ ∈ U (b)

x_{(1)} ⊗ y_{(1)}⟨x_{(2)} | y_{(2)}⟩ − ⟨x_{(1)} | y_{(1)}⟩y_{(2)} ⊗ x_{(2)}, x ∈ A, y ∈ U (c)

Then the algebra A ⋊ ⋉ U := T (A ⊗ U)/I is called the Drinfeld double of {A, U}. It is a Hopf algebra in a natural way:

Δ(a ⊗ u) = (a_{(1)} ⊗ u_{(1)}) ⊗ (a_{(2)} ⊗ u_{(2)}),
ϵ(a ⊗ u) = ϵ(a)ϵ(u), S(a ⊗ u) = (S(a) ⊗ 1)(1 ⊗ S(u)).

Notice for further use that A ⋊ ⋉ U can equally be defined by relations of type (a), (b), (c_{x,y}) or (a), (b), (c_{y,x}), where we set

(c_{x,y})  x ⊗ y = ⟨x_{(1)} | y_{(1)}⟩⟨x_{(3)} | S(y_{(3)})⟩y_{(2)} ⊗ x_{(2)}, x ∈ A, y ∈ U
(c_{y,x})  y ⊗ x = ⟨x_{(1)} | S(y_{(1)})⟩⟨x_{(3)} | y_{(3)}⟩x_{(2)} ⊗ y_{(2)}, x ∈ A, y ∈ U

Theorem 2.6. Let {A^{op}, U} be an L-bigraded dual pair, p be an antisymmetric bicharacter on L and p̆ be the induced bicharacter on Q. Then A ⋊ ⋉ U inherits an L-bigrading and there is a natural isomorphism of L-bigraded Hopf algebras:

(A ⋊ ⋉ U)_{p} ≅ A_{p} ⋊ ⋉ Up̆.

Proof. Recall that as a 𝕂-vector space A ⋊ ⋉ U identifies with A ⊗ U. Define an L-bigrading on A ⋊ ⋉ U by

∀α, β ∈ L, (A ⋊ ⋉ U)_{α,β} = ∑_{λ−γ̆=α,μ−δ̆=β} A_{λ,μ} ⊗ U_{γ,δ}.

To verify that this yields a structure of graded algebra on A ⋊ ⋉ U it suffices to check that the defining relations of A ⋊ ⋉ U are homogeneous. This is clear for relations of type (a) or (b). Let x_{λ,μ} ∈ A_{λ,μ} and y_{γ,δ} ∈ U_{γ,δ}. Then the corresponding relation of type (c) becomes

∑_{ν,ξ} x_{λ,ν}y_{γ,ξ}⟨x_{−ν,μ} | y_{−ξ,δ}⟩ − ⟨x_{λ,μ} | y_{γ,ξ}⟩y_{−ξ,δ}x_{−ν,μ}. (⋆)

When a term of this sum is non-zero we obtain −ν+μ = −ξ̆+δ̆, λ+ν = γ̆+ξ̆. Hence λ−γ̆ = −ν+ξ̆ = −μ+δ̆, which shows that the relation (⋆) is homogeneous. It is easy to see that the conditions (2), (3), (4) of 2.1 hold. Hence A ⋊ ⋉ U is an L-bigraded Hopf algebra.

Notice that (A_{p})^{op} ≅ (A^{op})_{p^{−1}}, so that Theorem 2.4 defines a suitable pairing between (A_{p})^{op} and Up̆. Thus A_{p} ⋊ ⋉ Up̆ is defined. Let ϕ be the natural surjective homomorphism from T (A ⊗ U) onto A_{p} ⋊ ⋉ Up̆. To check that ϕ induces an isomorphism it again suffices to check that ϕ vanishes on the defining relations of (A ⋊ ⋉ U)_{p}. Again, this is easy for relations of type (a) and (b). The relation (⋆) says that

p(λ, γ̆)p(−ν, ξ̆)⟨x_{−ν,μ} | y_{−ξ,δ}⟩x_{λ,ν} · y_{γ,ξ} − p(ξ̆, ν)p(δ̆, −μ)⟨x_{λ,μ} | y_{γ,ξ}⟩y_{−ξ,δ} · x_{−ν,μ} = 0

in (A ⋊ ⋉ U)_{p}. Multiply the left hand side of this equation by p(λ, −γ̆)p(μ, −δ̆) and apply ϕ. We obtain the following expression in A_{p} ⋊ ⋉ Up̆:

p(−ν, ξ̆)p(μ, −δ̆)⟨x_{−ν,μ} | y_{−ξ,δ}⟩x_{λ,ν}y_{γ,ξ} − p(λ, −γ̆)p(ν, −ξ̆)⟨x_{λ,μ} | y_{γ,ξ}⟩y_{−ξ,δ}x_{−ν,μ}

which is equal to

⟨x_{−ν,μ} | y_{−ξ,δ}⟩_{p}x_{λ,ν}y_{γ,ξ} − ⟨x_{λ,μ} | y_{γ,ξ}⟩_{p}y_{−ξ,δ}x_{−ν,μ}.

But this is a defining relation of type (c) in A_{p} ⋊ ⋉ Up̆, hence zero.

It remains to see that ϕ induces an isomorphism of Hopf algebras, which is a straightforward consequence of the definitions.

2.4. Cocycles. Let L be, in this section, an arbitrary free abelian group with basis {ω_{1}, …, ω_{n}} and set 𝔥^{∗} = ℂ ⊗_{ℤ} L. We freely use the terminology of [2]. Recall that H^{2}(L, ℂ^{∗}) is in bijection with the set H of multiplicatively antisymmetric n × n-matrices γ = [γ_{ij}]. This bijection maps the class [c] onto the matrix defined by γ_{ij} = c(ω_{i}, ω_{j})/c(ω_{j}, ω_{i}). Furthermore it is an isomorphism of groups with respect to component-wise multiplication of matrices.

Remark . The notation is as in 2.1. We recalled that the isomorphism class of the algebra A_{p} depends only on the cohomology class [p̃] ∈ H^{2}(L × L, 𝕂^{∗}). Let γ ∈ H be the matrix associated to p and γ^{−1} its inverse in H. Notice that the multiplicative matrix associated to [p̃] is then γ̃ = [ \begin{matrix}γ & 1 \\ 1 & γ^{−1} \end{matrix} ] in the basis given by the (ω_{i}, 0), (0, ω_{i}) ∈ L × L. Therefore the isomorphism class of the algebra A_{p} depends only on the cohomology class [p] ∈ H^{2}(L, 𝕂^{∗}).

Let ℏ ∈ ℂ^{∗}. If x ∈ ℂ we set q^{x} = exp(−xℏ/2). In particular q = exp(−ℏ/2). Let u : L × L → ℂ be a complex alternating ℤ-bilinear form. Define

p : L × L → ℂ^{∗}, p(λ, μ) = exp ( − \frac{ℏ}{4}u(λ, μ)) = q^{\frac{1}{2} u(λ,μ)}. (2.3)

Then it is clear that p is an antisymmetric bicharacter on L.

Observe that, since 𝔥^{∗} = ℂ ⊗_{ℤ} L, there is a natural isomorphism of additive groups between ∧^{2}𝔥 and the group of complex alternating ℤ-bilinear forms on L, where 𝔥 is the ℂ-dual of 𝔥^{∗}. Set Z_{ℏ} = {u ∈ ∧^{2}𝔥 | u(L × L) ⊂ \frac{4iπ}{ℏ} ℤ\}.

Theorem 2.7. There are isomorphisms of abelian groups:

H^{2}(L, ℂ^{∗}) ≅ H ≅ ∧^{2}𝔥/Z_{ℏ}.

Proof. The first isomorphism has been described above. Let γ = [γ_{ij}] ∈ H and choose u_{ij}, 1 ≤ i < j ≤ n such that γ_{ij} = exp(−\frac{ℏ}{2}u_{ij}). We can define u ∈ ∧^{2}𝔥 by setting u(ω_{i}, ω_{j}) = u_{ij}, 1 ≤ i < j ≤ n. It is then easily seen that one can define an injective morphism of abelian groups

φ : H^{2}(L, ℂ^{∗}) ≅ H ⟶ ∧^{2}𝔥/Z_{ℏ}, φ(γ) = [u]

where [u] is the class of u. If u ∈ ∧^{2}𝔥, define a 2-cocycle p by the formula (2.3). Then the multiplicative matrix associated to [p] ∈ H^{2}(L, ℂ^{∗}) is given by

γ_{ij} = p(ω_{i}, ω_{j})/p(ω_{j}, ω_{i}) = p(ω_{i}, ω_{j})^{2} = exp(−\frac{ℏ}{2}u(ω_{i}, ω_{j})).

This shows that [u] = φ([γ_{ij}]); thus φ is an isomorphism.

We list some consequences of Theorem 2.7. We denote by [u] an element of ∧^{2}𝔥/Z_{ℏ} and we set [p] = φ^{−1}([u]). We have seen that we can define a representative p by the formula (2.3).

1. [p] of finite order in H^{2}(L, ℂ^{∗}) ⇔ u(L × L) ⊂ \frac{iπ}{ℏ} ℚ, and q root of unity ⇔ ℏ ∈ iπℚ.

2. Notice that u = 0 is algebraic, whether q is a root of unity or not. Assume that q is a root of unity; then we get from 1 that

[p] of finite order ⇔ u is algebraic.

3. Assume that q is not a root of unity and that u ≠ 0. Then [p] of finite order implies (0) ≠ u(L×L) ⊂ \frac{iπ}{ℏ} ℚ. This shows that

0 ≠ u algebraic ⇒ [p] is not of finite order.

Definition . The bicharacter p : (λ, μ) ↦ q^{\frac{1}{2} u(λ,μ)} is called q-rational if u ∈ ∧^{2}𝔥 is algebraic.

The following technical result will be used in the next section. Recall the definition of Φ_{−} = Φ − I given in the Section 1.

Proposition 2.8. Let K = \{λ ∈ L : (Φ_{−}λ, L) ⊂ \frac{4iπ}{ℏ} ℤ\}. If q is not a root of unity, then K = 0.

Proof. Let λ ∈ K. We can define z : 𝔥_{ℚ}^{∗} → ℚ, by

∀ μ ∈ 𝔥_{ℚ}^{∗}, (Φ_{−}λ, μ) = \frac{4iπ}{ℏ} z(μ).

The map z is clearly ℚ-linear. It follows, since ( , ) is non-degenerate on 𝔥_{ℚ}^{∗}, that there exists ν ∈ 𝔥_{ℚ}^{∗} such that z(μ) = (ν, μ) for all μ ∈ 𝔥_{ℚ}^{∗}. Therefore Φ_{−}λ = \frac{4iπ}{ℏ} ν, and so Φλ = λ + \frac{4iπ}{ℏ} ν.

Now, (Φλ, λ) = u(λ, λ) = 0 implies that

\frac{4iπ}{ℏ} (ν, λ) = −(λ, λ)

If (λ, λ) ≠ 0 then ℏ ∈ iπℚ, contradicting the assumption that q is not a root of unity. Hence (λ, λ) = 0, which forces λ = 0 since λ ∈ L ⊂ 𝔥_{ℚ}^{∗}.

3. Multiparameter Quantum Groups

3.1. One-parameter quantized enveloping algebras. The notation is as in sections 1 and 2. In particular we fix a lattice L such that Q ⊂ L ⊂ P and we denote by G the connected semi-simple algebraic group with maximal torus H such that Lie(G) = 𝔤 and X(H) ≅ L.

Let q ∈ ℂ^{∗} and assume that q is not a root of unity. Let ℏ ∈ ℂ \ iπℚ such that q = exp(−ℏ/2) as in 2.4. We set

q_{i} = q^{d_{i}}, q̂_{i} = (q_{i} − q_{i}^{−1})^{−1}, 1 ≤ i ≤ n.

Denote by U^{0} the group algebra of X(H), hence

U^{0} = ℂ[k_{λ}; λ ∈ L], k_{0} = 1, k_{λ}k_{μ} = k_{λ+μ}.

Set k_{i} = k_{α_{i}}, 1 ≤ i ≤ n. The one parameter quantized enveloping algebra associated to this data, cf. [33], is the Hopf algebra

U_{q}(𝔤) = U^{0}[e_{i}, f_{i}; 1 ≤ i ≤ n]

with defining relations:

k_{λ}e_{j}k_{λ}^{−1} = q^{(λ,α_{j})}e_{j}, k_{λ}f_{j}k_{λ}^{−1} = q^{−(λ,α_{j})}f_{j}

e_{i}f_{j} − f_{j}e_{i} = δ_{ij}q̂_{i}(k_{i} − k_{i}^{−1})

∑_{k=0}^{1−a_{ij}}(−1)^{k} [ \begin{matrix}1−a_{ij} \\ k \end{matrix} ]_{q_{i}} e_{i}^{1−a_{ij}−k}e_{j}e_{i}^{k} = 0, if i ≠ j

∑_{k=0}^{1−a_{ij}}(−1)^{k} [ \begin{matrix}1−a_{ij} \\ k \end{matrix} ]_{q_{i}} f_{i}^{1−a_{ij}−k}f_{j}f_{i}^{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}} . The Hopf algebra structure is given by

Δ(k_{λ}) = k_{λ} ⊗ k_{λ}, ϵ(k_{λ}) = 1, S(k_{λ}) = k_{λ}^{−1}
Δ(e_{i}) = e_{i} ⊗ 1 + k_{i} ⊗ e_{i}, Δ(f_{i}) = f_{i} ⊗ k_{i}^{−1} + 1 ⊗ f_{i}
ϵ(e_{i}) = ϵ(f_{i}) = 0, S(e_{i}) = −k_{i}^{−1}e_{i}, S(f_{i}) = −f_{i}k_{i}.

We define subalgebras of U_{q}(𝔤) as follows

U_{q}(𝔫^{+}) = ℂ[e_{i}, ; 1 ≤ i ≤ n], U_{q}(𝔫^{−}) = ℂ[f_{i}, ; 1 ≤ i ≤ n]
U_{q}(𝔟^{+}) = U^{0}[e_{i}, ; 1 ≤ i ≤ n], U_{q}(𝔟^{−}) = U^{0}[f_{i}, ; 1 ≤ i ≤ n].

For simplicity we shall set U^{±} = U_{q}(𝔫^{±}). Notice that U^{0} and U_{q}(𝔟^{±}) are Hopf subalgebras of U_{q}(𝔤). Recall [23] that the multiplication in U_{q}(𝔤) induces isomorphisms of vector spaces

U_{q}(𝔤) ≅ U^{−} ⊗ U^{0} ⊗ U^{+} ≅ U^{+} ⊗ U^{0} ⊗ U^{−}.

Set Q_{+} = ⊕_{i=1}^{n}ℕα_{i} and

∀β ∈ Q_{+}, U_{β}^{±} = {u ∈ U^{±} | ∀λ ∈ L, k_{λ}uk_{λ}^{−1} = q^{(λ,±β)}u}.

Then one gets: U^{±} = ⊕_{β∈Q_{+}}U_{±β}^{±}.

3.2. The Rosso-Tanisaki-Killing form. Recall the following result, [28, 33].

Theorem 3.1. 1. There exists a unique non degenerate Hopf pairing

⟨ | ⟩ : U_{q}(𝔟^{+})^{op} ⊗ U_{q}(𝔟^{−}) ⟶ ℂ

satisfying the following conditions:

(i) ⟨k_{λ} | k_{μ}⟩ = q^{−(λ,μ)};

(ii) ∀λ ∈ L, 1 ≤ i ≤ n, ⟨k_{λ} | f_{i}⟩ = ⟨e_{i} | k_{λ}⟩ = 0;

(iii) ∀1 ≤ i, j ≤ n, ⟨e_{i} | f_{j}⟩ = −δ_{ij}q̂_{i}.

2. If γ, η ∈ Q_{+}, ⟨U_{γ}^{+} | U_{−η}^{−}⟩ ≠ 0 implies γ = η.

The results of §2.3 then apply and we may define the associated double:

D_{q}(𝔤) = U_{q}(𝔟^{+}) ⋊ ⋉ U_{q}(𝔟^{−}).

It is well known, e.g. [10], that

D_{q}(𝔤) = ℂ[s_{λ}, t_{λ}, e_{i}, f_{i}; λ ∈ L, 1 ≤ i ≤ n]

where s_{λ} = k_{λ} ⊗ 1, t_{λ} = 1 ⊗ k_{λ}, e_{i} = e_{i} ⊗ 1, f_{i} = 1 ⊗ f_{i}. The defining relations of the double given in §2.3 imply that

s_{λ}t_{μ} = t_{μ}s_{λ}, e_{i}f_{j} − f_{j}e_{i} = δ_{ij}q̂_{i}(s_{α_{i}} − t_{α_{i}}^{−1})
s_{λ}e_{j}s_{λ}^{−1} = q^{(λ,α_{j})}e_{j}, t_{λ}e_{j}t_{λ}^{−1} = q^{(λ,α_{j})}e_{j}, s_{λ}f_{j}s_{λ}^{−1} = q^{−(λ,α_{j})}f_{j}, t_{λ}f_{j}t_{λ}^{−1} = q^{−(λ,α_{j})}f_{j}.

It follows that

D_{q}(𝔤)/(s_{λ} − t_{λ}; λ ∈ L) →^{∼} U_{q}(𝔤), e_{i} ↦ e_{i}, f_{i} ↦ f_{i}, s_{λ} ↦ k_{λ}, t_{λ} ↦ k_{λ}.

Observe that this yields an isomorphism of Hopf algebras. The next proposition collects some well known elementary facts.

Proposition 3.2. 1. Any finite dimensional simple U_{q}(𝔟^{±})-module is one dimensional and R(U_{q}(𝔟^{±})) identifies with H via

∀h ∈ H, h(k_{λ}) = ⟨λ, h⟩, h(e_{i}) = 0, h(f_{i}) = 0.

2. R(D_{q}(𝔤)) identifies with H via

∀h ∈ H, h(s_{λ}) = ⟨λ, h⟩, h(t_{λ}) = ⟨λ, h⟩^{−1}, h(e_{i}) = h(f_{i}) = 0.

Corollary 3.3. 1. {U_{q}(𝔟^{+})^{op}, U_{q}(𝔟^{−})} is an L-bigraded dual pair. We have

k_{λ} ∈ U_{q}(𝔟^{±})_{−λ,λ}, e_{i} ∈ U_{q}(𝔟^{+})_{−α_{i},0}, f_{i} ∈ U_{q}(𝔟^{−})_{0,−α_{i}}.

2. D_{q}(𝔤) is an L-bigraded Hopf algebra where

s_{λ} ∈ D_{q}(𝔤)_{−λ,λ}, t_{λ} ∈ D_{q}(𝔤)_{λ,−λ}, e_{i} ∈ D_{q}(𝔤)_{−α_{i},0}, f_{i} ∈ D_{q}(𝔤)_{0,α_{i}}.

Proof. 1. Observe that for all h ∈ H,

l_{h}(k_{λ}) = h^{−1}(k_{λ}) = ⟨−λ, h⟩k_{λ}, r_{h}(k_{λ}) = h(k_{λ}) = ⟨λ, h⟩k_{λ},
l_{h}(e_{i}) = h^{−1}(k_{i})e_{i} = ⟨−α_{i}, h⟩e_{i}, r_{h}(e_{i}) = e_{i},
l_{h}(f_{i}) = f_{i}, r_{h}(f_{i}) = h(k_{i}^{−1})f_{i} = ⟨−α_{i}, h⟩f_{i}.

It is then clear that U_{−γ,0}^{+} = U_{γ}^{+} and U_{0,−γ}^{−} = U_{−γ}^{−} for all γ ∈ Q_{+}. The claims then follow from these formulas, Theorem 2.3, Theorem 3.1, and the definitions.

2. The fact that D_{q}(𝔤) is an L-bigraded Hopf algebra follows from Theorem 2.3. The assertions about the L × L degree of the generators is proved by direct computation using Proposition 3.2.

Remark . We have shown in Theorem 2.6 that, as a double, D_{q}(𝔤) inherits an L-bigrading given by:

D_{q}(𝔤)_{α,β} = ∑_{λ−γ=α,μ−δ=β} U_{q}(𝔟^{+})_{λ,μ} ⊗ U_{q}(𝔟^{−})_{γ,δ}.

It is easily checked that this bigrading coincides with the bigrading obtained in the above corollary by means of Theorem 2.3.

3.3. One-parameter quantized function algebras. Let M be a left D_{q}(𝔤)-module. The dual M^{∗} will be considered in the usual way as a left D_{q}(𝔤)-module by the rule: (uf)(x) = f(S(u)x), x ∈ M, f ∈ M^{∗}, u ∈ D_{q}(𝔤). Assume that M is an U_{q}(𝔤)-module. An element x ∈ M is said to have weight μ ∈ L if k_{λ}x = q^{(λ,μ)}x for all λ ∈ L; we denote by M_{μ} the subspace of elements of weight μ.

It is known, [13], that the category of finite dimensional (left) U_{q}(𝔤)-modules is a completely reducible braided rigid monoidal category. Set L^{+} = L ∩ P^{+} and recall that for each Λ ∈ L^{+} there exists a finite dimensional simple module of highest weight Λ, denoted by L(Λ), cf. [29] for instance. One has L(Λ)^{∗} ≅ L(w_{0}Λ) where w_{0} is the longest element of W. (Notice that the results quoted usually cover the case where L = Q. One defines the modules L(λ) in the general case in the following way. Let us denote temporarily the algebra U_{q}(𝔤) for a given choice of L by U_{q,L}(𝔤). Given a module L(λ) defined on U_{q,Q}(𝔤) we may define an action of U_{q,L}(𝔤) by setting k_{λ}.x = q^{(λ,μ)}x for all elements x of weight μ, where q^{(λ,μ)} is as defined in section 2.4.)

Let C_{q} be the subcategory of finite dimensional U_{q}(𝔤)-modules consisting of finite direct sums of L(Λ), Λ ∈ L^{+}. The category C_{q} is closed under tensor products and the formation of duals. Notice that C_{q} can be considered as a braided rigid monoidal category of D_{q}(𝔤)-modules where s_{λ}, t_{λ} act as k_{λ} on an object of C_{q}.

Let M ∈ obj(C_{q}), then M = ⊕_{μ∈L}M_{μ}. For f ∈ M^{∗}, v ∈ M we define the coordinate function c_{f,v} ∈ U_{q}(𝔤)^{∗} by

∀u ∈ U_{q}(𝔤), c_{f,v}(u) = ⟨f, uv⟩

where ⟨ , ⟩ is the duality pairing. Using the standard isomorphism (M ⊗ N)^{∗} ≅ N^{∗} ⊗ M^{∗} one has the following formula for multiplication,

c_{f,v}c_{f′,v′} = c_{f′⊗f,v⊗v′}.

Definition . The quantized function algebra ℂ_{q}[G] is the restricted dual of C_{q}: that is to say

ℂ_{q}[G] = ℂ[c_{f,v}; v ∈ M, f ∈ M^{∗}, M ∈ obj(C_{q})].

The algebra ℂ_{q}[G] is a Hopf algebra; we denote by Δ, ϵ, S the comultiplication, counit and antipode on ℂ_{q}[G]. If {v_{1}, …, v_{s}; f_{1}, …, f_{s}} is a dual basis for M ∈ obj(C_{q}) one has

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}. (3.1)

Notice that we may assume that v_{j} ∈ M_{ν_{j}}, f_{j} ∈ M_{−ν_{j}}^{∗}. We set

C(M) = ℂ⟨c_{f,v}; f ∈ M^{∗}, v ∈ M⟩, C(M)_{λ,μ} = ℂ⟨c_{f,v}; f ∈ M_{λ}^{∗}, v ∈ M_{μ}⟩.

Then C(M) is a subcoalgebra of ℂ_{q}[G] such that C(M) = ⨁_{(λ,μ)∈L×L} C(M)_{λ,μ}. When M = L(Λ) we abbreviate the notation to C(M) = C(Λ). It is then classical that

ℂ_{q}[G] = ⨁_{Λ∈L^{+}} C(Λ).

Since ℂ_{q}[G] ⊂ U_{q}(𝔤)^{∗} we have a duality pairing

⟨ , ⟩ : ℂ_{q}[G] × D_{q}(𝔤) ⟶ ℂ.

Observe that there is a natural injective morphism of algebraic groups

H ⟶ R(ℂ_{q}[G]), h(c_{f,v}) = ⟨μ, h⟩ϵ(c_{f,v}) for all v ∈ M_{μ}, M ∈ obj(C_{q}).

The associated automorphisms r_{h}, l_{h} ∈ Aut(ℂ_{q}[G]) are then described by

∀c_{f,v} ∈ C(M)_{λ,μ}, r_{h}(c_{f,v}) = ⟨μ, h⟩c_{f,v}, l_{h}(c_{f,v}) = ⟨λ, h⟩c_{f,v}.

Define

∀(λ, μ) ∈ L × L, ℂ_{q}[G]_{λ,μ} = {a ∈ ℂ_{q}[G] | r_{h}(a) = ⟨μ, h⟩a, l_{h}(a) = ⟨λ, h⟩a}.

Theorem 3.4. The pair of Hopf algebras {ℂ_{q}[G], D_{q}(𝔤)} is an L-bigraded dual pair.

Proof. It follows from (3.1) that ℂ_{q}[G] is an L-bigraded Hopf algebra. The axioms (1) to (4) of 2.3 are satisfied by definition of the Hopf algebra ℂ_{q}[G]. We take ˘ to be the identity map of L. The condition (2.2) is consequence of D_{q}(𝔤)_{γ,δ}M_{μ} ⊂ M_{μ−γ−δ} for all M ∈ C_{q}. To verify this inclusion, notice that

e_{j} ∈ D_{q}(𝔤)_{−α_{j},0}, f_{j} ∈ D_{q}(𝔤)_{0,α_{j}}, e_{j}M_{μ} ⊂ M_{μ+α_{j}}, f_{j}M_{μ} ⊂ M_{μ−α_{j}}.

The result then follows easily

Consider the algebras D_{q^{−1}}(𝔤) and ℂ_{q^{−1}}[G] and use ˆ to distinguish elements, subalgebras, etc. of D_{q^{−1}}(𝔤) and ℂ_{q^{−1}}[G]. It is easily verified that the map σ : D_{q}(𝔤) → D_{q^{−1}}(𝔤) given by

s_{λ} ↦ ŝ_{λ}, t_{λ} ↦ t̂_{λ}, e_{i} ↦ q_{i}^{1/2}f̂_{i}t̂_{α_{i}}, f_{i} ↦ q_{i}^{1/2}ê_{i}ŝ_{α_{i}}^{−1}

is an isomorphism of Hopf algebras.

For each Λ ∈ L^{+}, 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 we obtain 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}. (3.2)

Notice that

σ(D_{q}(𝔤)_{γ,δ}) = D_{q^{−1}}(𝔤)_{−γ,−δ} and σ(ℂ_{q^{−1}}[G]_{λ,μ}) = ℂ_{q}[G]_{−λ,−μ}. (3.3)

3.4. Deformation of one-parameter quantum groups. We continue with the same notation. Let [p] ∈ H^{2}(L, ℂ^{∗}). As seen in §2.4 we can, and we do, choose p to be an antisymmetric bicharacter such that

∀λ, μ ∈ L, p(λ, μ) = q^{\frac{1}{2} u(λ,μ)}

for some u ∈ ∧^{2}𝔥. Recall that p̃ ∈ Z^{2}(L × L, ℂ^{∗}), cf. 2.1.

We now apply the results of §2.1 to D_{q}(𝔤) and ℂ_{q}[G]. Using Theorem 2.1 we can twist D_{q}(𝔤) by p̃^{−1} and ℂ_{q}[G] by p̃. The resulting L-bigraded Hopf algebras will be denoted by D_{q,p^{−1}}(𝔤) and ℂ_{q,p}[G]. The algebra ℂ_{q,p}[G] will be referred to as the multi-parameter quantized function algebra. Versions of D_{q,p^{−1}}(𝔤) are referred to by some authors as the multi-parameter quantized enveloping algebra. Alternatively, this name can be applied to the quotient of D_{q,p^{−1}}(𝔤) by the radical of the pairing with ℂ_{q,p}[G].

Theorem 3.5. Let U_{q,p^{−1}}(𝔟^{+}) and U_{q,p^{−1}}(𝔟^{−}) be the deformations by p^{−1} of U_{q}(𝔟^{+}) and U_{q}(𝔟^{−}) respectively. Then the deformed pairing

⟨ | ⟩_{p^{−1}} : U_{q,p^{−1}}(𝔟^{+})^{op} ⊗ U_{q,p^{−1}}(𝔟^{−}) → ℂ

is a non-degenerate Hopf pairing satisfying:

∀ x ∈ U^{+}, y ∈ U^{−}, λ, μ ∈ L, ⟨x · k_{λ} | y · k_{μ}⟩_{p^{−1}} = q^{(Φ_{−}λ,μ)}⟨x | y⟩. (3.4)

Moreover,

U_{q,p^{−1}}(𝔟^{+}) ⋊ ⋉ U_{q,p^{−1}}(𝔟^{−}) ≅ (U_{q}(𝔟^{+}) ⋊ ⋉ U_{q}(𝔟^{−}))_{p^{−1}} = D_{q,p^{−1}}(𝔤).

Proof. By Theorem 2.4 the deformed pairing is given by

⟨a_{λ,μ} | u_{γ,δ}⟩_{p^{−1}} = p(λ, γ)p(μ, δ)⟨a_{λ,μ} | u_{γ,δ}⟩.

To prove (3.4) we can assume that x ∈ U_{−γ,0}^{+}, y ∈ U_{0,−ν}^{−}. Then we obtain

⟨x · k_{λ} | y · k_{μ}⟩_{p^{−1}} = p(λ + γ, μ)p(λ, μ − ν)⟨x · k_{λ} | y · k_{μ}⟩
= p(λ, 2μ)p(λ − μ, γ − ν)q^{−(λ,μ)}⟨x | y⟩

by the definition of the product · and [33, 2.1.3]. But ⟨x | y⟩ = 0 unless γ = ν, hence the result. Observe in particular that ⟨x | y⟩_{p^{−1}} = ⟨x | y⟩. Therefore [33, 2.1.4] shows that ⟨ | ⟩_{p^{−1}} is non-degenerate on U_{γ}^{+} × U_{−γ}^{−}. It then follows from (3.4) and Proposition 2.8 that ⟨ | ⟩_{p^{−1}} is non-degenerate. The remaining isomorphism follows from Theorem 2.6.

Many authors have defined multi-parameter quantized enveloping algebras. In [14, 25] a definition is given using explicit generators and relations, and in [1] the construction is made by twisting the comultiplication, following [26]. It can be easily verified that these algebras and the algebras D_{q,p^{−1}}(𝔤) coincide. The construction of a multi-parameter quantized function algebra by twisting the multiplication was first performed in the GL(n)-case in [2].

The fact that D_{q,p^{−1}}(𝔤) and ℂ_{q,p}[G] form a Hopf dual pair has already been observed in particular cases, see e.g. [14]. We will now deduce from the previous results that this phenomenon holds for an arbitrary semi-simple group.

Theorem 3.6. {ℂ_{q,p}[G], D_{q,p^{−1}}(𝔤)} is an L-bigraded dual pair. The associated pairing is given by

∀a ∈ ℂ_{q,p}[G]_{λ,μ}, ∀u ∈ D_{q,p^{−1}}(𝔤)_{γ,δ}, ⟨a, u⟩_{p} = p(λ, γ)p(μ, δ)⟨a, u⟩.

Proof. This follows from Theorem 2.4 applied to the pair {A, U} = {ℂ_{q}[G], D_{q}(𝔤)} and the bicharacter p^{−1} (recall that the map ˘ is the identity).

Let M ∈ obj(C_{q}). The left D_{q}(𝔤)-module structure on M yields a right ℂ_{q}[G]-comodule structure in the usual way. Let {v_{1}, …, v_{s}; f_{1}, …, f_{s}} be a dual basis for M. The structure map ρ : M → M ⊗ ℂ_{q}[G], is given by ρ(x) = ∑_{j} v_{j} ⊗ c_{f_{j},x} for x ∈ M. Using this comodule structure on M, one can check that

M_{μ} = {x ∈ M | ∀h ∈ H, r_{h}(x) = ⟨μ, h⟩x}.

Proposition 3.7. Let M ∈ obj(C_{q}). Then M has a natural structure of left D_{q,p^{−1}}(𝔤) module. Denote by M˘ this module and by (u, x) ↦ u · x the action of D_{q,p^{−1}}(𝔤). Then

∀u ∈ D_{q}(𝔤)_{γ,δ}, ∀x ∈ M_{λ}, u · x = p(λ, δ − γ)p(δ, γ)ux.

Proof. The proposition is a translation in this particular setting of Corollary 2.5.

Denote by C_{q,p} the subcategory of finite dimensional left D_{q,p^{−1}}(𝔤)-modules whose objects are the M˘ , M ∈ obj(C_{q}). It follows from Proposition 3.7 that if M ∈ obj(C_{q}), then M˘ = ⊕_{μ∈L}M_{μ̆}, where

M_{μ̆} = {x ∈ M | ∀α ∈ L, s_{α} · x = p(μ, 2α)q^{(μ,α)}x, t_{α} · x = p(μ, −2α)q^{(μ,α)}x}.

Notice that p(μ, ±2α)q^{(μ,α)} = q^{±(Φ_{±}μ,α)}.

Theorem 3.8. 1. The functor M → M˘ from C_{q} to C_{q,p} is an equivalence of rigid monoidal categories.

2. The Hopf pairing ⟨ , ⟩_{p} identifies the Hopf algebra ℂ_{q,p}[G] with the restricted dual of C_{q,p}, i.e. the Hopf algebra of coordinate functions on the objects of C_{q,p}.

Proof. 1. One needs in particular to prove that, for M, N ∈ obj(C_{q}), there are natural isomorphisms of D_{q,p^{−1}}(𝔤)-modules: φ_{M,N} : (M ⊗N)˘ → M˘ ⊗N˘ . These isomorphisms are given by x⊗y ↦ p(λ, μ)x⊗y for all x ∈ M_{λ}, y ∈ N_{μ}. The other verifications are elementary.

2. We have to show that if M ∈ obj(C_{q}), f ∈ M^{∗}, v ∈ M and u ∈ D_{q,p^{−1}}(𝔤), then ⟨c_{f,v}, u⟩_{p} = ⟨f, u · v⟩. It suffices to prove the result in the case where f ∈ M_{λ}^{∗}, v ∈ M_{μ} and u ∈ D_{q,p^{−1}}(𝔤)_{γ,δ}. Then

⟨f, u · v⟩ = p(μ, δ − γ)p(δ, γ)⟨f, uv⟩
= δ_{−λ+γ+δ,μ} p(−λ + γ + δ, δ − γ)p(δ, γ)⟨f, uv⟩
= p(λ, γ)p(μ, δ)⟨f, uv⟩
= ⟨c_{f,v}, u⟩_{p}

by Theorem 3.6.

Recall that we introduced in §3.3 isomorphisms σ : D_{q}(𝔤) → D_{q^{−1}}(𝔤) and σ : ℂ_{q}[G] → ℂ_{q^{−1}}[G]. From (3.3) it follows that, after twisting by p̃^{−1} or p̃, σ induces isomorphisms

D_{q,p^{−1}}(𝔤) →^{∼} D_{q^{−1},p^{−1}}(𝔤), ℂ_{q^{−1},p}[G] →^{∼} ℂ_{q,p}[G]

which satisfy (3.2).

3.5. Braiding isomorphisms. We remarked above that the categories C_{q,p} are braided. In the one parameter case this braiding is well-known. Let M and N be objects of C_{q}. Let E : M ⊗ N → M ⊗ N be the operator given by

E(m ⊗ n) = q^{(λ,μ)}m ⊗ n

for m ∈ M_{λ} and n ∈ N_{μ}. Let τ : M ⊗ N → N ⊗ M be the usual twist operator. Finally let C be the operator given by left multiplication by

C = ∑_{β∈Q_{+}} C_{β}

where C_{β} is the canonical element of D_{q}(𝔤) associated to the non-degenerate pairing U_{β}^{+} ⊗ U_{−β}^{−} → ℂ described above. Then one deduces from [33, 4.3] that the operators

θ_{M,N} = τ ∘ C ∘ E^{−1} : M ⊗ N → N ⊗ M

define the braiding on C_{q}.

As mentioned above, the category C_{q,p} inherits a braiding given by

ψ_{M,N} = φ_{N,M} ∘ θ_{M,N} ∘ φ_{M,N}^{−1}

where φ_{M,N} is the isomorphism (M ⊗ N)˘ →^{∼} M˘ ⊗ N˘ introduced in the proof of Theorem 3.8 (the same formula can be found in [1, §10] and in a more general situation in [24]). We now note that these general operators are of the same form as those in the one parameter case. Let M and N be objects of C_{q,p} and let E : M ⊗ N → M ⊗ N be the operator given by

E(m ⊗ n) = q^{(Φ_{+}λ,μ)}m ⊗ n

for m ∈ M_{λ} and n ∈ N_{μ}. Denote by C_{β} the canonical element of D_{q,p^{−1}}(𝔤) associated to the nondegenerate pairing U_{q,p^{−1}}(𝔟^{+})_{−β,0} ⊗ U_{q,p^{−1}}(𝔟^{−})_{0,−β} → ℂ and let C : M ⊗ N → M ⊗ N be the operator given by left multiplication by

C = ∑_{β∈Q_{+}} C_{β}.

Theorem 3.9. The braiding operators ψ_{M,N} are given by

ψ_{M,N} = τ ∘ C ∘ E^{−1}.

Moreover (ψ_{M,N})^{∗} = ψ_{M^{∗},N^{∗}}.

Proof. The assertions follow easily from the analogous assertions for θ_{M,N}.

The following commutation relations are well known [31], [21, 4.2.2]. We include a proof for completeness.

Corollary 3.10. Let Λ, Λ′ ∈ L^{+}, let g ∈ L(Λ′)_{−η}^{∗} and f ∈ L(Λ)_{−μ}^{∗} and let v_{Λ} ∈ L(Λ)_{Λ}. Then for any v ∈ L(Λ′)_{γ},

c_{g,v} · c_{f,v_{Λ}} = q^{(Φ_{+}Λ,γ)−(Φ_{+}μ,η)}c_{f,v_{Λ}} · c_{g,v} + q^{(Φ_{+}Λ,γ)−(Φ_{+}μ,η)} ∑_{ν∈Q_{+}} c_{f_{ν},v_{Λ}} · c_{g_{ν},v}

where f_{ν} ∈ (U_{q,p^{−1}}(𝔟^{+})f)_{−μ+ν} and g_{ν} ∈ (U_{q,p^{−1}}(𝔟^{−})g)_{−η−ν} are such that ∑ f_{ν} ⊗g_{ν} = ∑_{β∈Q^{+}\{0}} C_{β}(f ⊗ g).

Proof. Let ψ = ψ_{L(Λ),L(Λ′)}. Notice that

c_{f⊗g,ψ(v_{Λ}⊗v)} = c_{ψ^{∗}(f⊗g),v_{Λ}⊗v}.

Using the theorem above we obtain

ψ^{∗}(f ⊗ g) = q^{−(Φ_{+}μ,η)}(g ⊗ f + ∑ g_{ν} ⊗ f_{ν})

and

ψ(v_{Λ} ⊗ v) = q^{−(Φ_{+}Λ,γ)}(v ⊗ v_{Λ}). (3.5)

Combining these formulae yields the required relations.

4. Prime and Primitive Spectrum of ℂ_{q,p}[G]

In this section we prove our main result on the primitive spectrum of ℂ_{q,p}[G]; namely that the H orbits inside Prim_{w} ℂ_{q,p}[G] are parameterized by the double Weyl group. For completeness we have attempted to make the proof more or less self-contained. The overall structure of the proof is similar to that used in [16] except that the proof of the key 4.12 (and the lemmas leading up to it) form a modified and abbreviated version of Joseph’s proof of this result in the one-parameter case [18]. One of the main differences with the approach of [18] is the use of the Rosso-Tanisaki form introduced in 3.2 which simplifies the analysis of the adjoint action of ℂ_{q,p}[G]. The ideas behind the first few results of this section go back to Soibelman’s work in the one-parameter ‘compact’ case [31]. These ideas were adapted to the multi-parameter case by Levendorskii [20].

4.1. Parameterization of the prime spectrum. Let q, p be as in §3.4. For simplicity we set

A = ℂ_{q,p}[G]

and the product a · b as defined in (2.1) will be denoted by ab.

For each Λ ∈ L^{+} choose weight vectors

v_{Λ} ∈ L(Λ)_{Λ}, v_{w_{0}Λ} ∈ L(Λ)_{w_{0}Λ}, f_{−Λ} ∈ L(Λ)_{−Λ}^{∗}, f_{−w_{0}Λ} ∈ L(Λ)_{−w_{0}Λ}^{∗}

such that ⟨f_{−Λ}, v_{Λ}⟩ = ⟨f_{−w_{0}Λ}, v_{w_{0}Λ}⟩ = 1. Set

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

Recall the following result.

Theorem 4.1. The multiplication map A^{+} ⊗ A^{−} → A is surjective.

Proof. Clearly it is enough to prove the theorem in the one-parameter case. When L = P the result is proved in [31, 3.1] and [18, Theorem 3.7].

The general case can be deduced from the simply-connected case as follows. One first observes that ℂ_{q}[G] ⊂ ℂ_{q}[G̃] = ⨁_{Λ∈P^{+}} C(Λ). Therefore any a ∈ ℂ_{q}[G] can be written in the form a = ∑_{Λ′,Λ′′∈P^{+}} c_{f,v_{Λ′}} c_{g,v_{−Λ′′}} where Λ′ − Λ′′ ∈ L. Let Λ ∈ P and {v_{i}; f_{i}}_{i} be a dual basis of L(Λ). Then we have

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

Let Λ′ be as above and choose Λ such that Λ + Λ′ ∈ L^{+}. Then, for all i, c_{f,v_{Λ′}} c_{f_{i},v_{Λ}} ∈ C(Λ + Λ′) ∩ A^{+} and c_{v_{i},f_{−Λ}}c_{g,v_{−Λ′′}} ∈ C(−w_{0}(Λ + Λ′′)) ∩ A^{−}. The result then follows by inserting 1 between the terms c_{f,v_{Λ′}} and c_{g,v_{−Λ′′}}.

Remark . The algebra A is a Noetherian domain (this result will not be used in the sequel). The fact that A is a domain follows from the same result in [18, Lemma 3.1]. The fact that A is Noetherian is consequence of [18, Proposition 4.1] and [6, Theorem 3.7].

For each y ∈ W define the following ideals of A

I_{y}^{+} = ⟨c_{f,v_{Λ}} | f ∈ (U_{q,p^{−1}}(𝔟^{+})L(Λ)_{yΛ})^{⊥}, Λ ∈ L^{+}⟩,
I_{y}^{−} = ⟨c_{f,v_{w_{0}Λ}} | f ∈ (U_{q,p^{−1}}(𝔟^{−})L(Λ)_{yw_{0}Λ})^{⊥}, Λ ∈ L^{+}⟩

where ( )^{⊥} denotes the orthogonal in L(Λ)^{∗}. Notice that I_{y}^{−} = σ(Î_{y}^{+}), σ as in §3.4, and that I_{y}^{±} is an L × L homogeneous ideal of A.

Notation . For w = (w_{+}, w_{−}) ∈ W × W set I_{w} = I_{w_{+}}^{+} + I_{w_{−}}^{−}. For Λ ∈ L^{+} set c_{wΛ} = c_{f_{−w_{+}Λ},v_{Λ}} ∈ C(Λ)_{−w_{+}Λ,Λ} and c̃_{wΛ} = c_{v_{w_{−}Λ},f_{−Λ}} ∈ C(−w_{0}Λ)_{w_{−}Λ,−Λ}.

Lemma 4.2. Let Λ ∈ L^{+} and a ∈ A_{−η,γ}. Then

c_{wΛ}a ≡ q^{(Φ_{+}w_{+}Λ,η)−(Φ_{+}Λ,γ)}ac_{wΛ} mod I_{w_{+}}^{+}
c̃_{wΛ}a ≡ q^{(Φ_{−}Λ,γ)−(Φ_{−}w_{−}Λ,η)}ac̃_{wΛ} mod I_{w_{−}}^{−}.

Proof. The first identity follows from Corollary 3.10 and the definition of I_{w_{+}}^{+}. The second identity can be deduced from the first one by applying σ.

We continue to denote by c_{wΛ} and c̃_{wΛ} the images of these elements in A/I_{w}. It follows from Lemma 4.2 that the sets

E_{w_{+}} = {αc_{wΛ} | α ∈ ℂ^{∗}, Λ ∈ L^{+}}, E_{w_{−}} = {αc̃_{wΛ} | α ∈ ℂ^{∗}, Λ ∈ L^{+}}, E_{w} = E_{w_{+}}E_{w_{−}}

are multiplicatively closed sets of normal elements in A/I_{w}. Thus E_{w} is an Ore set in A/I_{w}. Define

A_{w} = (A/I_{w})_{E_{w}} .

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

Proposition 4.3. For all w ∈ W × W, A_{w} ≠ (0).

Proof. Notice first that since the generators of A_{w} and the elements of E_{w} are L × L homogeneous, it suffices to work in the one-parameter case. The proof is then similar to that of [15, Theorem 2.2.3] (written in the SL(n)-case). For completeness we recall the steps of this proof. The technical details are straightforward generalizations to the general case of [15, loc. cit.].

For 1 ≤ i ≤ n denote by U_{q}(𝔰𝔩_{i}(2)) the Hopf subalgebra of U_{q}(𝔤) generated by e_{i}, f_{i}, k_{i}^{±1}. The associated quantized function algebra A_{i} ≅ ℂ_{q}[SL(2)] is naturally a quotient of A. Let σ_{i} be the reflection associated to the root α_{i}. It is easily seen that there exist M_{i}^{+} and M_{i}^{−}, non-zero (A_{i})_{(σ_{i},e)} and (A_{i})_{(e,σ_{i})} modules respectively. These modules can then be viewed as non-zero A-modules.

Let w_{+} = σ_{i_{1}} … σ_{i_{k}} and w_{−} = σ_{j_{1}} … σ_{j_{m}} be reduced expressions for w_{±}. Then

M_{i_{1}}^{+} ⊗ ⋯ ⊗ M_{i_{k}}^{+} ⊗ M_{j_{1}}^{−} ⊗ ⋯ ⊗ M_{j_{m}}^{−}

is a non-zero A_{w}-module.

In the one-parameter case the proof of the following result was found independently by the authors in [16, 1.2] and Joseph in [18, 6.2].

Theorem 4.4. Let P ∈ Spec ℂ_{q,p}[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,p^{−1}}(𝔟^{+})f. This is a preordering which induces a partial ordering on the set of one dimensional weight spaces. Consider the set:

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

Let f be an element of F(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 Corollary 3.10 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^{(Φ_{+}Λ,Λ)−(Φ_{+}μ,μ′)}c_{f′,v_{Λ}}c_{f,v_{Λ}} = q^{2(Φ_{+}Λ,Λ)−(Φ_{+}μ,μ′)−(Φ_{+}μ′,μ)}c_{f,v_{Λ}}c_{f′,v_{Λ}}.

But, since u is alternating, 2(Φ_{+}Λ, Λ)−(Φ_{+}μ, μ′)−(Φ_{+}μ′, μ) = 2(Λ, Λ)−2(μ, μ′). 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} ∈ F(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′) for all Λ, Λ′ ∈ L^{+}. Then it is not difficult to show that this furnishes a unique w_{+} ∈ W such that w_{+}Λ = w_{Λ}Λ for all Λ ∈ L^{+}. Thus for each Λ ∈ L^{+},

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_{−}.

Definition . A prime ideal P such that P ⊃ I_{w} and P ∩ E_{w} = ∅ will be called a prime ideal of type w. We denote by Spec_{w} ℂ_{q,p}[G], resp. Prim_{w} ℂ_{q,p}[G], the subset of Spec ℂ_{q,p}[G] consisting of prime, resp. primitive, ideals of type w.

Clearly Spec_{w} ℂ_{q,p}[G] ≅ Spec A_{w} and σ(Specŵ ℂ_{q^{−1},p}[G]) = Spec_{w} ℂ_{q,p}[G]. The following corollary is therefore clear.

Corollary 4.5. One has

Spec ℂ_{q,p}[G] = ⊔_{w∈W ×W} Spec_{w} ℂ_{q,p}[G], Prim ℂ_{q,p}[G] = ⊔_{w∈W ×W} Prim_{w} ℂ_{q,p}[G].

We end this section by a result which is the key idea in [18] for analyzing the adjoint action of A on A_{w}. It says that in the one parameter case the quantized function algebra ℂ_{q}[B^{−}] identifies with U_{q}(𝔟^{+}) through the Rosso-Tanisaki-Killing form, [10, 17, 18]. Evidently this continues to hold in the multi-parameter case. For completeness we include a proof of that result.

Set ℂ_{q,p}[B^{−}] = A/I_{(w_{0},e)}. The embedding U_{q,p^{−1}}(𝔟^{−}) → D_{q,p^{−1}}(𝔤) induces a Hopf algebra map ϕ : A → U_{q,p^{−1}}(𝔟^{−})^{∘}, where U_{q,p^{−1}}(𝔟^{−})^{∘} denotes the cofinite dual. On the other hand the non-degenerate Hopf algebra pairing ⟨ | ⟩_{p^{−1}} furnishes an injective morphism θ : U_{q,p^{−1}}(𝔟^{+})^{op} → U_{q,p^{−1}}(𝔟^{−})^{∗}.

Proposition 4.6. 1. ℂ_{q,p}[B^{−}] is an L-bigraded Hopf algebra.

2. The map γ = θ^{−1}ϕ : ℂ_{q,p}[B^{−}] → U_{q,p^{−1}}(𝔟^{+})^{op} is an isomorphism of Hopf algebras.

Proof. 1. It is easy to check that I_{(w_{0},e)} is an L × L graded bi-ideal of the bialgebra A. Let μ ∈ L^{+} and fix a dual basis {v_{ν}; f_{−ν}}_{ν} of L(μ) (with the usual abuse of notation). Then

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

Taking γ = η = μ yields c̃_{μ}c_{μ} = 1 modulo I_{(w_{0},e)}. If γ = w_{0}μ and η ≠ w_{0}μ, the above relation shows that S(c_{f_{−η},v_{w_{0}μ}})c̃_{−w_{0}μ} ∈ I_{(w_{0},e)}. Thus I_{(w_{0},e)} is a Hopf ideal.

2. We first show that

(4.1) ∀ Λ ∈ L^{+}, c_{f,v_{Λ}} ∈ C(Λ)_{−λ,Λ}, ∃! x_{λ} ∈ U_{Λ−λ}^{+}, ϕ(c_{f,v_{Λ}}) = θ(x_{λ} · k_{−Λ}).

Set c = c_{f,v_{Λ}}. Then c(U_{−η}^{−}) = 0 unless η = Λ − λ; denote by c̄ the restriction of c to U^{−}. By the non-degeneracy of the pairing on U_{Λ−λ}^{+} × U_{λ−Λ}^{−} we know that there exists a unique x_{λ} ∈ U_{Λ−λ}^{+} such that c̄ = θ(x_{λ}). Then, for all y ∈ U_{λ−Λ}^{−}, we have

c(y · k_{μ}) = ⟨f, y · k_{μ} · v_{Λ}⟩ = q^{−(Φ_{−}Λ,μ)}c̄(y) = q^{−(Φ_{−}Λ,μ)}⟨x_{λ} | y⟩
= ⟨x_{λ} · k_{−Λ} | y · k_{μ}⟩_{p^{−1}}

by (3.4). This proves (4.1).

We now show that ϕ is injective on A^{+}. Suppose that c = c_{f,v_{Λ}} ∈ C(Λ)_{−λ,Λ} ∩ Ker ϕ, hence c = 0 on U_{q,p^{−1}}(𝔟^{−}). Since L(Λ) = U_{q,p^{−1}}(𝔟^{−})v_{Λ} = D_{q,p^{−1}}(𝔤)v_{Λ} it follows that c = 0. An easy weight argument using (4.1) shows then that ϕ is injective on A^{+}.

It is clear that Ker ϕ ⊃ I_{(w_{0},e)}, and that A^{+}A^{−} = A implies ϕ(A) = ϕ(A^{+}[c̃_{μ}; μ ∈ L^{+}]). Since c̃_{μ} = c_{μ}^{−1} modulo I_{(w_{0},e)} by part 1, if a ∈ A there exists ν ∈ L^{+} such that ϕ(c_{ν})ϕ(a) ∈ ϕ(A^{+}). The inclusion Ker ϕ ⊂ I_{(w_{0},e)} follows easily. Therefore γ is a well defined Hopf algebra morphism.

If α_{j} ∈ B, there exists Λ ∈ L^{+} such that L(Λ)_{Λ−α_{j}} ≠ (0). Pick 0 ≠ f ∈ L(Λ)_{−Λ+α_{j}}^{∗}. Then (4.1) shows that, up to some scalar, ϕ(c_{f,v_{Λ}}) = θ(e_{j} · k_{−Λ}). If λ ∈ L, there exists Λ ∈ W λ ∩ L^{+}; in particular L(Λ)_{λ} ≠ (0). Let v ∈ L(Λ)_{λ} and f ∈ L(Λ)_{−λ}^{∗} such that ⟨f, v⟩ = 1. Then it is easily verified that ϕ(c_{f,v}) = θ(k_{−λ}). This proves that γ is surjective, and the proposition.

4.2. The adjoint action. Recall that if M is an arbitrary A-bimodule one defines the adjoint action of A on M by

∀ a ∈ A, x ∈ M, ad(a).x = ∑ a_{(1)}xS(a_{(2)}).

Then it is well known that the subspace of ad-invariant elements M^{ad} = {x ∈ M | ∀a ∈ A, ad(a).x = ϵ(a)x} is equal to {x ∈ M | ∀a ∈ A, ax = xa}.

Henceforth we fix w ∈ W × W and work inside A_{w}. For Λ ∈ L^{+}, f ∈ L(Λ)^{∗} and v ∈ L(Λ) we set

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

Let {ω_{1}, …, ω_{n}} be a basis of L such that ω_{i} ∈ L^{+} for all i. Observe that c_{wΛ}c_{wΛ′} and c_{wΛ′}c_{wΛ} differ by a non-zero scalar (similarly for c̃_{wΛ}c̃_{wΛ′}). For each λ = ∑_{i} ℓ_{i}ω_{i} ∈ L we define normal elements of A_{w} by

c_{wλ} = ∏_{i=1}^{n} c_{wω_{i}}^{ℓ_{i}}, c̃_{wλ} = ∏_{i=1}^{n} c̃_{wω_{i}}^{ℓ_{i}}, d_{λ} = (c̃_{wλ}c_{wλ})^{−1}.

Notice then that, for Λ ∈ L^{+}, the “new” c_{wΛ} belongs to ℂ^{∗}c_{f_{−w_{+}Λ},v_{Λ}} (similarly for c̃_{wΛ}). Define subalgebras of A_{w} by

C_{w} = ℂ[z_{f}^{+}, z_{v}^{−}, c_{wλ}; f ∈ L(Λ)^{∗}, v ∈ L(Λ), Λ ∈ L^{+}, λ ∈ L]
C_{w}^{+} = ℂ[z_{f}^{+} ; f ∈ L(Λ)^{∗}, Λ ∈ L^{+}], C_{w}^{−} = ℂ[z_{v}^{−}; v ∈ L(Λ), Λ ∈ L^{+}].

Recall that the torus H acts on A_{λ,μ} by r_{h}(a) = μ(h)a, where μ(h) = ⟨μ, h⟩. 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}. The algebras C_{w} and C_{w}^{±} are obviously H-stable.

Theorem 4.7. 1. C_{w}^{H} = ℂ[z_{f}^{+}, z_{v}^{−}; f ∈ L(Λ)^{∗}, v ∈ L(Λ), Λ ∈ L^{+}].

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

3. For each λ ∈ L, let (A_{w})_{λ} = {a ∈ A_{w} | r_{h}(a) = λ(h)a}. Then A_{w} = ⨁_{λ∈L}(A_{w})_{λ} and (A_{w})_{λ} = A_{w}^{H}c_{wλ} . Moreover each (A_{w})_{λ} is an ad-invariant subspace.

Proof. Assertion 1 follows from

∀h ∈ H, r_{h}(z_{f}^{±}) = z_{f}^{±}, r_{h}(c_{wλ}) = λ(h)c_{wλ}, r_{h}(c̃_{wλ}) = λ(h)^{−1}c̃_{wλ}.

Let {v_{i}; f_{i}}_{i} 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}. Now by Theorem 4.1 any element of A_{w} can be written in the form c_{f_{1},v_{1}}c_{f_{2},v_{2}}d_{Λ}^{−1} where v_{1} = v_{Λ_{1}}, v_{2} = v_{−Λ_{2}} and Λ_{1}, Λ_{2}, Λ ∈ L^{+}. This element lies in (A_{w})_{λ} if and only if Λ_{1} − Λ_{2} = λ. In this case c_{f_{1},v_{1}}c_{f_{2},v_{2}}d_{Λ}^{−1} is equal, up to a scalar, to the element z_{f_{1}}^{+}z_{f_{2}}^{−}d_{Λ+Λ_{2}}^{−1}c_{wλ} ∈ (C_{w}^{H})_{D}c_{wλ}. Since the adjoint action commutes with the right action of H, (A_{w})_{λ} is an ad-invariant subspace. The remaining assertions then follow.

We now study the adjoint action of ℂ_{q,p}[G] on A_{w}. The key result is Theorem 4.12.

Lemma 4.8. Let T_{Λ} = {z_{f}^{+} | f ∈ L(Λ)^{∗}}. Then C_{w}^{+} = ⋃_{Λ∈L} T_{Λ}.

Proof. It suffices to prove that if Λ, Λ′ ∈ L^{+} and f ∈ L(Λ)^{∗}, then there exists a g ∈ L(Λ + Λ′)^{∗} such that z_{f}^{+} = z_{g}^{+}. Clearly we may assume that f is a weight vector. Let ι : L(Λ + Λ′) → L(Λ) ⊗ L(Λ′) be the canonical map. Then

c_{f,v_{Λ}}c_{f_{−w_{+}Λ′} ,v_{Λ′}} = c_{f_{−w_{+}Λ′} ⊗f,v_{Λ}⊗v_{Λ′}} = c_{g,v_{Λ+Λ′}}

where g = ι^{∗}(f_{−w_{+}Λ′} ⊗ f). Multiplying the images of these elements in A_{w} by the inverse of c_{w(Λ+Λ′)} ∈ ℂ^{∗}c_{wΛ}c_{wΛ′} yields the desired result.

Proposition 4.9. Let E be an object of C_{q,p} and let Λ ∈ L^{+}. Let σ : L(Λ) → E ⊗ L(Λ) ⊗ E^{∗} be the map (1 ⊗ ψ^{−1})(ι ⊗ 1) where ι : ℂ → E ⊗ E^{∗} is the canonical embedding and ψ^{−1} : E^{∗} ⊗ L(Λ) → L(Λ) ⊗ E^{∗} is the inverse of the braiding map described in §3.5. Then for any c = c_{g,v} ∈ C(E)_{−η,γ} and f ∈ L(Λ)^{∗}

ad(c).z_{f}^{+} = q^{(Φ_{+}w_{+}Λ,η)} z_{σ^{∗}(v⊗f⊗g)}^{+}.

In particular C_{w}^{+} is a locally finite ℂ_{q,p}[G]-module for the adjoint action.

Proof. Let {v_{i}; g_{i}}_{i} be a dual basis of E where v_{i} ∈ E_{ν_{i}}, g_{i} ∈ E_{−ν_{i}}^{∗}. Then ι(1) = ∑ v_{i} ⊗ g_{i}. By (3.5) we have

ψ^{−1}(g_{i} ⊗ v_{Λ}) = a_{i}(v_{Λ} ⊗ g_{i})

where a_{i} = q^{−(Φ_{+}Λ,ν_{i})} = q^{(Φ_{−}ν_{i},Λ)}. On the other hand the commutation relations given in Corollary 3.10 imply that c_{g,v_{i}}c_{wΛ}^{−1} = ba_{i}c_{wΛ}^{−1}c_{g,v_{i}}, where b = q^{(Φ_{+}w_{+}Λ,η)}. Therefore

ad(c).z_{f}^{+} = ∑ ba_{i}c_{wΛ}^{−1}c_{g,v_{i}}c_{f,v_{Λ}}c_{v,g_{i}} = bc_{wΛ}^{−1}c_{v⊗f⊗g,∑ a_{i}v_{i}⊗v_{Λ}⊗g_{i}} = bc_{wΛ}^{−1}c_{v⊗f⊗g,σ(v_{Λ})}.

Since the map σ is a morphism of D_{q,p^{−1}}(𝔤)-modules it is easy to see that c_{v⊗f⊗g,σ(v_{Λ})} = c_{σ^{∗}(v⊗f⊗g),v_{Λ}}.

Let γ : ℂ_{q,p}[G] → U_{q,p^{−1}}(𝔟^{+}) be the algebra anti-isomorphism given in Proposition 4.6.

Lemma 4.10. Let c = c_{g,v} ∈ ℂ_{q,p}[G]_{−η,γ}, f ∈ L(Λ)^{∗} be as in the previous theorem and x ∈ U_{q,p^{−1}}(𝔟^{+}) be such that γ(c) = x. Then

c_{S^{−1}(x).f,v_{Λ}} = c_{σ^{∗}(v⊗f⊗g),v_{Λ}}.

Proof. Notice that it suffices to show that

c_{S^{−1}(x).f,v_{Λ}}(y) = c_{σ^{∗}(v⊗f⊗g),v_{Λ}}(y)

for all y ∈ U_{q,p^{−1}}(𝔟^{−}). Denote by ⟨ | ⟩ the Hopf pairing ⟨ | ⟩_{p^{−1}} between U_{q,p^{−1}}(𝔟^{+})^{op} and U_{q,p^{−1}}(𝔟^{−}) as in §3.4. Let χ be the one dimensional representation of U_{q,p^{−1}}(𝔟^{+}) associated to v_{Λ} and let χ̃ = χ · γ. Notice that χ(x) = ⟨x | t_{−Λ}⟩; so χ̃(c) = c(t_{−Λ}). Recalling that γ is a morphism of coalgebras and using the relation (c_{xy}) of §2.3 in the double U_{q,p^{−1}}(𝔟^{+}) ⋊ ⋉ U_{q,p^{−1}}(𝔟^{−}), we obtain

c_{S^{−1}(x).f,v_{Λ}}(y) = f(xyv_{Λ})
= ∑⟨x_{(1)} | y_{(1)}⟩⟨x_{(3)} | S(y_{(3)})⟩ f(y_{(2)}x_{(2)}v_{Λ})
= ∑⟨x_{(1)} | y_{(1)}⟩⟨x_{(3)} | S(y_{(3)})⟩ χ(x_{(2)}) f(y_{(2)}v_{Λ})
= ∑⟨x_{(1)}χ(x_{(2)}) | y_{(1)}⟩⟨x_{(3)} | S(y_{(3)}⟩ f(y_{(2)}v_{Λ})
= ∑(c_{(1)}χ̃(c_{(2)}))(y_{(1)}) c_{(3)}(S(y_{(3)})) f(y_{(2)}v_{Λ})
= ∑ rχ̃(c_{(1)})(y_{(1)}) c_{f,v_{Λ}}(y_{(2)}) S(c_{(2)})(y_{(3)}).

Since rχ̃(c_{g,v_{i}}) = q^{(Φ_{−}ν_{i},Λ)}c_{g,v_{i}}, one shows as in the proof of Proposition 4.9 that

c_{S^{−1}(x).f,v_{Λ}}(y) = ∑ rχ̃(c_{(1)})(y_{(1)}) c_{f,v_{Λ}}(y_{(2)}) S(c_{(2)})(y_{(3)})
= ∑ q^{(Φ_{−}ν_{i},Λ)} (c_{g,v_{i}}c_{f,v_{Λ}}c_{v,g_{i}})(y)
= c_{σ^{∗}(v⊗f⊗g),v_{Λ}}(y),

as required.

Theorem 4.11. Consider C_{w}^{+} as a ℂ_{q,p}[G]-module via the adjoint action. Then 1. Soc C_{w}^{+} = ℂ.

2. Ann C_{w}^{+} ⊃ I_{(w_{0},e)}.

3. The elements c_{f_{−μ},v_{μ}}, μ ∈ L^{+}, act diagonalizably on C_{w}^{+}.

4. Soc C_{w}^{+} = {z ∈ C_{w}^{+} | Ann z ⊃ I_{(e,e)}}.

Proof. For Λ ∈ L^{+}, define a U_{q,p^{−1}}(𝔟^{+})-module by

S_{Λ} = (U_{q,p^{−1}}(𝔟^{+})v_{w_{+}Λ})^{∗} = L(Λ)^{∗}/(U_{q,p^{−1}}(𝔟^{+})v_{w_{+}Λ})^{⊥}.

It is easily checked that Soc S_{Λ} = ℂf_{−w_{+}Λ} (see [18, 7.3]). Let δ : S_{Λ} → T_{Λ} be the linear map given by f̄ ↦ z_{f}^{+}. Denote by ζ the one-dimensional representation of ℂ_{q,p}[G] given by ζ(c) = c(t_{−w_{+}Λ}). Let c = c_{g,v} ∈ C(E)_{−η,γ}. Then l_{ζ}(c) = q^{(Φ_{−}η,w_{+}Λ)}c = q^{−(Φ_{+}w_{+}Λ,η))}c. Then, using Proposition 4.9 and Lemma 4.10 we obtain,

ad(l_{ζ}(c)).δ(f̄) = q^{−(Φ_{+}w_{+}Λ,η)} ad(c).z_{f}^{+} = z_{S^{−1}γ(c).f}^{+} = δ(S^{−1}(γ(c))f̄).

Hence, ad(l_{ζ}(c)).δ(f̄) = δ(S^{−1}(γ(c))f̄) for all c ∈ A. This immediately implies part (2) since Ker γ ⊃ I_{(w_{0},e)} and l_{ζ}(I_{(w_{0},e)}) = I_{(w_{0},e)}. If S_{Λ} is given the structure of an A-module via S^{−1}γ, then δ is a homomorphism from S_{Λ} to the module T_{Λ} twisted by the automorphism l_{ζ}. Since δ(f_{−w_{+}Λ}) = 1 it follows that δ is bijective and that Soc T_{Λ} = δ(Soc S_{Λ}) = ℂ. Part (1) then follows from Lemma 4.8. Part (3) follows from the above formula and the fact that γ(c_{f_{−μ},v_{μ}}) = s_{−μ}. Since A/I_{(e,e)} is generated by the images of the elements c_{f_{−μ},v_{μ}}, (4) is a consequence of the definitions.

Theorem 4.12. Consider C_{w}^{H} as a ℂ_{q,p}[G]-module via the adjoint action. Then

Soc C_{w}^{H} = ℂ.

Proof. By Theorem 4.11 we have that Soc C_{w}^{+} = ℂ. Using the map σ, one obtains analogous results for C_{w}^{−}. The map C_{w}^{+} ⊗ C_{w}^{−} → C_{w}^{H} is a module map for the adjoint action which is surjective by Theorem 4.1. So it suffices to show that Soc C_{w}^{+} ⊗ C_{w}^{−} = ℂ. The following argument is taken from [18].

By the analog of Theorem 4.11 for C_{w}^{−} we have that the elements c_{f_{−Λ},v_{Λ}} act as commuting diagonalizable operators on C_{w}^{−}. Therefore an element of C_{w}^{+} ⊗C_{w}^{−} may be written as ∑ a_{i} ⊗b_{i} where the b_{i} are linearly independent weight vectors. Let c_{f,v_{Λ}} be a generator of I_{e}^{+}. Suppose that ∑ a_{i}⊗b_{i} ∈ Soc(C_{w}^{+}⊗C_{w}^{−}). Then

0 = ad(c_{f,v_{Λ}}).(∑_{i} a_{i} ⊗ b_{i}) = ∑_{i,j} ad(c_{f,v_{j}}).a_{i} ⊗ ad(c_{f_{j},v_{Λ}}).b_{i}

= ∑_{i} ad(c_{f,v_{Λ}}).a_{i} ⊗ ad(c_{f_{−Λ},v_{Λ}}).b_{i}

= ∑_{i} ad(c_{f,v_{Λ}}).a_{i} ⊗ α_{i}b_{i}

for some α_{i} ∈ ℂ^{∗}. Thus ad(c_{f,v_{Λ}}).a_{i} = 0 for all i. Thus the a_{i} are annihilated by the left ideal generated by the c_{f,v_{Λ}}. But this left ideal is two-sided modulo I_{(w_{0},e)} and Ann C_{w}^{+} ⊃ I_{(w_{0},e)}. Thus the a_{i} are annihilated by I_{(e,e)} and so lie in Soc C_{w}^{+} by Theorem 4.11. Thus ∑ a_{i} ⊗ b_{i} ∈ Soc(ℂ ⊗ C_{w}^{−}) = ℂ ⊗ ℂ.

Corollary 4.13. The algebra A_{w}^{H} contains no nontrivial ad-invariant ideals. Furthermore, (A_{w}^{H})^{ad} = ℂ.

Proof. Notice that Theorem 4.12 implies that C_{w}^{H} contains no nontrivial ad-invariant ideals. Since A_{w}^{H} is a localization of C_{w}^{H} the same must be true for A_{w}^{H}. Let a ∈ (A_{w}^{H})^{ad}\ℂ. Then a is central and so for any α ∈ ℂ, (a − α) is a non-zero ad-invariant ideal of A_{w}^{H}. This implies that a − α is invertible in A_{w}^{H} for any α ∈ ℂ. This contradicts the fact that A_{w}^{H} has countable dimension over ℂ.

Theorem 4.14. Let Z_{w} be the center of A_{w}. Then 1. Z_{w} = A_{w}^{ad};

2. Z_{w} = ⨁_{λ∈L} Z_{λ} where Z_{λ} = Z_{w} ∩ A_{w}^{H}c_{wλ};

3. If Z_{λ} ≠ (0), then Z_{λ} = ℂu_{λ} for some unit u_{λ};

4. The group H acts transitively on the maximal ideals of Z_{w}.

Proof. The proof of (1) is standard. Assertion (2) follows from Theorem 4.7. Let u_{λ} be a non-zero element of Z_{λ}. Then u_{λ} = ac_{wλ}, for some a ∈ A_{w}^{H}. This implies that a is normal and hence a generates an ad-invariant ideal of A_{w}^{H}. Thus a (and hence also u_{λ}) is a unit by Theorem 4.13. Since Z_{0} = ℂ, it follows that Z_{λ} = ℂu_{λ}. Since the action of H is given by r_{h}(u_{λ}) = λ(h)u_{λ}, it is clear that H acts transitively on the maximal ideals of Z_{w}.

Theorem 4.15. The ideals of A_{w} are generated by their intersection with the center, Z_{w}.

Proof. Any element f ∈ A_{w} may be written uniquely in the form f = ∑ a_{λ}c_{wλ} where a_{λ} ∈ A_{w}^{H}. Define π : A_{w} → A_{w}^{H} to be the projection given by π(∑ a_{λ}c_{wλ}) = a_{0} and notice that π is a module map for the adjoint action. Define the support of f to be Supp(f) = {λ ∈ L | a_{λ} ≠ 0}. Let I be an ideal of A_{w}. For any set Y ⊆ L such that 0 ∈ Y define

I_{Y} = {b ∈ A_{w}^{H} | b = π(f) for some f ∈ I such that Supp(f) ⊆ Y }

If I is ad-invariant then I_{Y} is an ad-invariant ideal of A_{w}^{H} and hence is either (0) or A_{w}^{H}.

Now let I′ = (I ∩ Z_{w})A_{w} and suppose that I ≠ I′. Choose an element f = ∑ a_{λ}c_{wλ} ∈ I\I′ whose support S has the smallest cardinality. We may assume without loss of generality that 0 ∈ S. Suppose that there exists g ∈ I′ with Supp(g) ⊂ S. Then there exists a g′ ∈ I′ with Supp(g′) ⊂ S and π(g′) = 1. But then f − a_{0}g′ is an element of I′ with smaller support than F. Thus there can be no elements in I′ whose support is contained in S. So we may assume that π(f) = a_{0} = 1. For any c ∈ ℂ_{q,p}[G], set f_{c} = ad(c).f − ϵ(c)f. Since π(f_{c}) = 0 it follows that |Supp(f_{c})| < |Supp(f)| and hence that f_{c} = 0. Thus f ∈ I ∩ A_{w}^{ad} = I ∩ Z_{w}, a contradiction.

Putting these results together yields the main theorem of this section, which completes Corollary 4.5 by describing the set of primitive ideals of type w.

Theorem 4.16. For w ∈ W ×W the subsets Prim_{w} ℂ_{q,p}[G] are precisely the H-orbits inside Prim ℂ_{q,p}[G].

Finally we calculate the size of these orbits in the algebraic case. Set L_{w} = {λ ∈ L | Z_{λ} ≠ (0)}. Recall the definition of s(w) from (1.3) and that p is called q-rational if u is algebraic. In this case we know by Theorem 1.7 that there exists m ∈ ℕ^{∗} such that Φ(mL) ⊂ L.

Proposition 4.17. Suppose that p is q-rational. Let λ ∈ L and y_{λ} = c_{wΦ_{−}mλ}c̃_{wΦ_{+}mλ}. Then 1. y_{λ} is ad-semi-invariant. In fact, for any c ∈ A_{−η,γ},

ad(c).y_{λ} = q^{(mσ(w)λ,η)}ϵ(c)y_{λ}.

where σ(w) = Φ_{−}w_{−}Φ_{+} − Φ_{+}w_{+}Φ_{−}

2. L_{w} ∩ 2mL = 2 Ker σ(w) ∩ mL

3. dim Z_{w} = n − s(w)

Proof. Using Lemma 4.2, we have that for c ∈ A_{−η,γ}

cy_{λ} = q^{(Φ_{+}w_{+}Φ_{−}mλ,−η)}q^{(Φ_{+}Φ_{−}mλ,γ)}q^{(Φ_{−}w_{−}Φ_{+}mλ,η)}q^{(Φ_{−}Φ_{+}mλ,−γ)}y_{λ}c
= q^{(mσ(w)λ,η)}y_{λ}c.

From this it follows easily that

ad(c).y_{λ} = q^{(mσ(w)λ,η)}ϵ(c)y_{λ}.

Since (up to some scalar) y_{λ} = d_{Φmλ}^{−1}d_{mλ}^{−1}c_{wmλ}^{−2} it follows from Theorem 4.7 that y_{λ} ∈ (A_{w})_{−2mλ}. However, as a ℂ_{q,p}[G]-module via the adjoint action, A_{w}^{H}y_{λ} ≅ A_{w}^{H} ⊗ ℂy_{λ} and hence Soc A_{w}^{H}y_{λ} = ℂy_{λ}. Thus Z_{−2mλ} ≠ (0) if and only if y_{λ} is ad-invariant; that is, if and only if mσ(w)λ = 0. Hence

dim Z_{w} = rk L_{w} = rk(L_{w} ∩ 2mL) = rk Ker_{mL} σ(w)
= dim Ker_{𝔥^{∗}} σ(w) = n − s(w)

as required.

Finally, we may deduce that in the algebraic case the size of the of the H-orbits Symp_{w} G and Prim_{w} ℂ_{q,p}[G] are the same, cf. Theorem 1.8.

Theorem 4.18. Suppose that p is q-rational and let w ∈ W × W. Then

∀P ∈ Prim_{w} ℂ_{q,p}[G], dim(H/ Stab_{H} P) = n − s(w).

Proof. This follows easily from theorems 4.15, 4.16 and Proposition 4.17.

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

   E-mail address: timothy.hodges@uc.edu

Université de Poitiers, 86022 Poitiers, France

   E-mail address: levasseu@mathlabo.univ-poitiers.fr

Universidad Nacional de Colombia, Apartado Aéreo, Medellin, Colombia

   E-mail address: mmtoro@perseus.unalmed.edu.co