ON THE q-COMMUTATIONS IN U_{q}(𝔫) AT ROOTS OF ONE

Philippe CALDERO

ABSTRACT. Let ε be a root of one and 𝔤 a semi-simple Lie algebra with triangular decomposition 𝔤 = 𝔫+𝔥+𝔫^{−}. Let U_{ε}^{+}, resp. U_{ε}^{res+}, the non restricted, resp. restricted, quantum enveloping algebra of 𝔫. We prove that FractU_{ε}^{+} is a quantum Weyl field. We then give a description of the ε-center of U_{ε}^{+}. Let U_{ε}^{fin+} be the finite part of U_{ε}^{res+}. Via the Drinfeld correspondence, the U_{ε}^{fin+}-covariant space of a Weyl module is ε-central. In case 𝔤 = 𝔰𝔩_{n}, this enables us to describe this space in term of semi-standard Young tableaux.

0. Introduction.

Let q be a non zero complex number. A ℂ-algebra defined by generators X_{i}, 1 ≤ i ≤ m, and relations X_{i}X_{j} = q^{a_{i,j}}X_{j}X_{i}, 1 ≤ i < j ≤ m, a_{i,j} ∈ ℤ, will be called ”algebra of regular functions on an affine quantum space”. Its skew field of fractions will be called quantum Weyl field. The X_{i}, 1 ≤ i ≤ m, will be called system of q-commuting generators.

Let 𝔤 be a semi-simple Lie ℂ-algebra of rank n. Let Δ be the root system associated to the choice of a Cartan subalgebra 𝔥 and let Δ^{+} be the set of positive roots. We fix a decomposition of the longest element w_{0} of the Weyl group.

Let q be an indeterminate and U_{q}(𝔤) be the simply connected quantized enveloping algebra, defined as in [16, 3.2.9]. Let ε be a l-th root of one, l ≠ 2, and l ≠ 3 if 𝔤 has a component of type G_{2}. We define U_{ε}, resp. U_{ε}^{res}, to be the non restricted, resp. restricted, form as in [11]. As in the classical case let U_{ε}^{+}, resp. U_{ε}^{res+} be its ”nilpotent” subalgebra. Let E_{α}, α ∈ Δ^{+}, be the root vectors of U_{ε}^{+} and GrU_{ε}^{+} be its associated graded algebra, see 1.5.

Generalizing results of [1], [17], [10], we prove in 3.2 that FractU_{ε}^{+} is a quantum Weyl field, i.e. it contains an algebra A_{w_{0}} of regular functions on an affine quantum space, such that FractA_{w_{0}} =FractU_{ε}^{+}. The method we use specifies the description of the E_{α}’s in term of a system of ε-commuting generators of A_{w_{0}}. Moreover, the algebra A_{w_{0}} is isomorphic to GrU_{ε}^{+}. The two main tools of the proof are the following. The first one is the notion of roots package, see definition 2.2, which arises in Weyl modules, see Lemma 2.1 and Lemma 2.2. The second tool is the Drinfeld correspondence which can be made precise by the universal R-matrix, see 1.6.

An element in U_{ε}^{+} will be called ε-central if it ε-commutes with the E_{α}’s. As in [1] for the generic case, the realization of U_{ε}^{+} as a quantum Weyl field is of great help in order to obtain its ε-center. As for U_{ε}, see [13], Corollary 4.2 asserts that the center Z of U_{ε}^{+} is generated by the specialization Z_{ε} of the generic center, see [7], and the algebra Z_{0} generated by the E_{α}^{l}, α ∈ Δ^{+}. For this result, we first study the center of GrU_{ε}^{+} and then lift our results by the (abstract) isomorphism GrU_{ε}^{+} ≃ A_{w_{0}} ⊂FractU_{ε}^{+}.

In Section 5, we give an application of the ε-center to some covariants of Weyl modules. Let U_{ε}^{fin+} be the subalgebra of U_{ε}^{res+} generated by E_{α}, α ∈ Δ^{+}. Let λ be a dominant weight and V_{ε}(λ) be the U_{ε}^{res+}-Weyl module as in 1.7. Then, as in the generic case, its dual V_{ε}(λ)^{∗}, viewed as a U_{ε}^{fin+}-module, can be naturally embedded in U_{ε}^{+}, endowed with the adjoint action of U_{ε}^{fin+}, via the Drinfeld isomorphism. Inside U_{ε}^{+} the U_{ε}^{fin+}-invariants of V_{ε}(λ)^{∗} are ε-central elements, cf. Proposition 5.1. For 𝔤 = 𝔰𝔩_{n}, this provides a complete description of the U_{ε}^{fin+}-invariants of V_{ε}(λ)^{∗} in terms of semi-standard Young tableaux, c.f. Theorem 5.3.

We are in debt to T. Levasseur for his remarks and corrections.

1. Preliminaries and notations.

1.1. Let 𝔤 be a semi-simple Lie ℂ-algebra of rank n. We fix a Cartan sub-algebra 𝔥 of 𝔤. Let 𝔤 = 𝔫^{−} + 𝔥 + 𝔫 be the triangular decomposition and {α_{i}}_{i} be a basis of the root system Δ resulting from this decomposition. We note 𝔟 = 𝔫+𝔥 and 𝔟^{−} = 𝔫^{−} +𝔥 the two opposed Borel sub-algebras. Let P be the weight lattice generated by the fundamental weights ϖ_{i}, 1 ≤ i ≤ n, and P^{+} := ∑_{i} ℕϖ_{i} the monoid of integral dominant weights. Let W be the Weyl group, generated by the reflections corresponding to the simple roots s_{i} := s_{α_{i}}. Let w_{0} be the longest element of W. We denote by ( , ) the W -invariant form on P. For each root we set d_{α} = \frac{(α,α)}{2}, and αˇ = \frac{1}{d_{α}} α. We have (α_{j}ˇ , ϖ_{i}) = δ_{ij}.

1.2. Let q be an indeterminate and U_{q}(𝔤) be the simply connected quantized enveloping algebra, defined as in [16, 3.2.9]. Let U_{q}^{+}, resp. U_{q}^{−}, be the subalgebra generated by the canonical generators E_{i} = E_{α_{i}}, resp. F_{i} = F_{α_{i}}, of positive, resp. negative, weights. For all λ in P, let τ(λ) be the corresponding element in the algebra U^{0} of the torus of U_{q}(𝔤). We have the triangular decomposition U_{q}(𝔤) = U_{q}^{−} ⊗ U^{0} ⊗ U_{q}^{+}. We set

(1.2.1)  U_{q}(𝔟) = U_{q}^{+} ⊗ U^{0},  U_{q}(𝔟^{−}) = U_{q}^{−} ⊗ U^{0}.

U_{q}(𝔤) is endowed with a structure of Hopf algebra with comultiplication Δ, and antipode S.

We define in U_{q}(𝔤) the left and right adjoint actions by

ad v.u = v_{(1)}uS(v_{(2)}), u.ãdv = S(v_{(1)})uv_{(2)},

where u, v ∈ U_{q}(𝔤) and Δ(v) = v_{(1)} ⊗ v_{(2)} with the Sweedler notation. In particular, if u is an element in U_{q}(𝔤) of weight μ, then :

(1.2.2)  ad E_{i}(u) = E_{i}u − q^{(α_{i},μ)}uE_{i},  1 ≤ i ≤ n.

If n is a non negative integer and α a positive root, we set :

[n]_{α} = \frac{1 − q^{nd_{α}}}{1 − q^{d_{α}}} , [n]_{α}! = [n]_{α}[n − 1]_{α} … [1]_{α}.

1.3. The dual space U_{q}(𝔤)^{∗} is endowed with the natural left and right regular actions of U_{q}(𝔤) : u.c(v) = c(uv), c.u(v) = c(vu), u, v ∈ U_{q}(𝔤), c ∈ U_{q}(𝔤)^{∗}. For all λ in P^{+}, let V_{q}(λ) be the simple U_{q}(𝔤)-module with highest weight λ. For any integral dominant weight λ, we fix a weight basis {v_{μ}}, of V_{q}(λ) (this is a misleading notation because there is generally more than one vector of weight μ). We denote by {v_{μ}^{∗}} its dual basis. V_{q}(λ)^{∗} is endowed with a natural right U_{q}(𝔤)-module structure. For all ξ in V_{q}(λ)^{∗} and v in V_{q}(λ), let c_{ξ,v}^{λ} in U_{q}(𝔤)^{∗} given by c_{ξ,v}^{λ}(u) = ξ(uv), u ∈ U_{q}(𝔤). Then, we have u.c_{ξ,v}^{λ} = c_{ξ,uv}^{λ} and c_{ξ,v}^{λ}.u = c_{ξu,v}^{λ}. If ξ, resp. v, has weight ν, resp μ, we set (if no confusion occurs), c_{ν,μ}^{λ} = c_{ξ,v}^{λ}. For any integral dominant weight λ, let C(λ), resp. C^{+}(λ), be the space generated by the c_{ξ,v}^{λ}, resp. c_{ξ,λ}^{λ}, ξ ∈ V_{q}(λ)^{∗}, v ∈ V_{q}(λ). We set R = ⨁_{λ∈P^{+}} C(λ), R^{+} = ⨁_{λ∈P^{+}} C^{+}(λ). R^{+} and R are subalgebras of the Hopf dual of U_{q}(𝔤). R^{+} is a P^{+}-graded algebra and the C^{+}(λ)’s are its graded components. R, resp. R^{+}, is a U_{q}(𝔤), resp. U_{q}^{+}, module for the left and right regular actions. Moreover, the left and right adjoint actions of U_{q}(𝔤) provide right and left adjoint actions of U_{q}(𝔤), resp. U_{q}^{+}, on R, resp. R^{+}.

1.4. [16, Lemma 7.1.9] asserts that the natural morphism U_{q}(𝔤) → R^{∗} is an embedding. Let λ in P^{+}. By [16, 1.4.11, Remark], the U_{q}(𝔤)-module structure on V_{q}(λ) extends to a R^{∗}-module structure. For all w in W, we define the corresponding element N_{w} in the quantum Weyl group, cf. [16, 10.2.1], [18]. Set N_{i} := N_{s_{α_{i}}}. Recall the following facts :

(i) N_{w} is an invertible element of R^{∗}, [16, Lemma 10.2.2]. (ii) The automorphism T_{w} : a ↦ N_{w}aN_{w}^{−1} restricts to an automorphism of U_{q}(𝔤), and

identifies with the Lusztig automorphism, [19], associated with w, [16, Theorem 10.2.6, 10.5.2].

(iii) Set T_{i} := T_{s_{α_{i}}}. If w = s_{i_{1}} … s_{i_{k}} is a reduced decomposition of w, then T_{w} =

T_{i_{1}} … T_{i_{k}}, [19]. (iv) Let M be a finite dimensional U_{q}(𝔤)-module, i in [1, n] and v an E_{i}-invariant element

of M with weight μ. Then, up to a multiplicative power of q, N_{i}.v = \frac{F_{i}^{m}}{[m]_{α_{i}}!}v, where m = (μ, α_{i}ˇ ), [16, 10.2.2 (9)].

1.5. We fix a decomposition of the longest element of the Weyl group w_{0} = s_{i_{1}} … s_{i_{N}}, where N = dim 𝔫. Set y_{0} = Id, y_{l} = s_{i_{1}}s_{i_{2}} … s_{i_{l}}, 1 ≤ l ≤ N, β_{l} = y_{l−1}(α_{i_{l}}), 1 ≤ l ≤ N. We endow an order into the set Δ^{+} of positive roots : β_{N} < … β_{2} < β_{1}. We now introduce the following elements in U_{q}^{+}, resp. U_{q}^{−} : E_{β_{l}} = T_{y_{l−1}}(E_{i_{l}}), 1 ≤ l ≤ N, resp. F_{β_{l}} = T_{y_{l−1}}(F_{i_{l}}). Fix w in W and a reduced decomposition s_{w} of w. We may assume without loss of generality that the decomposition of w_{0} verifies s_{w_{0}} = s_{w}s_{w′}, w′ ∈ W. We shall say that the reduced decomposition s_{w} of w defines the elements E_{β_{l}}, F_{β_{l}}, 1 ≤ l ≤ l(w). For each positive root α, and all non negative integer n, set E_{α}^{(n)} = \frac{1}{[n]_{dα}!}E_{α}^{n}, F_{α}^{(n)} = \frac{1}{[n]_{dα}!}F_{α}^{n}. For each ψ ∈ ℕ^{N}, let E^{ψ}, E^{(ψ)}, F^{ψ}, F^{(ψ)} be

the ordered products E^{ψ} := ∏_{l=N}^{1} E_{β_{l}}^{ψ_{l}}, E^{(ψ)} := ∏_{l=N}^{1} E_{β_{l}}^{(ψ_{l})}, F^{ψ} := ∏_{l=N}^{1} F_{β_{l}}^{ψ_{l}}, and

F^{(ψ)} := ∏_{l=N}^{1} F_{β_{l}}^{(ψ_{l})}. The components of ψ in ℕ^{N} will be indexed by the positive roots : ψ = (ψ_{β_{1}}, ψ_{β_{2}}, …, ψ_{β_{N}}) = (ψ_{1}, ψ_{2}, …, ψ_{N}). The set ℕ^{N} is endowed with a natural lexicographic ordering.

We know, cf. [19], that these elements generate a Poincaré-Birkoff-Witt basis of U_{q}^{+}. Moreover, the order on Δ^{+} defines a natural filtration on U_{q}^{+}. The associated graded GrU_{q}^{+} is generated by the GrE_{β}, β ∈ Δ^{+} and the relations, [20], [12, Lemma 1.7] :

(1.5.1)  GrE_{β}GrE_{β′} = q^{(β,β′)}GrE_{β}GrE_{β′},  β > β′.

For each w in W, set

𝔫_{w} := 𝔫 ∩ w(𝔫^{−}), U_{q}(𝔫_{w}) := U_{q}^{+} ∩ T_{w}(U_{q}(𝔟^{−})).

Lemma. For 1 ≤ i ≤ n, w ∈ W, w = s_{i_{1}} … s_{i_{k}}, we have

(i) wα_{i} ∈ Δ^{+} iff T_{w}E_{i} ∈ U_{q}^{+} (ii) −wα_{i} ∈ Δ^{+} iff T_{w}E_{i} ∈ U_{q}(𝔟^{−}). (iii) U_{q}(𝔫_{w}) is the algebra generated by the E_{β_{j}}, 1 ≤ j ≤ k, defined by the above decom-

position of w. Proof. (i) is given by [19], [12, 1.6]. The hypothesis of (ii) implies that there exists p, 1 ≤ p ≤ k, such that s_{i_{p+1}} … s_{i_{k}}(α_{i}) = α_{i_{p}}. From (i), and the triangular decomposition, T_{w}E_{i} = T_{i_{1}} … T_{i_{p}}(E_{i_{p}}) = T_{i_{1}} … T_{i_{p−1}}(−F_{i_{p}}τ(α_{i_{p}})) ∈ U_{q}(𝔟^{−}). Let’s prove (iii). By the definition, U_{q}(𝔫_{w}) is an algebra. Let B_{w} be the space generated by the polynomial basis provided by the E_{β_{j}}, 1 ≤ j ≤ k. As a consequence of (ii), B_{w} ⊂ U_{q}(𝔫_{w}). Moreover, the reduced decomposition of w may be completed into a reduced decomposition of w_{0}. Thus, we can complete our polynomial basis into a polynomial basis of U_{q}^{+}. By (ii) and the triangular decomposition, we obtain the reverse inclusion. ♢

1.6. We know that U_{q}(𝔤) is an almost cocommutative Hopf algebra, cf. [14]. Let R = R_{(1)} ⊗ R_{(2)}, be the universal R-matrix of U_{q}(𝔤). We recall the expression of the R-matrix as an ordered product [18, 3.3] :

(1.6.1)  R = (∏_{α∈Δ^{+}} exp_{α}((1 − q_{α}^{−2})E_{α} ⊗ F_{α}))q^{H_{i}⊗H_{i}},

where (H_{i}) is an orthonormal basis in 𝔥 and exp_{α}(x) := ∑_{n≥0} \frac{x^{n}}{[n]_{α}!}. Let φ be the restriction homomorphism from U_{q}(𝔤)^{∗} to U_{q}(𝔟^{−})^{∗} and J^{−} = Kerφ ∩ R. The following is well known, cf. [4], [21], [5], [6] Theorem. We have :

(i) There exists an injective algebra antihomomorphism β^{+} : R/J^{−} → U_{q}(𝔟), given by : c ↦ (Id ⊗ c)(R), c ∈ R/J^{−},

(ii) There exists an injective adU_{q}(𝔤)-homomorphism ζ : R → U_{q}(𝔤) (iii) The natural projection π : R → R/J^{−} restricts into an embedding R^{+} ↪ R/J^{−} and for x ∈ C^{+}(λ), ζ(x) = τ(−λ)β^{+}(x) ⊂ τ(−2λ)U_{q}^{+} ♢

1.7. Let A be the algebra ℂ[q] localized at the multiplicative set generated by (q − 1), (q − ω), where ω is a k-th root of one, for k = 2, and k = 3 if 𝔤 has a component of type G_{2}. We define the A-algebra U_{A} generated by the E_{i}, F_{i}, 1 ≤ i ≤ n, τ(λ), λ ∈ P^{+}. The specialization of U_{A} at a root of one ε, (ε^{k} ≠ 1) will be denoted U_{ε}. We define in a similar way the A-forms U_{A}(𝔟^{+}), U_{A}(𝔟^{−}), U_{A}^{+}, U_{A}^{−}, U_{A}^{0} and the specializations U_{ε}(𝔟), U_{ε}^{+}, etc. As before A-forms and ε-specialization have a natural PBW-basis, [12]. Let U_{A}^{res+} be the A-algebra generated by the E_{α}^{(n)}, α ∈ Δ^{+}, 𝔫 ∈ ℕ. We define in a similar way the algebra U_{A}^{res−} and U_{A}^{res}, see [11, 9.3.1]. For all λ in P^{+}, we set V_{A}(λ) := U_{A}^{res−}v_{λ} ⊂ V_{q}(λ). By [11, proposition 10.1.4], V_{A}(λ) is a U_{A}^{res}-module and ℂ(q) ⊗_{A} V_{A}(λ) = V_{q}(λ); moreover, it is a free A-module. By 1.4 (iv), the extremal vectors v_{wλ} := N_{w}v_{λ}, w ∈ W, belong to V_{A}(λ). Hence, as T_{w} restricts to U_{A}^{res}, it follows that N_{w} acts on V_{A}(λ). We define as above the specializations U_{ε}^{res+}, U_{ε}^{res−}, V_{ε}(λ).

2. Integral modules and root packages. 2.1. A decomposition of w_{0} defines a partition of Δ^{+} in the following way. Definition and notation. Fix w_{0} = s_{i_{1}} … s_{i_{N}}. For 1 ≤ j ≤ n, we call root packages the sets Δ_{j}^{+} := {β_{l}, i_{l} = j}. For m, 1 ≤ m ≤ k :=CardΔ_{j}^{+}, we define α_{j,m} to be the m-th element in the decreasing sequence of the roots of Δ_{j}^{+} : α_{j,1} > α_{j,2} > … > α_{j,m} > … > α_{j,k}. Set Δ_{j,m}^{+} := {α_{j,t}, 1 ≤ t ≤ m}.

Remark that the partition of Δ^{+} into root packages depends on the choice of the reduced decomposition. In the case 𝔤 = 𝔰𝔩_{4}, the decompositions s_{1}s_{2}s_{3}s_{1}s_{2}s_{1} and s_{2}s_{1}s_{3}s_{2}s_{1}s_{3} of w_{0} define different partitions of Δ^{+} which are respectively :

{{α_{1}, α_{2}, α_{3}}, {α_{1} + α_{2}, α_{2} + α_{3}}, {α_{1} + α_{2} + α_{3}}},

{{α_{2}, α_{1} + α_{2} + α_{3}}, {α_{1} + α_{2}, α_{3}}, {α_{2} + α_{3}, α_{1}}}.

Let M be a A-module, v, w ∈ M. In the sequel, v ≡ w means that v equals w up to a (multiplicative) invertible element of A. Lemma. With the notation above, suppose β_{i_{p}} = α_{j,k}, β_{i_{q}} = α_{j,k+1}. Let ψ^{(j,k)} ∈ ℕ^{N} such that ψ_{l}^{(j,k)} = 1 if β_{l} ∈ Δ_{j,k}^{+}, and ψ_{l}^{(j,k)} = 0 otherwise. Let ϕ^{(j,k)} ∈ ℕ^{N} such that

ϕ_{l}^{(j,k)} = (α_{j}, α_{l}ˇ ) if p < l < q and ϕ_{l}^{(j,k)} = 0 otherwise. Let y_{l} be as in 1.5. Then :

(i) Set ψ := ψ^{(j,k+1)} and ϕ := ψ^{(j,k−1)} + ϕ^{(j,k)}. In V_{A}(ϖ_{j}), one has F^{(ψ)}v_{ϖ_{j}} ≡ F^{(ϕ)}v_{ϖ_{j}} ≡ v_{y_{q}ϖ_{j}},

(ii) If α ≤ α_{j,k+1}, then v_{y_{p}ϖ_{j}}^{∗}S(E_{α}) = δ_{α,α_{j,k+1}}v_{y_{q}ϖ_{j}}^{∗}. (iii) y_{p}ϖ_{j} − α_{j,k+1} = y_{q}ϖ_{j}.

Proof. Let’s prove that F^{(ψ)}v_{ϖ_{j}} ≡ v_{y_{q}ϖ_{j}}. We have to prove that F^{(ψ^{(j,k)})}v_{ϖ_{j}} ≡ v_{y_{p}ϖ_{j}} ⇒ F^{(ψ^{(j,k+1)})}v_{ϖ_{j}} ≡ v_{y_{q}ϖ_{j}}. The result then follows by induction. Write y_{p} = w_{1}s_{j}w_{2}s_{j} … w_{k}s_{j}, and y_{q} = w_{1}s_{j}w_{2}s_{j} … w_{k+1}s_{j}, where the w_{l} have no s_{j} in their decomposition. F^{(ψ^{(j,k+1)})}v_{ϖ_{j}} = F_{α_{j,k+1)}}F^{(ψ^{(j,k})}v_{ϖ_{j}} ≡ F_{α_{j,k+1}}v_{y_{p}ϖ_{j}} = N_{w_{1}s_{j}w_{2}s_{j}...w_{k+1}}F_{j}N_{w_{1}s_{j}w_{2}s_{j}...w_{k+1}}^{−1}v_{y_{p}ϖ_{j}} ≡ N_{w_{1}s_{j}w_{2}s_{j}...w_{k+1}}F_{j}N_{w_{k+1}}^{−1}v_{ϖ_{j}} = N_{w_{1}s_{j}w_{2}s_{j}...w_{k+1}}F_{j}v_{ϖ_{j}} ≡ N_{w_{1}s_{j}w_{2}s_{j}...w_{k+1}}v_{s_{j}ϖ_{j}} ≡ v_{y_{q}ϖ_{j}}.

We now prove that F^{(ϕ)}v_{ϖ_{j}} ≡ v_{y_{q}ϖ_{j}}. By the previous assertion, it is enough to prove that F^{(ϕ^{(j,k)})}v_{w_{1}s_{j}w_{2}s_{j}...w_{k−1}s_{j}ϖ_{j}} ≡ v_{y_{q}ϖ_{j}}. For p < l < q, set z_{l} = s_{i_{p+1}} … s_{i_{l}}. We prove that F_{β_{l}}^{−(α_{j},α_{l}ˇ )}v_{y_{p}z_{l−1}s_{j}ϖ_{j}} ≡ v_{y_{p}z_{l}s_{j}ϖ_{j}}. The result will follow by induction.

F_{β_{l}}^{−(α_{j},α_{l}ˇ )}v_{y_{p}z_{l−1}s_{j}ϖ_{j}} = N_{y_{p}z_{l−1}}F_{i_{l}}^{−(α_{j},α_{l}ˇ )}N_{y_{p}z_{l−1}}^{−1}v_{y_{p}z_{l−1}s_{j}ϖ_{j}} ≡ N_{y_{p}z_{l−1}}F_{i_{l}}^{−(α_{j},α_{l}ˇ )}v_{s_{j}ϖ_{j}} ≡ N_{y_{p}z_{l−1}}v_{s_{i_{l}} s_{j}ϖ_{j}} ≡ v_{y_{p}z_{l}s_{j}ϖ_{j}}.

Let α < α_{j,k+1}. Then, acting by T_{y_{p}}^{−1} gives F_{α}v_{y_{p}ϖ_{j}} = 0. Hence, by (i), α ≤ α_{j,k+1} ⇒ F_{α}v_{y_{p}ϖ_{j}} = δ_{α,α_{j,k+1}}v_{y_{q}ϖ_{j}}. This is (ii) via the Chevalley automorphism and it gives (iii). ♢

Example. Let 𝔤 of type B_{2}, with the notation of [2, Planche II]. Fix the following decomposition : w_{0} = s_{1}s_{2}s_{1}s_{2}. Then, the order in Δ^{+} is given by

α_{1} > α_{1} + α_{2} > α_{1} + 2α_{2} > α_{2}.

So,

Δ_{1}^{+} = {α_{1,1} = α_{1}, α_{1,2} = α_{1} + 2α_{2}}, Δ_{2}^{+} = {α_{2,1} = α_{1} + α_{2}, α_{2,2} = α_{2}}.

(i) gives α_{1} + (α_{1} + 2α_{2}) = 2(α_{1} + α_{2}), and (α_{1} + α_{2}) + α_{2} = (α_{1} + 2α_{2}). (ii) gives that F_{α_{1}+2α_{2}}F_{α_{1}}v_{ϖ_{1}} and F_{α_{1}+α_{2}}^{(2)}v_{ϖ_{1}} are extremal vectors of weight s_{1}s_{2}s_{1}ϖ_{1} in V_{A}(ϖ_{1}). Similarly, F_{α_{2}}F_{α_{1}+α_{2}}v_{ϖ_{2}} and F_{α_{1}+2α_{2}}v_{ϖ_{2}} are extremal vectors of weight s_{1}s_{2}s_{1}s_{2}ϖ_{1} in V_{A}(ϖ_{2}).

2.2. The integers j, k, p, are fixed as above. An element Λ ∈ ℕ^{N} is said to satisfy P iff F^{(Λ)}v_{ϖ_{j}} = v_{y_{p}ϖ_{j}}, up to a non zero scalar. Set χ = ψ^{(j,k)}. By the previous lemma, χ verifies P. Moreover : Lemma. Let Λ be in ℕ^{N} and suppose that Λ verifies P. Then :

(i) χ ≥ Λ. (ii) β ∉ [α_{j,k}, α_{j,1}] implies that Λ_{β} = 0. (iii) Λ_{α_{j,k}} ≤ 1. Proof. First, remark that if α_{j,1} < β then F_{β}v_{ϖ_{j}} = 0. Indeed, the α_{j} component of F_{β} is zero, by construction of F_{β}. Now, let β such that α_{j,k+1} < β < α_{j,k}. Then, by the previous remark we have F_{β}F^{(χ)}v_{ϖ_{j}} = F_{β}v_{y_{p}ϖ_{j}} ≡ N_{y_{p}}^{−1}T_{y_{p}}^{−1}(F_{β})v_{ϖ_{j}} = 0, (for the last equality, we can repeat the argument above, with T_{y_{p}}^{−1}(F_{β}) instead of F_{β}). This provides (i) by induction. We now prove (ii). By the previous assertion, we have β > α_{j,1} ⇒ Λ_{β} = 0. Hence, we can decompose Λ = Λ^{0} + Λ^{1}, where the support of Λ^{0}, resp. Λ^{1}, is in [β_{N}, α_{j,k}[, resp. [α_{j,k}, α_{j,1}]. Remark that, by Lemma 1.5, F := T_{y_{p}}^{−1}(F^{(Λ_{0})}) has a (strictly) negative weight. Suppose now F^{(Λ_{0})}F^{(Λ_{1})}v_{ϖ_{j}} = F^{(Λ)}v_{ϖ_{j}} ≡ v_{y_{p}ϖ_{j}}, then acting on both sides by N_{y_{p}}^{−1}, we obtain that v_{ϖ_{j}} ∈ F.V_{A}(ϖ_{j}). Hence a contradiction and (ii) holds. Suppose Λ_{α_{j,k}} > 1. Let Λ′ in ℕ^{N} such that Λ′_{β} = Λ_{β} − 1 if β = α_{j,k} and Λ′ = Λ_{β} otherwise. Then F^{(Λ′)}v_{ϖ_{j}} is non zero of weight y_{p}ϖ_{j} + α_{j,k}. Let r such that y_{r}α_{j} = α_{j,k−1} if k ≠ 1 and r = 0 if k = 1. Then, r < p and Lemma 2.1 (iii), applied to k − 1 gives y_{p}ϖ_{j} = y_{r}ϖ_{j} − α_{j,k}. Hence, F^{(Λ′)}v_{ϖ_{j}} = v_{y_{r}ϖ_{j}}, up to a non zero scalar. By the hypothesis v_{y_{r}ϖ_{j}} ∈ F_{α_{j,k}}V_{A}(ϖ_{j}) = T_{y_{p}}(α_{j})V_{A}(ϖ_{j}). As above, acting by T_{y_{r}}^{−1} on both sides yield a contradiction. So, (iii) holds. ♢

3. Functions on an affine quantum space and quantum Weyl fields. In the sequel, we fix a decomposition of w_{0}. For each λ in P^{+}, we fix a basis {v_{μ}} of the A-module V_{A}(λ). If {v_{μ}^{∗}} is the dual basis, we then have the matrix coefficient c_{v_{ν}^{∗},v_{μ}}^{λ}. For each symbol Y_{α_{j,k}}, we will frequently use the notation Y_{j,k}. 3.1. An algebra defined by generators X_{i}, 1 ≤ i ≤ m, and relations X_{i}X_{j} = q^{a_{i,j}}X_{j}X_{i}, 1 ≤ i < j ≤ m, a_{i,j} ∈ ℤ, will be called ”algebra of regular functions on an affine quantum space”. Its skew field of fractions will be called quantum Weyl field. The X_{i}, 1 ≤ i ≤ m, will be called a system of q-commuting generators (SQCG). The (antisymmetric) matrix A := (a_{i,j}) will be called matrix associated to the SQCG. Example 1 : By 1.5, {GrE_{β}, β ∈ Δ^{+}} is a system of q-commuting generators for GrU_{q}^{+}. Let A(𝔫) be the associated matrix. Then, A(𝔫) is a N × N antisymmetric matrix with the lower triangular part given by A(𝔫)_{β,β′} = (β, β′). Definition : Let β ∈ Δ^{+}, β = β_{i_{p}} = α_{j,k}. We define as in [10, 2.3] the following elements c_{β}(= c_{j,k}) of FractR^{+} :

c_{β} = c_{y_{p}ϖ_{ip} ,ϖ_{ip}}^{ϖ_{ip}} , a_{β} = c_{j,k−1}^{−1}c_{j,k},

with the convention c_{j,0} = c_{ϖ_{ip} ,ϖ_{ip}}^{ϖ_{ip}}. Example 2 : We shall prove in Corollary 3.2 and (3.3.2), see also [10, 2.3, 3.1] for another proof, that the {a_{β}, β ∈ Δ^{+}} is the SQCG of an algebra of regular functions on an affine quantum space whose associated matrix is the transpose of A(𝔫). ♢

3.2. We still denote by c_{β} and a_{β} the elements of FractU_{A}(𝔟^{+}) corresponding to c_{β}, a_{β} ∈ FractR^{+} via the Drinfeld antihomomorphism β^{+}, cf. 1.6, (in particular c_{j,0} = τ(−ϖ_{i_{p}})). These elements satisfy the following properties. Proposition. The c_{β}, resp. a_{β}, are q-commuting elements in the A-form U_{A}(𝔟^{+}), resp. Fract U_{A}^{+}. For all β = β_{p} in Δ^{+}, we have c_{β} = (E_{β}P + Q)τ(−ϖ_{l}), where P and Q are polynomials in E_{α}, α > β. Moreover, the ε-specialization of P is non zero.

Proof. The fact that the c_{β} and the a_{β} are q-commuting elements is a consequence of [10, Proposition 2.3]. By Theorem 1.6 (iii), c_{j,k} = c_{β} ∈ β^{+}(C^{+}(ϖ_{j})) ⊂ τ(−ϖ_{j})U_{q}(𝔤). This also holds for c_{j,k−1}, hence a_{β} ∈ Fract U_{q}^{+}. Set X_{β} = X_{j,k} = τ(ϖ_{j})c_{β} ∈ U_{q}^{+} By 1.6, we have :

(3.2.1)  X_{β} := ∑_{ψ∈ℕ^{N}} A_{ψ}E^{ψ}, where A_{ψ} = v_{y_{p}ϖ_{j}}^{∗}(F^{(ψ)}v_{ϖ_{j}}) ∏(1 − q_{α}^{−2})^{ψ_{α}}.

By the Weyl character formula, v_{y_{p}ϖ_{j}}^{∗}(F^{(ψ)}v_{ϖ_{j}}) is non zero iff F^{(ψ)}v_{ϖ_{j}} = v_{y_{p}ϖ_{j}}, up to a non zero scalar. Hence, X_{β} is a sum of monomials A_{ψ}E^{ψ}, where ψ verifies P and A_{ψ} ∈ U_{A}^{+}. By [Lemma 2.2 (ii), (iii)], we have X_{β} = (E_{β}P + Q), where P and Q are polynomials in E_{α}, α > β. By [Lemma 2.2 (i)], A_{χ}E^{χ} is one of these monomials. By construction, it can be factorized on the left by E_{β} and by Lemma 2.1 (i), Aχ is invertible in A. Hence, the ε-specialization of P is non zero. ♢

Let β = α_{j,k}. We retain the notation of the proof above and we set X_{β} = X_{j,k} = τ(ϖ_{j})c_{β} ∈ U_{A}^{+}. Then, (3.2.1) and Lemma 2.2 (i) give (up to an unit in A) :

(3.2.2)  GrX_{j,k} = ∏_{m≤k} GrE_{α_{j,m}}.

In the following corollary, we fix an element w in W and its reduced decomposition s_{w}. We assume as in 1.5 that the decomposition of w_{0} verifies s_{w_{0}} = s_{w}s_{w′}, w′ ∈ W. We have : Corollary. Let A_{w} be the algebra generated by the a_{β}, where β runs over the weights of 𝔫_{w}. Then, A_{w} is an algebra of regular functions on an affine quantum space, with GK-dimension l(w). Fract U_{A}(𝔫_{w}) =Fract A_{w}, in particular Fract U_{A}^{+} =Fract A_{w_{0}} and the associated matrix of A_{w_{0}} is A(𝔫). This remains true after ε-specialization. Proof. A_{w} is generated by the a_{β}, which q-commute. From the assumption preceding the corollary, if β ∈ 𝔫_{w} and α > β, then α ∈ 𝔫_{w}. Then, the equality Fract U_{A}(𝔫_{w}) =Fract A_{w} follows from Lemma 1.5 (iii) and Proposition 3.2. Recalling that the E^{ψ}, ψ ∈ ℕ^{N} form a PBW-basis of U_{A}^{+}, we deduce that the only relations in A_{w} are the q-commutations. ♢

3.3. In this section, we make more precise the q-commutations inside U_{A}(𝔟^{+}). By [16, Proposition 9.1.5),

(3.3.1) c_{β}c_{β′} = q^{(s(β),s(β′))−(s(β),ϖ_{j′})−(ϖ_{j},s(β′))}c_{β′}c_{β},  β = α_{j,k} > β′ = α_{j′,k′},

where s(β) is the weight of c_{β}. By (3.2.2), we have s(α_{j,k+1}) = s(α_{j,k}) + α_{j,k+1} (*). It follows that :

(3.3.2) a_{β}a_{β′} = q^{−(β,β′)}a_{β′}a_{β}, X_{β}X_{β′} = q^{(s(β),s(β′)−2ϖ_{j′})}X_{β′}X_{β}, β > β′ = α_{j′,k′}.

Recall that for all λ in P, τ(λ) is a group-like element, and so, adτ(λ) is an automorphism of U_{q}(𝔤).

Lemma. Suppose α_{j,k} not almost minimal. Let σ = adτ(α_{j,k+1}) and D be a σderivation on U_{q}(𝔤), i.e. D(uv) = D(u)v + σ(u)D(v), u, v ∈ U_{q}(𝔤). Suppose D(c_{j,k}) = c_{j,k+1}. Then, D(c_{j,k}^{n}) = [n]_{α_{j,k+1}}c_{j,k}^{n−1}c_{j,k+1}, up to a power of q.

Proof. By (3.2.2) and (1.5.1), we have X_{j,k}X_{j,k+1} = q^{(s(α_{j,k}),α_{j,k+1})}X_{j,k+1}X_{j,k} (**). Comparing with (3.3.2), we obtain (s(α_{j,k}), s(α_{j,k})) = −2(s(α_{j,k}), ϖ_{j}). Hence, (s(α_{j,k+1}), s(α_{j,k+1})) = −2(s(α_{j,k+1}), ϖ_{j}). Using (*) in the left hand side gives

(3.3.3)  (α_{j,k+1}, s(α_{j,k}) = (α_{j,k+1}, ϖ_{j}) − \frac{1}{2}(α_{j,k+1}, α_{j,k+1}).

Using (3.3.3) and (**), we obtain σ(c_{j,k})c_{j,k+1} = q^{(α_{j,k+1},s(α_{j,k}))}c_{j,k}c_{j,k+1} = q^{m}c_{j,k+1}c_{j,k}, with m = (α_{j,k+1}, s(α_{j,k})) − (α_{j,k+1}, ϖ_{j}) − \frac{1}{2}(α_{j,k+1}, α_{j,k+1}). Using (3.3.3) again, we have m = (α_{j,k+1}, α_{j,k+1}). We have proved that σ(c_{j,k})c_{j,k+1} = q^{(α_{j,k+1},α_{j,k+1})}c_{j,k+1}c_{j,k}. This implies the lemma by induction on n.

4. Center of U_{ε}^{+}.

Recall that by convention, we denote by the same symbol an element and its ε-specia-lization (when it exists). For example, we set X_{β} = X_{j,k} = τ(ϖ_{j})c_{β} ∈ U_{ε}^{+} and these elements exist by Proposition 3.2. 4.1. We suppose in the sequel that l is odd, and not divisible by 3 if 𝔤 has a componant of type G_{2}. Let ε be a l-th root of one.

Let β = α_{j,k} ∈ Δ_{j}^{+}. We say that β is an almost minimal root if β is minimal in Δ_{j}^{+}, i.e β = α_{j,m}, m =CardΔ_{j}^{+}. Set X_{j,0} = 1 and X_{j,k+1} = 0 if α_{j,k} is almost minimal. If β is almost minimal, then let z_{j} := X_{β} = X_{j,k}. For λ ∈ P^{+}, λ = ∑ n_{j}ϖ_{j}, set z_{λ} = ∏ z_{j}^{n_{j}}.

Definition. An ε-central element a in FractU_{ε}^{+} is an element which ε-commute with the generators E_{i}, i.e there exists a weight η (unique modulo lP) such that aE_{i} = ε^{(α_{i},η)}E_{i}a. The class of η modulo lP will be called the commutation weight of a. Lemma. Let β = α_{j,k}, λ ∈ P^{+} :

(i) if β is not almost minimal and γ = α_{j,k+1}, α ≤ γ, then : adE_{α}(c_{β}) = δ_{γ,α}c_{γ} in R^{+},

(ii) z_{λ} has weight λ − w_{0}λ and commutation weight (Id + w_{0})(λ) modulo lP. (iii) Grz_{j} = ∏_{α∈Δ_{j}^{+}} GrE_{α}.

Proof. (i) is a consequence of Lemma 2.1 (ii) and the definition of c_{β}, cf. 3.1. (ii) is provided by [8, 2.2.1].

(iii) is a particular case of (3.2.2) ♢

Remark. If u = z_{0}z_{μ}, z_{0} ∈ Z_{0}, has weight λ and commutation weight λ′, then, by (ii), μ = \frac{1}{2}(λ + λ′) modulo lP. The elements z_{λ} and z_{μ} have the same commutation weight iff λ = μ + lP + Ker(Id + w_{0}). 4.2. The group H generated by −w_{0} acts naturally on [1, n] by h(α_{i}) = α_{h(i)}, h ∈ H. Set −w_{0}(i) = i^{∗}. Let I be a subset of [1, n] containing exactly one representative for each class modulo H and \overline{I} its complementary subset. Let Z_{ε} be the algebra generated by the z_{i}z_{i^{∗}}, if i ≠ i^{∗}, and the z_{i}, if i = i^{∗}. Then, by (ii) of the previous lemma, Z_{ε} is central in U_{ε}^{+}. Remark that Z_{ε} may be seen as the specialization of the generic center, see [7]. As ε is a l-th root of one, we know, [12], that E_{β}^{l}, β ∈ Δ^{+}, is also a central in U_{ε}^{+}. Let Z_{0} be the algebra generated by those elements.

Let \overline{P} the lattice P/lP + Ker (Id + w_{0}). As in [8, 2.2], each class of elements in \overline{P} has a representative in P^{+}. For all λ in P^{+}, let C^{λ} be the line in U_{ε}^{+} generated by z_{λ}. For all \overline{π} dans \overline{P}, set C\overline{π} = ⨁_{\overline{λ}=\overline{π}} C^{λ} (this is a direct sum by Lemma 4.1 (iii)). The following theorem describes the ε-central elements in U_{ε}^{+}.

Theorem. Let C_{ε} be the set of ε-central elements in U_{ε}^{+}. Then C_{ε} = ⋃_{\overline{λ}∈\overline{P}} Z_{0}C\overline{λ}. In particular, C_{ε} ⊂ ⨁_{μ∈P^{+}} Z_{0}z_{μ}. Proof. Step 1. Let a ∈ C_{ε} be of the form ∏ X_{β}^{n_{β}}, 0 ≤ n_{β} < l then a ∈ ⋃_{λ∈P^{+}} C^{λ}. Proof. We have to show that n_{β} = 0 for β non almost minimal. We prove this assertion by (reverse) induction on β. Suppose that α_{j,k} is non almost minimal and n_{β} = 0 for β > α_{j,k}. Let γ = α_{j,k+1} and γ^{>} be the smallest root greater than γ. By Lemma 4.1 (i) and Theorem 1.6, β in Δ_{i}^{+}, β > α_{j,k} implies that τ(−2ϖ_{i})X_{β} is adE_{γ^{>}}-invariant. Hence, by [18, 2.4.1], see also [10, 2.1], adE_{γ} acts as a twisted derivation on the algebra generated by those τ(−2ϖ_{i})X_{β}. To be more precise, on this algebra, adE_{γ} is a derivation twisted by the automorphism adτ(γ). Let τ = ∏_{1≤i≤n,β∈Δ_{i}^{+}} τ(−2n_{β}ϖ_{i}). On the one hand, by the (quantum) Leibniz formula, Lemma 4.1 (i) and Lemma 3.3 give : adE_{γ}(τa) = [n_{j,k}]_{γ}τaX_{j,k}^{−1}X_{j,k+1}. On the other hand, as a is ε-central, we have adE_{j,k+1}(τa) = E_{j,k+1}τa, up to a multiplicative scalar (which may be zero). This gives [n_{j,k}]_{γ}X_{j,k+1} = E_{j,k+1}X_{j,k}, up to a scalar. By (3.2.1) and Lemma 2.1 (i), the left hand side contains E^{(ϕ)} in its PBW-decomposition, and the right hand side does not. Hence, [n_{j,k}]_{γ} = 0. Thus, n_{j,k} = 0 and the assertion follows inductively. ♢ Step 2. Each monomial b in GrU_{ε}^{+} such that bGrE_{β} = ε^{(β,ν)}GrE_{β}b, ν ∈ P, lifts to ⋃_{\overline{λ}∈\overline{P}} Z_{0}C\overline{λ}.

Proof. Let b = ∏ GrE_{β}^{n_{β}}. As GrE_{β}^{l} lifts to Z_{0}, we can restrict to the case 0 ≤ n_{β} < l. Let c := ∏ a_{β}^{n_{β}} ∈ FractU_{ε}^{+}. By Corollary 3.2, the hypothesis on b implies that ca_{β} = ε^{(β,ν)}a_{β}c. This easily implies by Proposition 3.2 that cE_{β} = ε^{(β,ν)}E_{β}c. In particular, c is ε-central in FractU_{ε}^{+}. Remark now that c = ∏ X_{j,k}^{n_{j,k}−n_{j,k+1}}, with the convention n_{j,k+1} = 0 if n_{j,k} is almost minimal. Define r_{α_{j,k}} = r_{j,k} by r_{j,k} ≡ n_{j,k} − n_{j,k+1} [l] and 0 ≤ r_{j,k} < l. Remark that |n_{j,k} − n_{j,k+1}| < l. Hence, multiplying c by an (unique) element z_{0} in Z_{0}, we obtain that ∏ X_{j,k}^{r_{j,k}} is ε-central in U_{ε}^{+}. So, Step 1 gives z_{0}c ∈ ⋃_{λ∈P^{+}} C^{λ}. In particular, r_{β} = 0, for β non almost minimal. Thus, z_{0} = 1. Gr(c) = b by (3.2.2) and c is the required element. ♢ We can now finish the proof of the theorem. Let a ∈ C_{ε} and b :=Gra. Then, b verifies the hypothesis of Step 2. Let c_{0} in ⋃_{\overline{λ}∈\overline{P}} Z_{0}C\overline{λ} such that Grc_{0} = b. a and c_{0} having the same commutation weight, c_{1} := a − c_{0} is in C_{ε} and has a lower degree. Hence, the theorem follows by induction. ♢

Remark that if w_{0} = −Id, then C_{ε} is the center of U_{ε}^{+}. In general, we have :

Corollary. Let Z be the center of U_{ε}^{+}. Then,

(i) GrZ is the center of GrU_{ε}^{+}, (ii) Z is generated by Z_{ε} and Z_{0}, i.e. Z = Z_{0}[Z_{ε}]. (iii) U_{ε}^{+} is projective over its center, of rank l^{N−#I}. Proof. (i) and (ii) are direct consequences of Step 2 and the previous theorem. Let S be the multiplicative set in GrU_{ε}^{+} generated by the GrE_{β}, β non almost minimal or β = α_{i,k}, i ∈ \overline{I}. By Lemma 4.1 (iii), S^{−1}GrU_{ε}^{+} is free over S^{−1}GrZ, of rank l^{N−#I}. This yields (iii). ♢

Remark 1. Recall that for this result, we assumed l odd. If l is even, then Z_{0} has to be replaced by a larger algebra Z′_{0}. This algebra can easily be obtained for each simple Lie algebra 𝔤. In fact, every argument in our proofs remains true except for the last assertion of the proof of Step 1. Remark 2. We conjecture that the assertion (i) is true for U_{ε}(𝔤), i.e. the center of GrU_{ε} is the graded of the center. This conjecture has been verified with Maple V for ”small” classical Lie algebras 𝔤 and all exceptional Lie algebras.

4.3. The center of an algebra of functions on a quantum space is directly connected with the kernel of its associated matrix. So, we deduce the following result, which may be written in term of root systems (and roots packages). Corollary. Fix a decomposition of w_{0}. Let A(𝔫) as in [3.1, Example 1]. The rank of A(𝔫) is N − #I. Let B(𝔫) be the submatrix of A(𝔫) obtained by omitting the almost minimal roots of Δ_{i}^{+}, i ∈ I. Then the determinant of B(𝔫) is a power of 2 (and 3 if 𝔤 has a component of type G_{2}). ♢

5. U_{ε}^{+}-invariants in V_{ε}(λ)^{∗}.

5.1. By [11, 9.3.6], U_{ε}^{res+}, c.f. 1.7, is generated by the E_{i}, E_{i}^{(l)}, 1 ≤ i ≤ n. By [Loc cit], the E_{i} generate a finite dimensional Hopf sub-algebra U_{ε}^{fin+}. Define the U_{A}^{res+}-module morphism, c.f. Theorem 1.6 :

ζ_{A}^{λ} : V_{A}(λ)^{∗} → V_{A}(λ)^{∗} ⊗ v_{λ} ↪ C^{+}(λ) \underrightarrow{ζ} τ(−2λ)U_{q}^{+}.

The following proposition gives a characterization of the U_{ε}^{res+}-invariants in V_{ε}(λ)^{∗}. Proposition. The image of ζ_{A}^{λ} is contained in τ(−2λ)U_{A}^{+} and, by specialization, ζ_{A}^{λ} defines an injective morphism of U_{ε}^{res+}-module : ζ_{ε}^{λ} : V_{ε}(λ)^{∗} ↪ τ(−2λ)U_{ε}^{+}. Moreover, setting P_{λ}^{+} := P^{+} ∩ (λ + lP), we have :

V_{ε}(λ)^{∗U_{ε}^{fin+}} = (ζ_{ε}^{λ})^{−1}( τ(−2λ) ∑_{μ∈P_{λ}^{+}} Z_{0}z_{μ}),

Proof. Let u^{∗} ∈ V_{A}(λ)^{∗}. By 1.6, ζ_{A}^{λ}(u^{∗}) = τ(−2λ) ∑ a_{ψ}E^{ψ}u^{∗}(F^{(ψ)}v_{λ}), where the a_{ψ} are units in A. If v^{∗} belongs to a basis of V_{A}(λ)^{∗}, then v^{∗}(F^{(ψ)}v_{λ}) ∈ A. Hence, ζ_{A}^{λ}(v^{∗}) ∈ τ(−2λ)U_{A}^{+}. Suppose ζ_{A}^{λ}(u^{∗}) ∈ (q − ε)τ(−2λ)U_{A}^{+}, then u^{∗}(F^{(ψ)}v_{λ}) ∈ (q − ε)A for all ψ. Thus, u^{∗} ∈ (q − ε)V_{A}(λ)^{∗}. This proves the injectivity of the specialization of the U_{ε}^{res+}-module morphism ζ_{ε}^{λ}. We now claim that (ζ_{ε}^{λ})^{−1}( τ(−2λ) ∑_{μ∈P_{λ}^{+}} Z_{0}z_{μ}) ⊂

V_{ε}(λ)^{∗U_{ε}^{fin+}}. Indeed, τ(−2λ) ∑_{μ∈P_{λ}^{+}} Z_{0}z_{μ} is adE_{i}-invariant by (1.2.2) and Lemma 4.1 (ii).

We now prove the reverse inclusion. Suppose u^{∗} ∈ V_{ε}(λ)^{∗U_{ε}^{fin+}} has weight ν. Then, τ(−2λ)a := ζ_{ε}^{λ}(u^{∗}) is adE_{i}-invariant by Theorem 1.6, with weight λ + ν. This implies by (1.2.2), that a is ε-central and that the commutation weight of a is equal to λ − ν. By Theorem 4.2 and Remark 4.1 , we have a ∈ ∑_{μ∈P_{λ}^{+}} Z_{0}z_{μ}. This gives the claimed equality. ♢

5.2. Let τ(−2λ)K_{λ}^{+} be the image of V_{ε}(λ)^{∗} by the embedding ζ_{ε}^{λ}. By [6, Proposition 4.1 (iii)], we have

(5.2.1)  K_{λ}^{+}K_{μ}^{+} ⊂ K_{λ+μ}^{+}.

By Proposition 5.1, the vector space V_{ε}(λ)^{∗U_{ε}^{fin+}} is isomorphic to the intersection of K_{λ}^{+} and ∑_{μ∈P_{λ}^{+}} Z_{0}z_{μ}. Both spaces can be made explicit, but the intersection is not quite clear. In return, the intersection of the associated graded spaces can be made explicit. We shall prove in 5.3 that, when 𝔤 = 𝔰𝔩_{n}, the associated graded of the intersection is the intersection of the associated graded spaces. First, we give the following lemma.

Lemma. Let λ ∈ P^{+} and X be an extremal vector in K_{λ}^{+}, i.e. τ(−2λ)X corresponds via ζ_{ε}^{λ} to an extremal vector in V_{ε}(λ)^{∗}. Then, X^{l} ∈ ∑_{μ∈lP^{+}} Z_{0}[z_{μ}].

Proof. It is enough to prove that X^{l} is central. Indeed, by Remark 4.1 and Theorem 4.2, a central element whose weight belongs to lQ^{+} is in ∑_{μ∈ \frac{l}{2} Q^{+}∩P^{+}} Z_{0}[z_{μ}] which is a subset of ∑_{μ∈lP^{+}} Z_{0}[z_{μ}] (recall that l is odd).

First, we prove this assertion for λ = ϖ_{j}, 1 ≤ j ≤ n. Suppose that X corresponds to the extremal vector v_{wϖ_{j}}^{∗}, w ∈ W, with w minimal for the length. Thus we have a reduced decomposition s_{w} = w_{1}s_{j} … w_{k}s_{j}, where the w_{l} have no s_{j} in their decomposition. There exists an element w′ of the Weyl group with reduced decomposition s_{w′} such that s_{w}s_{w′} is a reduced decomposition of w_{0}. Fix this decomposition of w_{0} and set β := w_{1}s_{j} … w_{k}(α_{j}). Then, X is the X_{β} defined in the proof of 3.2. By (3.3.2) and Corollary 3.2, X^{l} commutes with the generators of A_{w_{0}}, thus it commutes with U_{ε}^{+} and we have the claimed assertion.

Suppose that λ = ∑_{i} λ_{i}ϖ_{i}. Suppose also that X corresponds to the extremal vector v_{wλ}^{∗} ∈ V_{ε}(λ)^{∗} and that X_{j} corresponds to v_{wϖ_{j}}^{∗} ∈ V_{ε}(ϖ_{j})^{∗}. Then, ∏ X_{j}^{lλ_{j}} and X^{l} are equal by (5.2.1) because they both correspond to v_{wlλ}^{∗} ∈ V_{ε}(lλ)^{∗}. The lemma follows. ♢

5.3. In this section, we suppose 𝔤 of type A_{n}, see [2, Table I]. Let ϵ_{i}, 1 ≤ i ≤ n + 1 be such that α_{i} = ϵ_{i} − ϵ_{i+1}, 1 ≤ i ≤ n, ∑_{i=1}^{n+1} ϵ_{i} = 0. Fix the decomposition of w_{0} :

w_{0} = (s_{n} … s_{1})(s_{n} … s_{2}) … (s_{n}).

This decomposition settles the following order on Δ^{+}, c.f. 1.5 :

α_{n} > α_{n} + α_{n−1} > … > α_{n} + … + α_{1} > α_{n−1} > … > α_{n−1} + … + α_{1}

> … > α_{2} + α_{1} > α_{1}.

The roots packages, c.f. 2.1, are defined by

(5.3.1)  α_{i,k} = α_{i−k+1} + … + α_{n−k+1},  1 ≤ k ≤ i ≤ n.

The principal symbols of the generators of the center are given by :

(5.3.2).  z_{i} = ∏_{k=1}^{i} E_{α_{i,k}}, 1 ≤ i ≤ n, z_{λ} = ∏_{i} E_{α_{i,k}}^{λ_{i}}

Let λ = ∑ λ_{i}ϖ_{i} ∈ P^{+}, and a_{i} = ∑_{j=i}^{n} λ_{j}. A Young diagram of shape λ is a left justified sequence of rows with a_{i} boxes in the i-th row. A semi-standard Young tableau of shape λ is a filling of the boxes with numbers 1 to n + 1 such that (R) : the entries are not decreasing in the rows (C) : the entries are strictly increasing in the columns.

Let 𝕋, resp. 𝕋_{λ}, be the set of semi-standard Young tableaux, resp. semi-standard Young tableaux of shape λ. We know that #𝕋_{λ} =dimV_{ε}(λ)^{∗}. Moreover, there exists a basis B_{λ} of V_{ε}(λ)^{∗} and a one to one correspondence 𝕋_{λ} → B_{λ} such that a semi-standard tableau T maps to a vector of weight− ∑_{i} k_{i}ϵ_{i}, where k_{i} is the number of i in T. We can define 𝕋_{λ}^{μ} to be the set of semi-standard tableaux of shape λ with weight μ. If n ≥ i_{1} ≥ … ≥ i_{k} ≥ 1, T_{m} ∈ 𝕋_{ϖ_{im}}, and λ := ∑_{m} ϖ_{i_{m}}. Then, the T_{m} are single columns with i_{m} rows, and we naturally define the concatenation T_{1} ∗ … ∗ T_{k} which belongs to 𝕋_{λ} if (R) and (C) are verified.

Let η : 𝕋 →GrU_{ε}^{+} such that η(T) = ∏_{β∈Δ^{+}} GrE_{β}^{n_{β}}, where n_{α_{i}+...+α_{j}}, 1 ≤ i ≤ j ≤ n is the number of j + 1 in the i-th row of T. Let η_{λ} be its restriction to 𝕋_{λ}.

Example. Let 𝔤 be of type A_{2}. The following tableau T

\begin{matrix}| 1 & | 1 & | 2 & | 3 & | 3 & | \\ | 2 & | 3 & | \end{matrix}

is of shape λ = 3ϖ_{1} + 2ϖ_{2} and corresponds to a vector of weight μ = −2ϵ_{1} − 2ϵ_{2} − 3ϵ_{3}. Let T_{1}, …, T_{5} be its single columns from left to right. We have T_{1}, T_{2} ∈ 𝕋_{ϖ_{2}}, T_{3}, T_{4}, T_{5} ∈ 𝕋_{ϖ_{1}}. The concatenation T_{1} ∗…∗T_{5} gives T. η(T) =GrE_{α_{1}}GrE_{α_{1}+α_{2}}^{2}GrE_{α_{2}}, η(T_{1}) = 1, η(T_{2}) =GrE_{α_{2}}, η(T_{3}) =GrE_{α_{1}}, η(T_{4}) = η(T_{5}) =GrE_{α_{1}+α_{2}}.

Lemma. For all λ in P^{+}, η_{λ} yields a one to one correspondence between 𝕋_{λ} and a basis of GrK_{λ}^{+}. Moreover, if T ∈ 𝕋_{λ}^{μ}, then the weight of η(T) is μ + λ. Proof. We have #𝕋_{λ} =dimGrK_{λ}^{+}. Moreover, the numbers m in the i-th row belong to [i, n + 1], thus η_{λ} is an embedding. So, by a cardinality argument, it is enough to prove that η maps 𝕋_{λ} into GrK_{λ}^{+}.

We sketch a proof of this assertion for λ = ϖ_{i}. With the notation of 1.6, let {v_{i}}, be a basis of V_{q}(ϖ_{1}) and C := (c_{v_{i}^{∗},v_{j}}) be the matrix with coefficients in C(ϖ_{1}) |_{U_{q}(𝔟)} corresponding to this basis. As in [5, 4.3], we can calculate β^{+}(C) from the representation V_{q}(ϖ_{1}). Taking the graded, we have, for an appropriate basis :

Grτ(ϖ_{1})β^{+}(C) = ( \begin{matrix}1 \\ GrE_{1} & 1 & & (0) \\ GrE_{12} & GrE_{2} & 1 \\ ⋮ & ⋮ & ⋮ & ... \\ GrE_{1...n} & GrE_{2...n} & … & GrE_{n} & 1 \end{matrix} ) , up to a power of q.

Now, as in [loc. cit.], K_{ϖ_{i}}^{+}, 1 ≤ i ≤ n, are generated by the quantum minors whose columns are the i first columns of τ(ϖ_{1})β^{+}(C). Let T ∈ 𝕋_{ϖ_{i}} and t_{1} < t_{2} … < t_{i} be its components. It is easily verified that η(T) is the graduate of the quantum minor Δ_{{t_{1},...,t_{i}},{1,...,i}}. This gives the result.

Now, for all λ, we can decompose T ∈ 𝕋_{λ} into single columns T = T_{1} ∗ … ∗ T_{k}. We have η(T) = ∏ η(T_{m}) ∈ ∏ K_{ϖ_{im}}^{+} ⊂ K_{λ}^{+}, by (5.2.1). The assertion on weights can be verified for T in 𝕋_{ϖ_{i}} as above and then generalized for all λ. ♢

Theorem. For T ∈ 𝕋, 1 ≤ i ≤ n, 1 ≤ k ≤ n − i + 1, let m_{i,k} = m_{i,k}(T) be the number of k + i on the i-th row. Then, for λ = ∑ λ_{i}ϖ_{i} ∈ P^{+}, the character formula of the U_{ε}^{fin+} invariants in V_{ε}(λ)^{∗} is given by :

dimV_{ε}(λ)_{ν}^{∗U_{ε}^{fin+}} = #{T ∈ 𝕋_{λ}^{ν} | ∀i, k, m_{i,k} ≡ λ_{n−k+1} [l]}.

Proof. First remark that, by the definition and by (5.3.1), η(T) = ∏ GrE_{α_{i,k}}^{m_{i−k+1,n−i+1}}. We prove the theorem by a double inequality.

≤ : Fix a basis {k_{ν}} of K_{λ}^{+U_{ε}^{fin+}} := τ(2λ)ζ_{ε}^{λ}(V_{ε}(λ)^{∗U_{ε}^{fin+}}) such that {Grk_{ν}} is a basis

of GrK_{λ}^{+U_{ε}^{fin+}}. We suppose k_{ν} of weight ν + λ, i.e. k_{ν} identifies with a vector of weight

ν in V_{ε}(λ)_{ν}^{∗U_{ε}^{fin+}}, and Grk_{ν} = ∏GrE_{α_{i,k}}^{a_{i,k}}. Then, by Proposition 5.1, Grk_{ν} is in Gr(∑_{μ∈P_{λ}^{+}} Z_{0}z_{μ}) and this imply a_{i,k} ≡ λ_{i} [l] by (5.3.2). By the first remark and the previous Lemma, η_{λ}^{−1}(Grk_{ν}) is in {T ∈ 𝕋_{λ}^{ν} | ∀i, k, m_{i,k} ≡ λ_{n−k+1} [l]}. So, ≤ holds. ≥ : let T in the set defined above. By Proposition 5.1 and the previous Lemma, it is enough to prove that η(T) ∈GrK_{λ}^{+} can be lifted into K_{λ}^{+} ∩ ∑_{μ∈P_{λ}^{+}} Z_{0}z_{μ}. We now show this assertion.

Let m_{k} :=Inf_{i}m_{i,k}. From T and i ∈ [1, n], we construct a new tableau as follows: on the j-th row of T, 1 ≤ j ≤ i, cancel the rightmost m_{n−i+1} boxes containing n−i+j +1, the (non empty) boxes remaining on the right handside being moved to the left, see 5.4.

By [5.4, Claim 1], we obtain a semi-standard tableau T′ ∈ 𝕋_{λ−m_{n−i+1}ϖ_{i}}. Write T \underrightarrow{i} T′. By (5.3.2) and the first remark of the proof, we have

(5.3.3)  η(T) = η(T′)Grz_{i}^{m_{n−i+1}}.

Now, let T = T_{n} \underrightarrow{n} T_{n−1} … \underrightarrow{1} T_{1} = T̃. By hypothesis on T, T̃ is a semi-standard tableau of shape λ̃ := λ − ∑_{i} m_{n−i+1}ϖ_{i}, and m_{i,k}(T̃) ≡ 0 [l]. On the one hand, this implies, by [5.4, Claim 2], that T̃ is a concatenation of l-columns, where a l-column is a the l-th exponent (for concatenation) of a single column. On the other hand, the fondamental weights are minuscule, hence, by the previous Lemma, the image by η of a single column (in 𝕋_{ϖ_{i}}) can be lifted into an extremal vector (in K_{ϖ_{i}}^{+}). By Lemma 5.2, η(T̃) can be lifted into Kλ̃ ∩ ∑_{μ∈lP^{+}} Z_{0}[z_{μ}]. By (5.3.3), η(T) = η(T̃)Grz_{λ−λ̃}, and the assertion follows. ♢

5.4. This section is an appendix for section 5.3.

Let T be in 𝕋_{λ}, a_{i} = ∑_{j=i}^{n} λ_{i}, and t_{i,j}, 1 ≤ i ≤ n, 1 ≤ j ≤ a_{i}, be the entry in the i-th row, j-th column of T. Fix i_{0}, 1 ≤ i_{0} ≤ n. Let m be an integer such that 1 ≤ m ≤ a_{i_{0}} and k : [1, i_{0}] → ℕ, such that

a) 1 ≤ k(i) ≤ k(i) + m − 1 ≤ a_{i},

b) k is decreasing,

c) t_{i_{0}+1,k(i_{0})} = ∅. Let T′ be the tableau obtained from T by cancelling t_{i,j} 1 ≤ i ≤ i_{0}, k(j) ≤ j ≤ k(j) + m − 1, the boxes remaining on the right handside being moved to the left. In clear, for 1 ≤ i ≤ i_{0}, t′_{i,j} = t_{i,j} if j < k(i) and t′_{i,j} = t_{i,j+m−1}, otherwise. Then, Claim 1. T′ is a semi-standard Young tableau of shape λ − mϖ_{i}. Proof. Clearly, T′ verifies (R) and, by b), c) it verifies (C). The number of entries in the i-th row of T′, is λ_{i} − m if 1 ≤ i ≤ i_{0} and λ_{i} otherwise. This gives λ of shape λ − mϖ_{i}. ♢ Example. In this example, 𝔤 is of type A_{5}, λ = 2ϖ_{1} +ϖ_{2} +2ϖ_{3} +3ϖ_{4}. In the following tableau T, we have underlined the entries to cancel.

| 1 | 1 | 2 | 3 | \underline{3} | \underline{4} | 4 | 5 |
| 2 | 3 | 3 | \underline{4} | \underline{5} | 5 |
| 3 | 4 | 5 | \underline{5} | \underline{6} |
| 5 | 5 | 6 |

The cancellation corresponds to m = 2, i_{0} = 3, k(1) = 5, k(2) = k(3) = 4. We then obtain T′ :

| 1 | 1 | 2 | 3 | 4 | 5 |
| 2 | 3 | 3 | 5 |
| 3 | 4 | 5 |
| 5 | 5 | 6 |

Claim 2. With the notation of Theorem 5.3, suppose T ∈ 𝕋_{λ}, λ ∈ lP^{+}, such that n_{i,k}(T) ∈ lℕ, 1 ≤ i ≤ n, 1 ≤ k ≤ n − i + 1. Then, T is a concatenation of l-columns, where a l-column is the l-th exponent (for concatenation) of a single column. Proof. By construction, the entries t of the i-th row of a semi-standard tableau verify i ≤ t ≤ n + 1. By hypothesis, n_{i,k}(T) ∈ lℕ, hence each entry t_{i}, i + 1 ≤ t_{i} ≤ n + 1, in the i-th row occurs a l-multiple of times. As λ ∈ lP^{+}, the number of boxes in the i-th row is a multiple of l. So, each entry in the i-th row occurs a l-multiple of times. By (R), the l first columns are the l-th exponant of a single column. The result is obtained by reverse induction. ♢

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