Adapted algebras for the Berenstein-Zelevinsky conjecture

Philippe CALDERO

ABSTRACT. Let G be a simply connected semi-simple complex Lie group and fix a maximal unipotent subgroup U^{−} of G. Let q be an indeterminate and let’s denote by B^{∗} the dual canonical basis, [18], of the quantized algebra ℂ_{q}[U^{−}] of regular functions on U^{−}. Following [19], fix a ℤ_{≥0}^{N}-parametrization of this basis, where N =dimU^{−}. In [2], A. Berenstein and A. Zelevinsky conjecture that two elements of B^{∗} q-commute if and only if they are multiplicative, i.e. their product is an element of B^{∗} up to a power of q. For all reduced decomposition w̃_{0} of the longest element of the Weyl group of 𝔤, we associate a subalgebra Aw̃_{0}, called adapted algebra, of ℂ_{q}[U^{−}] such that 1) Aw̃_{0} is a q-polynomial algebra which equals ℂ_{q}[U^{−}] up to localization, 2) Aw̃_{0} is spanned by a subset of B^{∗}, 3) the Berenstein-Zelevinsky conjecture is true on Aw̃_{0}. Then, we test the conjecture when one element belongs to the q-center of ℂ_{q}[U^{−}].

This research has been partially supported by the EC TMR network ”Algebraic Lie Representations”, contract no. ERB FMTX-CT97-0100.

0. Introduction.

0.1. Let G be a simply connected semi-simple complex group, and let 𝔤 be its Lie algebra. Fix a Cartan subalgebra 𝔥 of 𝔤 and let 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫^{+} be the corresponding triangular decomposition. For all λ in the set of integral dominant weights P^{+}, let V (λ) be the simple 𝔤-module with highest weight λ.

For all Lie algebra 𝔞, the enveloping algebra of 𝔞 will be denoted by U(𝔞). The module V (λ) is naturally endowed with a structure of U(𝔤)-module, and it is known that V (λ) = U(𝔫^{−})v_{λ}, where v_{λ} is a highest weight vector of V (λ). By the work of Lusztig, [19], there exists a basis B of U(𝔫^{−}) with the following property :

Set B(λ) := {b ∈ B, bv_{λ} ≠ 0}. Then, for all λ in P^{+}, B(λ).v_{λ} is a basis of V (λ). Let U^{−} be the maximal unipotent subgroup of G such that Lie(U^{−}) = 𝔫^{−} and let ℂ[U^{−}] be the algebra of regular functions on U^{−}. It is well known that ℂ[U^{−}] is isomorphic to the symmetric algebra S(𝔫^{−}^{∗}), and thus to S(𝔫) via the Killing form. The left multiplication of U^{−} on itself provides a natural morphism ϕ from U(𝔫^{−}) on the algebra of differential operators on ℂ[U^{−}]. The pairing on U(𝔫^{−})×ℂ[U^{−}] such that (u, f) is the value of ϕ(u)(f) on the identity of U^{−}, is non degenerate. Let B^{∗} be the basis of ℂ[U^{−}] dual to B for this pairing. 0.2. Berenstein and Zelevinsky have a conjecture in order to calculate the basis B^{∗}. This conjecture involves the standard q-deformation of the basis B^{∗}. Let’s present some notations. The objects introduced in the previous section have a q-deformation. Let q be an indeterminate and let U_{q}(𝔤) be the quantized enveloping algebra on ℂ(q) as defined by Drinfeld and Jimbo. Let U_{q}(𝔫) and U_{q}(𝔫^{−}) be the quantized subalgebra corresponding to U(𝔫) and U(𝔫^{−}), see 1.2, and let ℂ_{q}[U^{−}] be the quantized algebra of regular functions on U^{−}. For all λ in P^{+}, let V_{q}(λ) be the simple U_{q}(𝔤)-module, 1.3, with highest weight vector v_{λ}. Then, there exists

1) A canonical basis, still denoted by B, of U_{q}(𝔫^{−}), with the following property Set B(λ) := {b ∈ B, bv_{λ} ≠ 0}. Then, for all λ in P^{+}, B(λ).v_{λ} is a basis of V_{q}(λ).

2) A pairing on U_{q}(𝔫^{−}) × ℂ_{q}[U^{−}] which quantifies the pairing in 0.1 and which permits to define the dual canonical basis B^{∗} in ℂ_{q}[U^{−}]. Remark that we can recover both basis of 0.1 by specializing q on 1. Remark also that ℂ_{q}[U^{−}] is isomorphic to U_{q}(𝔫).

We say that two elements u and u′ in ℂ_{q}[U^{−}] q-commute if there exists an integer m such that uu′ = q^{m}u′u. Two elements b^{∗} and b^{′∗} in B^{∗} are said to be multiplicative if there exists an m such that q^{m}b^{∗}b^{′∗} ∈ B^{∗}. We can formulate the Berenstein-Zelevinsky conjecture.

Conjecture. Two elements of the dual canonical basis B^{∗} q-commute if and only if they are multiplicative.

Let A be a subalgebra of ℂ_{q}[G] such that A is generated as a space by a subset of B^{∗}. In the sequel, we say that the Berenstein-Zelevinsky conjecture is true on A if the assertion is true when both elements are in A. 0.3. We present results about the conjecture. Recall that the elements of the dual canonical basis are labelled by elements in ℤ_{≥0}^{N}, where N is the dimension of 𝔫. This is the so-called Lusztig parametrization of B^{∗} which depends on the choice of a reduced decomposition w̃_{0} of the longest element w_{0} of the Weyl group.

In [21], the author uses the quiver approach of quantum groups. First of all, he obtains the only if part of the Berenstein-Zelevinsky conjecture, [loc. cit., Corollary 4.5], in the simply laced case. Maybe the most remarkable result of [21] is obtained in the case A_{n}. It can be interpreted as follows : for some quiver orientation Ω of the diagram of A_{n}, the author proves that there exists a subalgebra A_{Ω} of maximal GK-dimension of ℂ_{q}[U^{−}] which is compatible with B^{∗} and on which the Berenstein-Zelevinsky conjecture is true. Moreover, each pair of elements in B^{∗} ∩A_{Ω} q-commute (and so is multiplicative) and the previous parametrization yields a correspondance between B^{∗} ∩ A_{Ω} and the set C_{Ω} of integral points of a simplicial cone in ℝ^{N}. Let 𝔤 be of type A_{n}. The non zero classical minors in ℂ[U^{−}] belong to the (classical) dual canonical basis. They have a quantum analogue in ℂ_{q}[U^{−}] which belong to the (quantum) dual canonical basis. A natural question comes up : is the assertion of the Conjecture true when both elements are quantum minors. In [13], Leclerc, Nazarov and Thibon give a positive answer to this question. 0.5. The main results of this article are

1) a generalization of the result of [21] for all semi-simple Lie algebra 𝔤, see Theorem 2.2. An adapted subalgebra of ℂ_{q}[U^{−}], see Definition 2.2, is an algebra which is spanned by a maximal subset P^{∗} of B^{∗} such that two elements in P^{∗} are multiplicative. For all reduced decomposition w̃_{0} of w_{0}, we provide an adapted algebra Aw̃_{0} which generalizes the algebra A_{Ω} discussed in 0.3. Recall that in [5] and [6], we provide, for all reduced decomposition of w_{0}, an subalgebra Aw̃_{0} of ℂ_{q}[U^{−}] such that a) the algebra Aw̃_{0} is generated by a family Fw̃_{0} of N algebraically independant elements which pairwise q-commute b) Aw̃_{0} and ℂ_{q}[U^{−}] are equal up to localization by the multiplicative part generated by Fw̃_{0}. Using Kashiwara’s crystal basis and some standard monomial theory, we prove that Aw̃_{0} is spanned by a subset of B^{∗} and that the Conjecture is true on Aw̃_{0}. Then, we prove that Aw̃_{0} is an adapted algebra.

2) we test the Berenstein-Zelevinsky conjecture on the q-center of ℂ_{q}[U^{−}], i.e. the space generated by elements which q-commute with all homogeneous elements of ℂ_{q}[U^{−}]. Indeed, as the q-center of ℂ_{q}[U^{−}] is spanned by a part of the dual canonical basis, a natural test for the conjecture can be stated as follows : is the conjecture true if one element belongs to the q-center. We give a positive answer to this question. Moreover, it reduces the conjecture to a subset of B^{∗}, see Corollary 3.2.

3) we study the case when 𝔤 is of type B_{2}, see 3.4, and we remark that in this case, ℂ_{q}[U^{−}] is a direct sum of adapted algebras.

To conclude, we give a presentation of these results in terms of cones in ℤ_{≥0}^{N}, which are sets of parametrizations for the part P∗ ; w̃_{0} associated to Aw̃_{0}. These cones generalize the cones ℂ_{Ω} discussed in 0.4.

1. Notations and recollection on global basis.

1.1. Let 𝔤 be a semi-simple Lie ℂ-algebra of rank n. We fix a Cartan subalgebra 𝔥 of 𝔤. Let 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫 be the triangular decomposition and let {α_{i}}_{i} be a basis of the root system Δ resulting from this decomposition. Let Δ^{+} be the set of positive roots. Let P be the weight lattice generated by the fundamental weights ϖ_{i}, 1 ≤ i ≤ n. Let P^{+} := ∑_{i} ℤ_{≥0}ϖ_{i} and P := ∑_{i} ℤϖ_{i} be respectively the semigroup of integral dominant weights and the group of integral weights. We endow P^{+} with the ordering ≤ defined by λ ≤ μ ⇔ μ − λ ∈ P^{+}. Let W be the Weyl group, generated by the reflections corresponding to the simple roots s_{i} := s_{α_{i}}, with longest element w_{0}. We note < , > the W -invariant form on P.

1.2. Let d be an integer such that < P, P >⊂ (2/d)ℤ. Let q be a indeterminate and set 𝕂 = ℂ(q^{1/d}). Let U_{q}(𝔤) be the simply connected quantized enveloping algebra on 𝕂, as defined in [10]. Set d_{i} =< α_{i}, α_{i} > /2 and q_{i} = q^{d_{i}} for all i. Let U_{q}(𝔫), resp. U_{q}(𝔫^{−}), be the subalgebra generated by the canonical generators E_{α_{i}}, resp. F_{α_{i}}, of positive, resp. negative, weights and the quantum Serre relations. For all λ in P, let K_{λ} the corresponding element in the algebra U_{q}^{0} = 𝕂[P] of the torus of U_{q}(𝔤). Recall the triangular decomposition U_{q}(𝔤) = U_{q}(𝔫^{−}) ⊗ U_{q}^{0} ⊗ U_{q}(𝔫). We define the following subalgebras of U_{q}(𝔤) :

U_{q}(𝔟) = U_{q}(𝔫) ⊗ U_{q}^{0}, U_{q}(𝔟^{−}) = U_{q}(𝔫^{−}) ⊗ U_{q}^{0}.

U_{q}(𝔤) is endowed with a structure of Hopf algebra and the comultiplication Δ, the antipode S and the augmentation ε are given by

ΔE_{i} = E_{i} ⊗ 1 + K_{α_{i}} ⊗ E_{i}, ΔF_{i} = F_{i} ⊗ K_{α_{i}}^{−1} + 1 ⊗ F_{i}, ΔK_{λ} = K_{λ} ⊗ K_{λ}
S(E_{i}) = −K_{α_{i}}^{−2}E_{i}, S(F_{i}) = −F_{i}K_{α_{i}}^{2}, S(K_{λ}) = K_{−λ}
ε(E_{i}) = ε(F_{i}) = 0, ε(K_{λ}) = 1.

Let’s denote by ad the adjoint action. Remark that there exists an algebra isomorphism U_{q}(𝔫) ≃ U_{q}(𝔫^{−}) which sends E_{α_{i}} on F_{α_{i}}.

If α = ∑_{i} m_{i}α_{i} is in Q^{+} := ∑ ℤ_{≥0}α_{i}, resp. Q^{−} := ∑ ℤ_{≤0}α_{i}, then an element X of the subspace of U_{q}(𝔫), resp. U_{q}(𝔫^{−}), generated by {E_{i_{1}}^{n_{1}} … E_{i_{k}}^{n_{k}}, n_{i_{1}}α_{i_{1}} + … + n_{i_{k}}α_{i_{k}} = α}, resp. {F_{i_{1}}^{n_{1}} … F_{i_{k}}^{n_{k}}, n_{i_{1}}α_{i_{1}} +…+n_{i_{k}}α_{i_{k}} = −α}, will be called homogeneous element of weight α, resp. −α. In this case, we set wt(X) = α, resp. wt(X) = −α, and Tr(X) = ∑ m_{i}.

For each u in U_{q}(𝔤) we set u_{(1)} ⊗ u_{(2)} = Δ(u) ∈ U_{q}(𝔤) ⊗ U_{q}(𝔤). There exists a unique form, [22], [24], ( , ) on U_{q}(𝔟) × U_{q}(𝔟^{−}) such that :

(E_{i}, F_{i}) = \frac{1}{1 − q_{i}^{−2}} δ_{ij},

(u^{+}, u_{1}^{−}u_{2}^{−}) = (Δ(u^{+}), u_{1}^{−} ⊗ u_{2}^{−}) , u^{+} ∈ U_{q}(𝔟) ; u_{1}^{−}, u_{2}^{−} ∈ U_{q}(𝔟^{−})
(u_{1}^{+}u_{2}^{+}, u^{−}) = (u_{2}^{+} ⊗ u_{1}^{+}, Δ(u^{−})) , u^{−} ∈ U_{q}(𝔟^{−}) ; u_{1}^{+}, u_{2}^{+} ∈ U_{q}(𝔟)
(K_{λ}, K_{μ}) = q^{−(λ,μ)} , (K_{λ}, F_{i}) = 0 , (E_{i}, K_{λ}) = 0 , λ, μ ∈ P

For all β in Q^{+}, let U_{q}(𝔫)_{β}, resp. U_{q}(𝔫^{−})_{−β}, be the subspace of U_{q}(𝔫), resp. U_{q}(𝔫^{−}), with weight β, resp. −β. The form ( , ) is non degenerate on U_{q}(𝔫)_{β} × U_{q}(𝔫^{−})_{−β}, β ∈ Q^{+}. We have, :

(XK_{λ}, Y K_{μ}) = q^{−(λ,μ)}(X, Y) , X ∈ U_{q}(𝔫), Y ∈ U_{q}(𝔫^{−})

We can define a bilinear form < , > on U_{q}(𝔤) × U_{q}(𝔤) by :

< X_{1}K_{λ}S(Y_{1}), Y_{2}K_{μ}S(X_{2}) >= (X_{1}, Y_{2})(X_{2}, Y_{1})q^{−(λ,μ)/2}

where X_{1}, X_{2} ∈ U_{q}(𝔫), Y_{1}, Y_{2} ∈ U_{q}(𝔫^{−}), λ, μ ∈ P. Moreover, this form is non degenerate.

We define the ring automorphism x ↦ η(x) of U_{q}(𝔫), resp. U_{q}(𝔫^{−}), such that η(q) = q^{−1} and η(E_{i}) = E_{i}, resp. η(F_{i}) = F_{i} for all i, 1 ≤ i ≤ n. We also define the antihomomorphism σ of the ℂ(q)-algebra U_{q}(𝔫), resp. U_{q}(𝔫^{−}), such that σ(E_{i}) = E_{i}, resp. σ(F_{i}) = F_{i}, for all i, 1 ≤ i ≤ n. 1.3. For all U_{q}^{0}-module M, set M_{λ} := {m ∈ M, K_{μ}.m = q^{(μ,λ)}m}. Elements of M_{μ} are called elements of weight μ. For all λ in P^{+}, V_{q}(λ) denotes the simple U_{q}(𝔤)-module with highest weight λ, with highest weight vector v_{λ}. It is known, see [10, 4.3.6], that this module verifies the Weyl character formula. In particular, for all w in W, dimV_{q}(λ)_{wλ} = 1, and we can define extremal vectors v_{wλ} in V_{q}(λ) as in the classical case.

The dual space V (λ)^{∗} is endowed with a structure of left U_{q}(𝔤)-module by twisting with the antipode.

1.4. Fix a reduced decomposition w̃_{0} = s_{i_{1}} … s_{i_{N}} of w_{0}. Set β_{s} := s_{i_{1}} … s_{i_{s−1}}(α_{i_{s}}), 1 ≤ s ≤ N. Recall that {β_{s}, 1 ≤ s ≤ N} is exactly the set of positive roots. We define a total ordering of the positive roots as follows :

β_{1} > β_{2} > … > β_{N}.

For i, 1 ≤ i ≤ n, let T_{i} be the Lusztig’s automorphisms, [19, 37.1.3], [23], associated to i. We introduce the following elements in U_{q}(𝔫) : E_{β_{s}} = T_{i_{1}} … T_{i_{s−1}}(E_{i_{s}}), 1 ≤ s ≤ N. Remark that these elements depend on w̃_{0}. We index the so-called Poincaré-Birkoff-Witt basis of U_{q}(𝔫) by m in ℤ_{≥0}^{N} as follows

E(m) = Ew̃_{0}(m) = ∏_{i=N}^{1} \frac{1}{[m_{i}]_{q_{i}}!}E_{β_{i}}^{m_{i}},

where [n]_{q_{i}}! = [n]_{q_{i}}[n − 1]_{q_{i}} … [1]_{q_{i}}, [n]_{q_{i}} = \frac{q_{i}^{n}−q_{i}^{−n}}{q_{i}−q_{i}^{−1}} q_{i} = q^{\frac{(β_{i},β_{i})}{2}} . We define in the same

way a PBW-basis (F (m))_{m∈ℤ_{≥0}^{N}} of U_{q}(𝔫^{−}) via the isomorphism in 1.2.

Let (E(m)^{∗})_{m∈ℤ_{≥0}^{N}} be the dual basis of the PBW-basis (F (m))_{m∈ℤ_{≥0}^{N}} for ( , ). By [14], we have that E(m)^{∗} is equal to E(m) up to a multiplicative scalar. We know by [9, Lemma 1.7] that the lexicographical ordering on the PBW-basis provides a filtration on U_{q}(𝔫) whose associated algebra is a q-polynomial algebra :

Theorem. Fix a reduced decomposition w̃_{0} of w_{0} and set

F_{m}(U_{q}(𝔫)) = ⊕_{n≺m}𝕂Ew̃_{0}(n), m ∈ ℤ_{≥0}^{N},

where ≺ is the lexicographical ordering of ℤ_{≥0}^{N}. Then, the associated graded algebra Grw̃_{0}(U_{q}(𝔫)) is generated by Grw̃_{0}(E_{α}), α ∈ Δ^{+} and with relations (up to a sign) :

Grw̃_{0}(E_{α})Grw̃_{0}(E_{β}) = q^{<α,β>}Grw̃_{0}(E_{β})Grw̃_{0}(E_{α}), α < β.

In particular Grw̃_{0}(E(m)^{∗})Grw̃_{0}(E(n)^{∗}) = Grw̃_{0}(E(m + n)^{∗}) up to a multiplicative scalar. ♢

1.5. Let’s define now the following sub-ℤ[q]-lattices of U_{q}(𝔫) :

L = ⨁_{m∈ℤ_{≥0}^{N}} ℤ[q]E(m), L^{∗} = ⨁_{m∈ℤ_{≥0}^{N}} ℤ[q]E(m)^{∗}

A remarkable result of Lusztig states that these lattices do not depend on a reduced decomposition of w_{0}. The following theorem is due to Lusztig [17]. It introduces the so-called canonical basis B of U_{q}(𝔫) which will be identified with the canonical basis of of U_{q}(𝔫^{−}). Theorem. Fix a reduced decomposition w̃_{0} of w_{0}. Then, for each m in ℤ_{≥0}^{N}, there exists a unique element B(m) = Bw̃_{0}(m) in U_{q}(𝔫) such that η(B(m)) = B(m) and B(m) ∈ E(m) + qL. The set B := {B(m), m ∈ ℤ_{≥0}^{N}} is a basis of U_{q}(𝔫) which does not depend on w̃_{0}. ♢ Hence, for a fixed w̃_{0}, the natural bijection ℤ_{≥0}^{N} → B gives rise to a parametrization of B. This will be called the Lusztig’s parametrization of the canonical basis B.

Let B^{∗} = (B(m)^{∗})_{m∈ℤ_{≥0}^{N}} be the dual canonical basis of U_{q}(𝔫). As in [13, Proposition 16], we have Proposition. Fix a reduced decomposition w̃_{0} of w_{0}. Then, for each m in ℤ_{≥0}^{N}, the element B(m)^{∗} is the unique homogeneous element X of U_{q}(𝔫) such that

η(X) = (−1)^{tr(X)}q^{<wt(X),wt(X)>/2}q_{X}σ(X), X = E(m)^{∗} + qL^{∗},

where q_{X} = ∏_{i} q_{i}^{m_{i}}, wt(X) = ∑_{i} m_{i}α_{i}.

Remark. Suppose that w̃_{0} corresponds to a quiver orientation. Then, by [21, Lemma 4.1], see also [20, par. 7] for the non simply laced case,

(1.5.1)  Grw̃_{0}(B(m)^{∗}) = Grw̃_{0}(E(m)^{∗}).

Recall that two elements of the dual canonical basis are called multiplicative if their product is an element of the dual canonical basis, up to a power of q. The following corollary of the proposition is due to Reineke, [21, Corollary 4.5]. It is an implication in the Berenstein-Zelevinsky conjecture. His proof is also available for the non simply laced case.

Corollary. If two elements of the dual canonical basis are multiplicative, then they q-commute. ♢

1.6. In this section we consider the canonical basis B in U_{q}(𝔫^{−}).

Let Ẽ_{i}, F̃_{i} : U_{q}(𝔫^{−}) → U_{q}(𝔫^{−}) be the Kashiwara operators, [19]. For b ∈ B, Ẽ_{i}(b), resp. F̃_{i}(b), equals some b′ in B ∪ {0} modulo qℤ[q]B. The rule b ↦ b′ defines maps ẽ_{i} and f̃_{i} from B to B ∪ {0}. For b ∈ B, 1 ≤ i ≤ n, set ε_{i}(b) = Max{r, ẽ_{i}^{r}(b) ≠ 0}, and E(b) = ∑_{i=1}^{n} ε_{i}(b)ϖ_{i}.

Fix λ in P^{+} and define the following subset of B

B(λ) = {b ∈ B, E(σ(b)) ≤ λ}.

Let L(λ) be the ℂ[q]-sublattice of V_{q}(λ) generated by {bv_{λ}, b ∈ B(λ)} and define the following subset of L(λ)/qL(λ)

B(λ) = {b mod qL(λ), b ∈ B(λ)}.

In the sequel, the element b mod qL(λ) will be denoted [b]_{λ} or [b] if no confusion occurs. The following is well known, [11, 12.3] :

Theorem. For all λ ∈ P^{+}, we have :

(i) the set B(λ) is the set of elements b of B such that bv_{λ} is non zero in V_{q}(λ), moreover the set B(λ)v_{λ} is a basis of V_{q}(λ),

(ii) the pair (L(λ), B(λ)) is a crystal basis of V_{q}(λ), see [11, par 4]. ♢

1.7. Recall, [10, 6.4.27] that for all λ in P^{+}, the Kashiwara crystal B(λ) is in bijection with the Littelmann path crystal C(λ). Moreover, through this isomorphism, the tensor product can be defined in terms of concatenation of paths [15]. The following propositions are then direct consequences of [16, Theorem 10.1].

Proposition A. Let λ_{i} in P^{+}, y_{i} in W, b_{i} in B(λ_{i}), with weight y_{i}λ_{i}, 1 ≤ i ≤ r. Suppose y_{1} ≤ y_{2} ≤ … ≤ y_{r} for the Bruhat ordering. Then, b_{r} ⊗b_{r−1} ⊗…⊗b_{1} belongs to the component of type B(λ_{r}+λ_{r−1}+…+λ_{1}) in the crystal B(λ_{r})⊗B(λ_{r−1})⊗…⊗B(λ_{1}). ♢

Proposition B. Let λ, μ in P^{+}, b in B(λ), c in B(μ) with weight w_{0}μ. Then, c ⊗ b belongs to the component of type B(μ + λ) in the crystal B(μ) ⊗ B(λ). ♢

For all λ in P^{+} and w in W, let B_{w}(λ) the subset of B(λ) corresponding to the Demazure module associated to w, see [11, 12.4].

Proposition C. Let λ, μ in P^{+}, w in W, b in B_{w}(λ), c in B_{w}(μ) with weight wμ. Then, c ⊗ b belongs to the component of type B(μ + λ) in the crystal B(μ) ⊗ B(λ). ♢

1.8. In this section, we define the string parametrization of the canonical basis. For b in B and i, 1 ≤ i ≤ n, set ẽ_{i}^{max}(b) = ẽ_{i}^{n_{i}(b)}(b), where n_{i}(b) := max{k, ẽ_{i}^{k}(b) ≠ 0}. Let w̃_{0} = s_{i_{1}} … s_{i_{N}} be a reduced decomposition of w_{0}. To an element b in B, we associate the element Sw̃_{0}(b) in ℤ_{≥0}^{N} such that the k-th component of Sw̃_{0}(b) is n_{i_{k}}(ẽ_{i_{k−1}}^{max}(… (ẽ_{i_{1}}^{max}(b))). It is known, see [1], that Sw̃_{0} is injective and that Sw̃_{0}(B) is the set of integral points of a cone in ℝ^{N}. Hence, the map Sw̃_{0} defines a parametrization of the canonical basis. It is called the string parametrization of B.

Proposition. We have

(i) Fix a reduced decomposition w̃_{0} of the longest element of the Weyl group. Let b, b′, b′′ be in B with Sw̃_{0}(b) + Sw̃_{0}(b′) = Sw̃_{0}(b′′). Then, the b^{′′∗}-component of b^{∗}b^{′∗} in the dual canonical basis B^{∗} is a power of q.

(ii) In the framework of Propositions 1.7, there exists a reduced decomposition of w_{0} for which the string parametrization of the tensor product is the sum of the string parametrizations of the factors.

Proof. (i) is a part of [8, Theorem 2.3]. Let’s prove (ii). It is enough to prove it in the framework of Proposition 1.7. C. Indeed, the case A is obtained by induction from the case C and the case B is obtained from the case C by setting w = w_{0}. First suppose that c and b are respectively in B(μ) and B(λ) and that for some i, f̃_{i}(c) = 0. Then , by [11, 4.3],

(∗)  ε_{i}(c ⊗ b) = ε_{i}(c) + ε_{i}(b)

and

(∗∗)  ẽ_{i}^{max}(c ⊗ b) = ẽ_{i}^{max}(c) ⊗ ẽ_{i}^{max}(b).

Let w̃ = s_{i_{1}} … s_{i_{k}} be a reduced decomposition of w and let w̃_{0} = s_{i_{1}} … s_{i_{k}}s_{i_{k+1}} … s_{i_{N}} be the reduced decomposition of w_{0} which completes w̃ on the right hand. For all w′ in W, let c_{w′} be the extremal element of B(μ) corresponding to w′. By [11, 12.4], we know that f̃_{i_{1}}(c_{w}) = 0 and ẽ_{i_{1}}^{max}(c_{w}) = c_{s_{i_{2}} ...s_{i_{k}}}. Using (**) by induction, we obtain that ẽ_{i_{k}}^{max}(… (ẽ_{i_{1}}^{max}(c_{w} ⊗ b))) = ẽ_{i_{k}}^{max}(… (ẽ_{i_{1}}^{max}(c_{w}))) ⊗ ẽ_{i_{k}}^{max}(… (ẽ_{i_{1}}^{max}(b))). By [11, Theorem 12.4], the hypothesis b ∈ B_{w}(λ) implies that ẽ_{i_{k}}^{max}(… (ẽ_{i_{1}}^{max}(b))) is the highest weight vector b_{λ}. In particular, ẽ_{i_{k}}^{max}(… (ẽ_{i_{1}}^{max}(c_{w}))) is the highest weight vector b_{μ}. Using (*) by induction, we obtain the proposition. ♢

1.9. The following proposition is given in [13, Proposition 33]. In [loc. cit.], the authors study the A_{n} case, but the generalization of their proof for all 𝔤 is straightforward.

Proposition. Fix r in ℤ_{≥0} and λ_{i}, 1 ≤ i ≤ r, in P^{+}. Let b_{i}, 1 ≤ i ≤ r, and b respectively in B(λ_{i}), B(∑_{i} λ_{i}), such that

(i) [b_{1}] ⊗ [b_{2}] … ⊗ [b_{r}] is in the connected component of the crystal graph of B(λ_{1}) ⊗ B(λ_{2}) ⊗ … ⊗ B(λ_{r}) of type B(∑_{i} λ_{i}),

(ii) [b] identifies with [b_{1}] ⊗ [b_{2}] ⊗ … ⊗ [b_{r}] in B(∑_{i} λ_{i}), then there exists an integer m such that the multiplication rule holds in U_{q}(𝔫)

q^{m}b_{1}^{∗}b_{2}^{∗} … b_{r}^{∗} = b^{∗} mod qL^{∗}.

Corollary. Let b_{i}, 1 ≤ i ≤ r, b, m as in the previous proposition. Moreover, suppose that (iii) the b_{i} pairwise q-commute (iv) for some w̃_{0}, the string parametrization of [b] is the sum of the string parametriza-

tions of the [b_{i}]. Then, q^{m}b_{1}^{∗}b_{2}^{∗} … b_{r}^{∗} = b^{∗}. Proof. From the previous proposition, we obtain

(1.9.1)  q^{m}b_{1}^{∗}b_{2}^{∗} … b_{r}^{∗} = b^{∗} + q ∑_{b^{′∗}∈B^{∗}} c_{b^{′∗}}(q)b^{′∗},

where c_{b^{′∗}}(q) ∈ ℤ[q]. Clearly, the elements of the dual canonical basis of the right hand term of (1.9.1) have the same weight equal to ∑_{i} wt(b_{i}^{∗}). Applying the automorphism η and the antiautomorphism σ on (1.9.1), Proposition 1.5 and (iii) give :

q^{m}b_{1}^{∗} … b_{r−1}^{∗}b_{r}^{∗} = q^{k}b^{∗} + q^{k−1} ∑_{b^{′∗}∈B^{∗}} c_{b^{′∗}}(q^{−1})b^{′∗},

for some integer k. Now, using (iv) and Proposition 1.8 (i), we obtain that the sum in the right hand side of (1.9.1) does not contain any term in b^{∗}. Hence, by comparison, k = 0 and then we obtain c_{b^{′∗}}(q) = 0. This gives the corollary. ♢

Remark that, by a weight argument, the condition (iv) of the corollary implies (but is not equivalent to) the condition (i) of the proposition.

2. Adapted algebras

2.1. For each reduced decomposition of w_{0}, we define a set of pairwise q-commuting elements of the dual canonical basis.

Lemma. Fix a reduced decomposition w̃_{0} = s_{i_{1}} … s_{i_{N}} of w_{0}. For all r, 1 ≤ r ≤ N, set y_{r} = s_{i_{1}} … s_{i_{r}}. Then,

(i) there exists a unique Xr ; w̃_{0} in U_{q}(𝔫) such that v_{y_{r}ϖ_{ir}}^{∗} (uv_{ϖ_{ir}}) = (K_{−ϖ_{ir}} Xr ; w̃_{0}, u), for all u in U_{q}(𝔟^{−}),

(ii) up to a multiplicative scalar Xr ; w̃_{0} is in B^{∗}, (iii) the Xr ; w̃_{0}, 1 ≤ r ≤ N, pairwise q-commute. Proof. the existence of the q-commuting family of elements {Xr ; w̃_{0}, 1 ≤ r ≤ N} is proved in [6, Proposition 3.2]. By [loc. cit., Theorem 1.6], K_{−2ϖ_{ir}} Xr ; w̃_{0} is in adU_{q}(𝔫).K_{−2ϖ_{ir}}. Hence, by [7, Theorem 1.6], Xr ; w̃_{0} is in the space generated by B(ϖ_{i_{r}})^{∗}. By the Weyl character formula, see 1.3, Xr ; w̃_{0} is equal to an element of B(ϖ_{i_{r}})^{∗} up to a multiplicative scalar. This gives the lemma. ♢

Remark and Definition. We can choose the extremal vectors v_{y_{r}ϖ_{ir}} such that Xr ; w̃_{0} ∈ B^{∗}. Let m^{r}, such that B(m^{r})^{∗} = Xr ; w̃_{0}, 1 ≤ r ≤ N. The elements m^{r} can be easily calculated at least when the reduced decomposition w̃_{0} corresponds to some quiver orientation of the Dynkin diagram of 𝔤. Let m_{k}^{r} be the k-th entry of m^{r}. By [6, (3.2.2)] and (1.5.1) :

m_{k}^{r} = \{ \begin{matrix}1 & if i_{k} = i_{r} and k ≤ r \\ 0 & if not \end{matrix}

It is likely that this formula generalizes for all decomposition w̃_{0}.

Let Aw̃_{0} be the algebra generated by the Xr ; w̃_{0}, 1 ≤ r ≤ N. Then, by [6, Corollary 3.2]

Proposition. Fix a reduced decomposition w̃_{0} of w_{0}. The algebra Aw̃_{0} is an algebra of regular functions on a quantum space. Moreover, let Tw̃_{0} be the multiplicative part generated by the Xr ; w̃_{0}, 1 ≤ r ≤ N. Then Tw̃_{0} is an Ore set in Aw̃_{0} and T −1 ; w̃_{0}Aw̃_{0} =

T −1 ; w̃_{0}U_{q}(𝔫). ♢

2.2. Let’s start with a definition.

Definition. Fix a reduced decomposition w̃_{0} of w_{0}. A subalgebra A of ℂ_{q}[U^{−}] will be called adapted if it is spanned by a subset P^{∗} of B^{∗} which verifies the following properties :

(i) (multiplicativity) For all p_{i}^{∗} in P^{∗}, 1 ≤ i ≤ k, and all (n_{i}) ∈ ℤ_{≥0}^{k}, we have ∏_{i}(p_{i}^{∗})^{n_{i}} ∈ B^{∗} up to a power of q,

(ii) (maximality) for all q^{∗} in B^{∗}\P^{∗}, there exists p^{∗} in P^{∗} such that p^{∗} and q^{∗} are not

multiplicative. We now prove the main theorem of this article. It is a generalization of [21, Theorem 6.1] for all semi-simple Lie algebra 𝔤 and for all reduced decomposition of w_{0}.

Theorem. Fix a reduced decompositions w̃_{0} of w_{0}. The algebra Aw̃_{0} is adapted. Moreover, the Ore set Tw̃_{0}, see Proposition 2.1, is a subset of B^{∗} which spans Aw̃_{0}. Proof. Let’s prove that the algebra Aw̃_{0} is adapted. We know from Lemma 2.1 (iii) that the Xr ; w̃_{0} q-commute. Hence, the multiplicativity part is a consequence of Corollary 1.9. Indeed, the conditions of the corollary are satisfied by Proposition A in 1.7 and Proposition 1.8 (ii). The last assertion of the theorem holds.

We now prove the maximality property. Fix a reduced decomposition w̃′_{0} which is compatible with a quiver orientation. Let Cw̃′_{0} ; w̃_{0} be the set of parametrizations of elements of Tw̃_{0} for the Lusztig’s parametrization associated to w̃′_{0}. By Theorem 1.4, (1.5.1) and Proposition 2.1, Cw̃′_{0} ; w̃_{0} is the set of integral points of a simplicial cone in ℝ^{N}. Suppose q^{∗} in B^{∗}\Tw̃_{0}. Then, by Proposition 2.1, there exists an element p^{∗} in Tw̃_{0} such that u := p^{∗}q^{∗} ∈ Aw̃_{0}. Suppose that u is a monomial, i.e. belongs to Tw̃_{0} up to a multiplicative scalar. Then, Theorem 1.4 and (1.5.1) imply m + n ∈ Cw̃′_{0} ; w̃_{0}, where m and n are respectively the w̃′_{0}-parametrizations of p^{∗} and q^{∗}. As m is in the simplicial cone Cw̃′_{0} ; w̃_{0}, we have n ∈ Cw̃′_{0} ; w̃_{0}, which contradicts the hypothesis q^{∗} in B^{∗}\Tw̃_{0}. This ends the proof of the Theorem. ♢

In view of Proposition 2.1 and the previous theorem, we can see that the Berenstein-Zelevinsky conjecture is true ”up to localization”.

Remark. By [19, 14.2.5 and Lemma 1.2.8 (b)], the dual canonical basis is stable by σ. Hence, for all adapted algebra A, the algebra σ(A) is adapted. In the section 4.2, we give some examples for 𝔤 of type A_{2} and B_{2}.

3. The multiplicativity property of the q-center.

3.1. Let Z_{q} be the q-center of U_{q}(𝔫), i.e. the subspace of U_{q}(𝔫) generated by elements which q-commute with all homogeneous elements of U_{q}(𝔫). Let’s describe Z_{q} as a space and as an algebra.

Lemma. We have :

(i) For all λ in P^{+}, there exists an element z_{λ} in U_{q}(𝔫) such that v_{w_{0}λ}^{∗}(uv_{λ}) = (K_{−λ}z_{λ}, u), for all u in U_{q}(𝔟^{−}),

(ii) z_{λ} q-commutes with all homogeneous elements of U_{q}(𝔫) and Z_{q} is generated as a space by z_{λ}, λ ∈ P^{+}, (iii) z_{λ}z_{μ} = z_{λ+μ} and so, the algebra Z_{q} is generated by the z_{ϖ_{k}}, 1 ≤ k ≤ n, (iv) up to a multiplicative scalar z_{λ} belongs to the dual canonical basis. Proof. (i), (ii), and (iii) are proved in [4] and the proof of (iv) is similar to the proof of Lemma 2.1 by remarking that z_{λ} corresponds to the extremal vector v_{w_{0}λ}^{∗}. ♢

In the sequel, we choose v_{w_{0}λ} such that z_{λ} is in B^{∗}. Fix a reduced decomposition w̃_{0} = s_{i_{1}} … s_{i_{N}} of w_{0} and for all k, 1 ≤ k ≤ n, let r(k) be the greatest integer such that i_{r(k)} = k. Then, it is easily seen that z_{ϖ_{k}} = Xr(k) ; w̃_{0}. Set n_{k} = m_{r(k)}. Then, when w̃_{0} corresponds to a quiver orientation, Remark 2.1 gives :

(3.1.1)  z_{λ} = B∗ ; w̃_{0}(∑_{k} λ_{k}n_{k}), where λ = ∑ λ_{k}ϖ_{k} ∈ P^{+}.

3.2. As in the proof of Theorem 2.2, the following Proposition is a consequence of Corollary 1.9, Proposition B in 1.7 and Proposition 1.8 (ii).

Proposition. Let b^{∗} be in B^{∗} and let c^{∗} be in Z_{q} ∩B^{∗}, then b^{∗} and c^{∗} are multiplicative. ♢

Let J_{q} be the ideal of U_{q}(𝔫) generated by the z_{ϖ_{k}}, 1 ≤ k ≤ n. The previous proposition implies that J_{q} is generated as a space by a part B^{∗}(J_{q}) of B^{∗}. Let H_{q} be the subspace generated by the complementary set B^{∗}(H_{q}) = B^{∗}\B^{∗}(J_{q}).

Corollary The Z_{q}-module U_{q}(𝔫) is free with (canonical) basis B^{∗}(H_{q}), i.e. U_{q}(𝔫) = Z_{q} ⊗ H_{q}. The Berenstein-Zelevinsky conjecture is true on B^{∗} if and only if it is true on B^{∗}(H_{q}). Proof. Fix a reduced decomposition of w̃_{0} = s_{i_{1}} … s_{i_{N}} which is compatible with a quiver orientation. For all k, 1 ≤ k ≤ n, set I_{k} := {r, i_{r} = k}. remark that {1, …, N} is the disjoint union of the I_{k}, 1 ≤ i ≤ n. Let Pw̃_{0}(Z_{q}), resp. Pw̃_{0}(H_{q}), be the set of parametrizations of Z_{q} ∩ B^{∗}, resp. B^{∗}(H_{q}), for w̃_{0}. By Proposition 3.2, (3.1.1) and Remark 1.5, it is easy to describe Pw̃_{0}(Z_{q}) and Pw̃_{0}(H_{q}) : The semigroup Pw̃_{0}(Z_{q}) is generated by the n_{k}, 1 ≤ k ≤ n, whose r-th entry is one if r ∈ I_{k} and 0 if not. The set Pw̃_{0}(H_{q}) is the set of elements m of ℤ_{≥0}^{N} such that for all k, 1 ≤ k ≤ n, there exists r in I_{k} such that the r-th entry of m is 0. Now, for all m in ℤ_{≥0}^{N}, and k, 1 ≤ k ≤ n, let m(k) be the minimal r-th entry of m when r runs over I_{k}. It is clear that m = (∑_{k} m(k)n_{k}) + (m − ∑_{k} m(k)n_{k}) is the unique decomposition of m into a sum of an element of Pw̃_{0}(Z_{q}) and an element of Pw̃_{0}(H_{q}). This fact and the proposition implies that U_{q}(𝔫) is a free Z_{q}-module.

Now, suppose that the Berenstein-Zelevinsky conjecture is true on B^{∗}(H_{q}). Suppose that two elements b^{∗} and b^{′∗} of B^{∗} q-commute. Let’s prove that they are multiplicative. By the previous proposition and the previous assertion, we can decompose b^{∗} = b_{z}^{∗}b_{h}^{∗}, b^{′∗} = b_{z}^{′∗}b_{h}^{′∗}, up to a power of q, where b_{z}^{∗}, b_{z}^{′∗} ∈ Z_{q} ∩ B^{∗}, b_{h}^{∗}, b_{h}^{′∗} ∈ B^{∗}(H_{q}). By Theorem 1.4, two elements in U_{q}(𝔫) commute up to a scalar imply that they q-commute. If we suppose that b_{h}^{∗} and b_{h}^{′∗} do not q-commute, then we obtain that there exists u ∉ 𝕂b_{h}^{∗}b_{h}^{′∗} such that b_{h}^{∗}b_{h}^{′∗} = q^{m}b_{h}^{′∗}b_{h}^{∗} + u. This implies that b^{∗}b^{′∗} = q^{m_{1}}b^{′∗}b^{∗} + q^{m_{2}}ub_{z}^{∗}b_{z}^{′∗}, and ub_{z}^{∗}b_{z}^{′∗} ∉ 𝕂b^{∗}b^{′∗}. This is in contradiction with the fact that b^{∗} and b^{′∗} q-commute. Hence, b_{h}^{∗} and b_{h}^{′∗} q-commute. By the hypothesis, this implies that b_{h}^{∗}b_{h}^{′∗} belongs to B^{∗} up to a power of q. Hence, this also holds for b_{h}^{∗}b_{h}^{′∗}b_{z}^{∗}b_{z}^{′∗} by the previous proposition. This implies that b^{∗} and b^{′∗} are multiplicative. ♢

3.3. In this section, we prove that the intersection of all adapted algebras is Z_{q}. To be more precise :

Proposition. Each adapted algebra contains Z_{q}. Moreover, ⋂w̃_{0} Aw̃_{0} = Z_{q}, where w̃_{0} runs over the set of reduced decomposition of w_{0}. Proof. By Proposition 3.2 and by the maximality condition, each adapted algebra contains Z_{q}.

Remark that ⋂w̃_{0} Aw̃_{0} is spanned by its intersection with B^{∗}. Now, let b^{∗} ∈ ⋂w̃_{0} Aw̃_{0} ∩ B^{∗}. Then, by construction, b^{∗} q-commutes with the Xr ; w̃_{0}. In particular, b^{∗} q-commutes with X1 ; w̃_{0} = E_{i_{1}}, where w̃_{0} = s_{i_{1}}s_{i_{2}} … s_{i_{N}}. Since each s_{k}, 1 ≤ k ≤ n can be the first factor of a reduced decomposition of w_{0}, it implies that b^{∗} is in Z_{q}. So, the proposition holds. ♢

Remark that in the intersection formula, we can choose a subset of cardinal n of the set w̃_{0} of reduced decomposition of w_{0}.

4. Adapted semigroups and examples.

4.1. Fix a reduced decomposition w̃_{0} of w_{0} which corresponds to a quiver orientation. In this section, we study the set of parametrizations of A ∩ B^{∗}, where A is an adapted algebra, for w̃_{0}.

Proposition. Let A be a adapted algebra, then the set CA ; w̃_{0} of parametrizations of A∩B^{∗} for w̃_{0} is a subsemigroup of ℤ_{≥0}^{N}. Proof. This is a direct consequence of Theorem 1.4 and (1.5.1). ♢

In the sequel, CA ; w̃_{0} will be called adapted semigroup for w̃_{0}. For any reduced decompo-

sition w̃′_{0} of w_{0}, set Cw̃′_{0} ; w̃_{0} := CA_{}_{w̃′_{0}} ; w̃_{0} . Proposition 2.1 and Theorem 2.2 give :

Proposition. The subset Cw̃′_{0} ; w̃_{0} of ℤ_{≥0}^{N} is the set of integral points of a simplicial cone

of ℝ^{N}. Moreover, Cw̃′_{0} ; w̃_{0} generates ℤ^{N} as a group. ♢

We do not know if an adapted semigroup is in general the set of integral points of a real cone. If this case occurs, it will be called adapted cone.

4.2. In this section, we study the cases where 𝔤 is of type A_{2} and B_{2}. For each case, we fix a reduced decomposition w̃_{0} for the parametrization of the dual canonical basis and then, we give explicitly the generators of the cones ℂw̃′_{0} ; w̃_{0} for all w̃′_{0}. The generators

of ℂ\begin{matrix} & w̃′_{0} \\ & w̃′_{0} \end{matrix} are calculated from Remark 2.1 and the ”reparametrization” ℂw̃′_{0} ; w̃_{0} is calculated

with the help of [1, Proposition 7.1 (3)]. The images σ(ℂw̃′_{0} ; w̃_{0}) can be easily guessed by using [11, Proposition 8.2].

Example 1. Let 𝔤 be of type A_{2}. Fix w̃_{0} = s_{1}s_{2}s_{1}, w̃′_{0} = s_{2}s_{1}s_{2}. Then, ℂw̃_{0} ; w̃_{0} is generated by {(1, 0, 0), (0, 1, 0), (1, 0, 1)}. ℂw̃_{0} ; w̃′_{0} is generated by {(0, 0, 1), (0, 1, 0), (1, 0, 1)}. Both

cones are stable by σ and the union ℂw̃_{0} ; w̃_{0} ∪ ℂw̃_{0} ; w̃′_{0} is the whole semigroup ℤ_{≥0}^{N}.

Example 2. Let 𝔤 be of type B_{2} with minuscule weight ϖ_{2}. Fix w̃_{0} = s_{2}s_{1}s_{2}s_{1}, w̃′_{0} = s_{1}s_{2}s_{1}s_{2}. The previous construction gives four cones, with intersection Pw̃_{0}(Z_{q}) generated by (1, 0, 1, 0) and (0, 1, 0, 1). In the following, we present the four cones ℂw̃_{0} ; w̃_{0}, ℂw̃_{0} ; w̃′_{0}, σ(ℂw̃_{0} ; w̃_{0}),

σ(ℂw̃_{0} ; w̃′_{0}) and their canonical set of generators :

ℂw̃_{0} ; w̃_{0} {(1, 0, 0, 0), (0, 1, 0, 0), (1, 0, 1, 0), (0, 1, 0, 1)},

ℂw̃′_{0} ; w̃_{0} {(0, 0, 0, 1), (1, 0, 0, 1), (1, 0, 1, 0), (0, 1, 0, 1)},

σ(ℂw̃_{0} ; w̃_{0}) {(1, 0, 0, 0), (2, 0, 0, 1), (1, 0, 1, 0), (0, 1, 0, 1)},

σ(ℂw̃′_{0} ; w̃_{0}) {(0, 0, 0, 1), (0, 0, 1, 0), (1, 0, 1, 0), (0, 1, 0, 1)},

where σ is the map which commutes with the parametrization corresponding with w̃_{0}. With the same methods used in 2.1 and 2.2, we obtain that the semigroup D generated by

D {(0, 1, 0, 0), (0, 0, 1, 0), (1, 0, 1, 0), (0, 1, 0, 1)}

is an adapted cone. Hence, σ(D), which is generated by

σ(D) {(2, 0, 0, 1), (1, 0, 0, 1), (1, 0, 1, 0), (0, 1, 0, 1)},

is also an adapted cone. Let’s give a few details.

Lemma. The cones D and σ(D) are adapted. sketch of the proof. It is enough to prove that D is an adapted cone, see Remark 2.2. Set a = (0, 0, 1, 0), b = (0, 1, 0, 0), c = (1, 0, 1, 0), d = (0, 1, 0, 1). By (3.1.1), B∗ ; w̃_{0}(c) B∗ ; w̃_{0}(d) generate the q-center Z_{q}. By 3.2, in order to prove the multiplicativity part, it is enough to prove that, up to a power of q, B∗ ; w̃_{0}(a)^{a}B∗ ; w̃_{0}(b)^{b} is in B^{∗} for all non negative integer a and b. Through Corollary 1.9 and Proposition 1.7. C, this is implied by the following assertions :

1) B∗ ; w̃_{0}(a) and B∗ ; w̃_{0}(b) q-commute,

2) B∗ ; w̃_{0}(a)^{a} is in B^{∗} up to a power of q,

3) In the W -stratification of B^{∗}, see [11, 12.4], B∗ ; w̃_{0}(a)^{a} belongs to the s_{2}s_{1} component of B^{∗} and B∗ ; w̃_{0}(b) corresponds to the extremal vector v_{s_{2}s_{1}ϖ_{1}}^{∗} of B(ϖ_{1})^{∗}. Let sketch the proof for 1) 2) and 3). By Remark 1.5, we can describe the parametrizations of B(ϖ_{i})^{∗} for i = 1, 2. We then obtain that B∗ ; w̃_{0}(a) and B∗ ; w̃_{0}(b) are in B(ϖ_{1})^{∗}. Let v^{∗} and w^{∗} be weight vectors in V_{q}(ϖ_{1})^{∗} corresponding respectively to B∗ ; w̃_{0}(a) and B∗ ; w̃_{0}(b). It is easy to prove that for all positive root α, then E_{α}.v^{∗} is non zero implies that F_{α}.w^{∗} is zero. By [5, 1.5], this implies 1). Now, by [11, Proposition 8.2], σ(B∗ ; w̃_{0}(a)) is in B(ϖ_{2})^{∗}, so it corresponds to an extremal vector. By Proposition 1.7. A, Proposition 1.8 (ii) and Corollary 1.9, it gives that σ(B∗ ; w̃_{0}(a))^{a} is in B^{∗}. This implies 2). Moreover, we can obtain by considering the crystal graph of V_{q}(ϖ_{1}) that B∗ ; w̃_{0}(a) belongs to the s_{2}s_{1} component of B^{∗}. This implies 3). The maximality property is proved as in Theorem 2.2 by remarking that D generates ℤ^{N} as a group. ♢

Let G^{∗} be the finite subset of B^{∗} parametrized via w̃_{0} by the union of the canonical generators of the cones ℂw̃_{0} ; w̃_{0}, ℂw̃′_{0} ; w̃_{0}, σ(ℂw̃_{0} ; w̃_{0}), σ(ℂw̃′_{0} ; w̃_{0}), D, σ(D), then it is easily verified that

Proposition. We have the decomposition ℤ_{≥0}^{N} = ℂw̃_{0} ; w̃_{0}∪ℂw̃′_{0} ; w̃_{0}∪σ(ℂw̃_{0} ; w̃_{0})∪σ(ℂw̃′_{0} ; w̃_{0})∪D∪σ(D). Hence, each element of B^{∗} can be decomposed into a product of elements of G^{∗}. ♢

ACKNOWLEDGMENTS

We wish to thank Bernard Leclerc for useful conversations on the conjecture. We are grateful to Olivier Mathieu for an interpretation of the condition (i) of Proposition 1.9 in terms of Littelmann’s path model.

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Keywords : Quantum groups, Canonical Basis.