ON HARMONIC ELEMENTS FOR SEMI-SIMPLE LIE ALGEBRAS

Philippe CALDERO

ABSTRACT. Let 𝔤 be a semi-simple complex Lie algebra, 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫 its triangular decomposition. Let U(𝔤), resp. U_{q}(𝔤), be its enveloping algebra, resp. its quantized enveloping algebra. This article gives a quantum approach to the combinatorics of (classical) harmonic elements and Kostant’s generalized exponents for 𝔤. A quantum analogue of the space of harmonic elements has been given in [21]. On the one hand, we give specialization results concerning harmonic elements, central elements of U_{q}(𝔤), and the Joseph and Letzter’s decomposition, see [21]. For 𝔤 = 𝔰𝔩_{n+1}, we describe the specialization of quantum harmonic space in the ℕ-filtered algebra U(𝔰𝔩_{n+1}) as the materialization of a theorem of Lascoux-Leclerc-Thibon, [29]. This enables us to study a Joseph-Letzter decomposition in the algebra U(𝔰𝔩_{n+1}). On the other hand, we prove that highest weight harmonic elements can be calculated in terms of the dual of Lusztig’s canonical basis. In the simply laced case, we parametrize a basis of 𝔫-invariants of minimal primitive quotients by the set C_{0} of integral points of a convex cone.

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 V be a complex finite dimensional space and G a Lie subgroup of GL(V). Assume that V is completely reducible as a G-module. Then G acts semi-simply on the symmetric ℂ-algebra S(V). Let J be the ideal generated by the non-constant homogeneous G-invariant elements of S(V). Let V^{∗} be the dual of V. There exists a natural non-degenerate duality between S(V) and S(V^{∗}). Let H^{∗} ⊂ S(V^{∗}) be the orthogonal of J, H^{∗} is the space of G-harmonic polynomials of V.

Suppose now that G is a complex simply connected semi-simple Lie group acting by the adjoint action on its associated Lie algebra 𝔤 = V. Then, the Killing form maps the harmonic space H^{∗} onto H ⊂ S(𝔤). We have S(𝔤) = H ⊕ J, cf [27].

Let U(𝔤) be the enveloping algebra of 𝔤 and Z(𝔤) be its center. Recall that U(𝔤) is naturally filtred and the associated graded algebra of U(𝔤) is S(𝔤), by the Poincaré-Birkhoff-Witt theorem. As in [27], we can choose a sub-ad𝔤-module H of U(𝔤) such the graded space associated to H is H. The space H will be called harmonic space in U(𝔤). Kostant’s separation theorem states that U(𝔤) ≃ Z(𝔤) ⊗ H via multiplication. 0.2. Fix a Cartan subalgebra 𝔥 of 𝔤 and let 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫 be the triangular decomposition of 𝔤. Let P be the group of integral weights of 𝔤 and P^{+} be the semi-group of dominant weights of 𝔤, generated by the fundamental weights ϖ_{i}, 1 ≤ i ≤ n. The set Δ, resp. Δ^{+}, is the root system, resp. the set of positive roots, and W is the Weyl group. For all λ in P^{+}, let V (λ) be the simple 𝔤-module with highest weight λ.

Let’s sketch a few results on H that can be found in [27]. First of all, H is an ad𝔤-module and the multiplicity of V (λ) in H is

(0.2.1)  k_{λ} := [H : V (λ)] = dim V (λ)_{0},

where V (λ)_{0} is the 0-weight subspace of V (λ). Let H(λ) be the isotypical component of V (λ) in H and let S(𝔤)^{n} be the n-th graded component of S(𝔤). Then, there are integers m_{λ}^{1} ≤ … ≤ m_{λ}^{k_{λ}} such that

(0.2.2)  ∑_{n≥0}[H(λ) ∩ S(𝔤)^{n} : V (λ)]t^{n} = ∑_{i=1}^{k_{λ}} t^{m_{λ}^{i}},

as polynomials in t. The m_{λ}^{i} are called Kostant’s generalized exponents. Indeed, they generalize the usual exponents of 𝔤 because they can be realized as exponents of the eigenvalue of the Coxeter element acting on the zero weight space of V (λ), [27]. Two problems arise : Problem 1. Find some combinatorial rules that calculates the generalized exponents. Problem 2. Calculate the harmonic elements of U(𝔤). 0.3. Let’s sketch results about Problem 1. For μ ∈ P, let P(μ) be the dimension of the space of weight μ in S(𝔫). P is Kostant’s partition function. In the completed group algebra ℂ < P >, cf. [14, 7.5], we have ∑_{μ∈P} P(μ)e^{−μ} = ∏_{α∈Δ^{+}}(1 − e^{−α})^{−1}. Lusztig’s q-analogue of Kostant’s partition function is given by ∑_{μ∈P} P_{q}(μ)e^{−μ} = ∏_{α∈Δ^{+}}(1 − qe^{−α})^{−1}, where q is an indeterminate. For λ ∈ P^{+} and μ ∈ P, the Kostka-Foulkes polynomial is defined by

(0.3.1)  K_{λμ}(q) = ∑_{w∈W} (−1)^{l(w)}P_{q}(w(λ + ρ) − μ − ρ),

where l(w) is the length of w and ρ is the sum of the fundamental weights. By [14, 7.5.10], K_{λμ}(1) = dim V (λ)_{μ}. R. Brylinski gives the following interpretation of Hesselink formula :

Theorem.[17], [7] For λ ∈ P^{+}, K_{λ 0}(q) = ∑_{i=1}^{k_{λ}} q^{m_{λ}^{i}}.

But this is not an answer to Problem 1 because of the signs in (0.3.1).

We define the multi-variable Kostka-Foulkes function 𝕂_{λ 0}(X_{1}, …, X_{n}) in terms of the set B(λ)_{0} of zero weight elements in the crystal basis of V (λ) by

𝕂_{λ 0}(X_{1}, …, X_{n}) = ∑_{b∈B(λ)_{0}} ∏_{i=1}^{n} X_{i}^{ε_{i}(b)},

where ε_{i} are the usual parameters of the crystal basis, [24]. This function can be easily calculated, at least for the classical case, by the combinatorics described in [25]. In [29], there is given some nice combinatorics for the generalized exponents in the case A_{n} : the Kostka-Foulkes function K_{λ 0}(q) is recovered by specializing X_{i} on q^{i} in 𝕂_{λ 0}(X_{1}, …, X_{n}) :

Theorem. [29] For λ ∈ P^{+}, K_{λ 0}(q) = 𝕂_{λ 0}(q, q^{2}, …, q^{n}).

So much for Problem 1 in the A_{n} case. Now, Problem 2 has no known general solution even in 𝔰𝔩_{3}.

0.4. The article aims to present a quantum theory of harmonic spaces for enveloping algebras of semi-simple Lie algebras, in order to use it to get results in the classical case.

Let q be an indeterminate. Let U_{q}(𝔤) be the q-deformation of the enveloping algebra U(𝔤). For λ in P^{+}, let V_{q}(λ) be the quantum simple module with highest weight λ.

The algebra U_{q}(𝔤) contains the algebra F (U_{q}(𝔤)) of ad-finite elements of U_{q}(𝔤), see 1.4. We use Joseph-Letzter’s decomposition theorem as a basic tool, [21]. It asserts that, see Theorem 1.4,

(0.4.1)  F (U_{q}(𝔤)) = ⨁_{λ∈P^{+}} F_{q}^{λ},

where F_{q}^{λ} are sub-adU_{q}(𝔤)-modules of F (U_{q}(𝔤)) isomorphic to V_{q}(λ)^{∗} ⊗ V_{q}(λ) endowed with the diagonal action. This can be seen, [9], as a Peter-Weyl theorem embedded in U_{q}(𝔤) via the Rosso form, [34]. The center Z of U_{q}(𝔤) is as in the classical case a polynomial algebra with dimension n = rk(𝔤). The invariant elements z_{i} of F_{q}^{ϖ_{i}}, where 1 ≤ i ≤ n, are algebraically independent generators of Z. Let J_{q} be the ideal in F (U_{q}(𝔤)) generated by the z_{i}. We will define an harmonic space H_{q} to be a complementary adU_{q}(𝔤)-module of J_{q} in F (U_{q}(𝔤)), such that H_{q} = ⊕_{λ∈P^{+}}F_{q}^{λ} ∩ H_{q}, with some compatibility conditions, see 1.8, Definition 1.

0.5. The second tool is what we call Ringel’s filtration : a Poincaré-Birkhoff-Witt basis of U_{q}(𝔤) can be lexicographically ordered in such a way that the graded algebra associated to this ordering, see 1.3, is q-commutative. We prove that, for this ordering : Theorem 1. The set {z_{i}, 1 ≤ i ≤ n} is a Gröbner basis of the ideal J_{q}. We obtain that U_{q}(𝔤) is free over its center, with a set of explicit generators given in terms of roots packages, [11]. We prove an analogue of this result for quantized enveloping algebras at a root of one.

Theorem 2. When q = ε is a root of one, then the algebra U_{ε}, see [12, 1.5], is free over its center. As before, an explicit set of generators can be given.

0.6. For λ ∈ P^{+}, let H_{q}(λ) be the isotypical component of type V_{q}(λ) in H_{q}. A result of Joseph and Letzter asserts that (0.2.1) holds in the quantum case. Now, Joseph-Letzter’s decomposition, see (0.4.1), affords a natural P^{+}-filtration on F (U_{q}(𝔤)). By analogy with equation (0.2.2), we define P^{+}-exponents of λ as the family μ_{λ}^{1},…, μ_{λ}^{k_{λ}} of dominant weights such that

∑_{μ∈P^{+}}[H_{q}(λ) ∩ F_{q}^{μ} : V_{q}(λ)]e^{μ} = ∑_{i=1}^{k_{λ}} e^{μ_{λ}^{i}}

in the group algebra of P. We prove, see Proposition 1.8 and Proposition 3.4, that the quantum analogues of Problem 1 and Problem 2 can be solved in terms of Lusztig’s canonical basis and its dual : Proposition. Let μ in P^{+}. Let B_{H} be the set of elements b of the canonical basis such that ε_{i}(b) ≤ −(wt(b), α_{i}^{∨}), 1 ≤ i ≤ n. Then,

(i) ∑_{λ}[H_{q} ∩ F_{q}^{λ} : V_{q}(μ)] ∏_{i} X_{i}^{<λ,α_{i}^{∨}>} = 𝕂_{μ 0}(X_{1}, …, X_{n}), (ii) a basis of the space of 𝔫-invariants of H_{q} is labelled by B_{H}. Note that (i) can be also easily deduced from a result of Baumann, [1]. In particular, the multi-variable Kostka-Foulkes function 𝕂_{λ 0}(X_{1}, …, X_{n}) can be described as the Poincaré polynomial of H_{q}(λ) in the P^{+}-Joseph-Letzter filtration.

0.7. Another related problem is the following : Problem 3. Describe a Joseph-Letzter decomposition inside the classical enveloping algebra U(𝔤) endowed with its natural filtration. This includes the problem of specialization of harmonic elements as well as the specialization of the center. For general 𝔤, we give some preparation theorems. Let A be the algebra ℂ[q] localized at q = 1. We present a A-form H_{A} which links the quantum harmonic space with the classical one. Indeed, the spaces H_{q} = ℂ(q) ⊗_{A} H_{A} and H_{1} = ℂ ⊗_{A} H_{A} both verify the separation theorem at, respectively, the quantum and the classical level. We also prove a separation theorem on A-forms.

For 𝔤 = 𝔰𝔩_{n+1}, the understanding of the specialization of a) the quantum harmonic space and b) the center, see [8], enables us, by the separation theorem, to give an answer to Problem 3, see Theorem 4.3.

0.8. Recall that any reduced decomposition of the longest element w_{0} ∈ W in the Weyl group gives rise to a Poincaré-Birkhoff-Witt basis of U_{q}(𝔫), whose elements are labelled by points in ℕ^{N}, where N =dim𝔫. When 𝔤 has type A-D-E, then by [31], [32], the dual of the canonical basis is in bijection with the PBW basis, with nice multiplicative properties, see (3.5.1)-(3.5.3). Hence, each element of the canonical basis corresponds to an integral point in ℝ^{N}, where N = dim𝔫. Theorem. Let C_{0} ⊂ ℕ^{N} be the image of B_{H} via the correspondence above. Then, C_{0} is the set of integral points in a convex cone.

We describe a degenerescence of the variety Specℂ[C_{0}] which is related with the combinatorics of Point 2, see Theorem 3.5.

1. Preliminaries and notations.

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 {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 opposite Borel sub-algebras. Let Δ^{+} be the set of positive roots, Q be the lattice of radical weights, P be the weight lattice generated by the fundamental weights ϖ_{i}, 1 ≤ i ≤ n, and P^{+} := ∑_{i} ℕϖ_{i} the semigroup of integral dominant weights. P is endowed with the ordering ⪯

λ ⪯ μ ⇔ μ − λ ∈ P^{+}.

We fix a total additive ordering ≤ on P such that μ − λ ∈ Q^{+} ⇒ λ ≤ μ for all λ, μ in P. Such an ordering always exists. For example, for λ = ∑_{i} λ_{i}α_{i}, μ = ∑_{i} μ_{i}α_{i}, λ ≤ μ iff (∑_{i} λ_{i}, λ_{1}, …, λ_{n}) preceeds (∑_{i} μ_{i}, μ_{1}, …, μ_{n}) for the lexicographical ordering of ℚ^{n+1} verifies the hypothesis. 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 note ( , ) a W -invariant form on P. For β ∈ Δ^{+}, ht(β) will mean the height of β.

1.2. Let q be an indeterminate and U_{q}(𝔤) be the simply connected quantized enveloping algebra, defined as in [13, 0.2-0.3]. 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. For all λ in P, let K_{λ} be the corresponding element in the algebra U_{q}^{0} = ℂ(q)[P] of the torus of U_{q}(𝔤). As in the classical case, we have the triangular decomposition : U_{q}(𝔤) ≃ U_{q}(𝔫^{−}) ⊗ U_{q}^{0} ⊗ U_{q}(𝔫).

U_{q}(𝔤) is endowed with a structure of Hopf algebra with comultiplication Δ, antipode S and augmentation ε, [18, 3.2.9].

We define in U_{q}(𝔤) the left adjoint action by ad v.u = v_{(1)}uS(v_{(2)}), where Δ(v) = v_{(1)} ⊗ v_{(2)} with the Sweedler notations.

Let M be a U_{q}(𝔤)-module and μ ∈ P. Let M_{μ} be the space of elements of weight μ, i.e M_{μ} := {u ∈ M, K_{α}m = q^{(α,μ)}m, ∀α ∈ P }.

1.3. 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 β_{k} = s_{i_{1}} … s_{i_{k−1}}(α_{i_{k}}), 1 ≤ k ≤ N. We endow with an order the set Δ^{+} of positive roots : β_{N} < … β_{2} < β_{1} = α_{i_{1}}, see [12, 1.7]. For all β in Δ^{+}, let E_{β}, resp. F_{β}, be the root elements of U_{q}(𝔫), resp. U_{q}(𝔫^{−}). Set E_{i} = E_{α_{i}}, F_{i} = F_{α_{i}}, for each simple root. For each ψ in ℕ^{N}, set E_{ψ} := ∏_{l=N}^{1} E_{β_{l}}^{ψ_{l}}, resp. F_{ψ} := ∏_{l=N}^{1} F_{β_{l}}^{ψ_{l}}. For Γ = (ϕ, λ, ψ) ∈ ℕ^{N} × P × ℕ^{N}, set X_{Γ} := F_{ϕ}K_{−λ}E_{ψ}. We know, cf. [12, 1.6], that the E_{ψ}, resp. F_{ϕ}, resp. X_{Γ}, form a Poincaré-Birkhoff-Witt basis of U_{q}(𝔫), resp U_{q}(𝔫^{−}), resp. U_{q}(𝔤), for ψ ∈ ℕ^{N}, resp. ϕ ∈ ℕ^{N}, resp. Γ ∈ ℕ^{N} × P × ℕ^{N}. To Γ = (ϕ, λ, ψ) ∈ ℕ^{N} × P × ℕ^{N}, we associate Γ̃ = (∑(ϕ_{l} + ψ_{l})ht(β_{l}), ϕ, ψ) ∈ ℕ^{2N+1}. The set of subspaces

{ ⨁_{Γ̃′≤Γ̃} ℂ(q)X_{Γ′}, Γ ∈ ℕ^{N} × P × ℕ^{N}},

for the lexicographic ordering on ℕ^{2N+1} define a filtration of algebra on U_{q}(𝔤). In the sequel, this will be called the Ringel filtration, [33]. The associated graded GrU_{q}(𝔤) is generated by the GrF_{β}, GrK_{λ}, GrE_{β}, where β ∈ Δ^{+}, λ ∈ P, and q-commuting relations, [12, Proposition 1.7]. In particular, we have

(1.3.1)  GrE_{α}GrE_{β} = q^{(α,β)}GrE_{β}GrE_{α}, if β < α

1.4. For all λ in P^{+}, let V_{q}(λ) be the simple U_{q}(𝔤)-module with highest weight λ and highest weight vector v_{λ}. Let V_{q}(λ)^{∗} be its dual, endowed with a structure of left U_{q}(𝔤)-module twisted by the antipode. We define the ad-finite part of U_{q}(𝔤) by

F (U_{q}(𝔤)) := {u ∈ U_{q}(𝔤), dim ad U_{q}(𝔤)(u) < +∞}.

We have the theorem, [18, 7.1], [9] :

Theorem. F (U_{q}(𝔤)) has the following properties :

(i) F (U_{q}(𝔤)) is an algebra and a semi-simple module for the adjoint action, (ii) F (U_{q}(𝔤)) = ⨁_{λ∈P^{+}} adU_{q}(𝔤).K_{−2λ}, (iii) The module adU_{q}(𝔤).K_{−2λ} is isomorphic to V_{q}(λ)^{∗} ⊗ V_{q}(λ). ♢

1.5. As in [11, 2.1], we define the root packages from a decomposition of w_{0}.

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 any j, we enumerate the elements of Δ_{j}^{+} in the decreasing order : α_{j,1} > α_{j,2} > … > α_{j,m} > … > α_{j,k}, where k =CardΔ_{j}^{+}. Remark that for some decompositions of w_{0}, the map α_{j,m} ↦ α_{j,m+1} corresponds to the translation functor in the Auslander-Reiten quiver, [3]. 1.6. Let ( , ) be the canonical pairing between U_{q}(𝔫^{−}) and U_{q}(𝔫), see [35], B be Lusztig’s canonical basis of U_{q}(𝔫^{−}), [31, par 14], and B^{∗} ⊂ U_{q}(𝔫) be the dual basis, i.e. (b^{∗}, b′) = δ_{b,b′}. Let u ↦ \overline{u} be the antihomomorphism of U_{q}(𝔫^{−}) such that \overline{F}_{i} = F_{i} and \overline{q} = q. Let Ẽ_{i}, F̃_{i} : U_{q}(𝔫^{−}) → U_{q}(𝔫^{−}) be the Kashiwara operators, [loc. cit.]. For b ∈ B, Ẽ_{i}(b), resp. F̃_{i}(b), equals some b′ in B ∪{0} modulo q^{−1}ℤ[q^{−1}]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}. The following is well known, [24, Proposition 8.2], [18, 6.2.18-6.2.19] :

Theorem. For all λ ∈ P^{+}, we have :

(i) Via the isomorphism of Theorem 1.4 (iii), adU_{q}(𝔫)K_{−2λ} is isomorphic to V_{q}(λ)^{∗} ⊗ v_{λ} as a U_{q}(𝔫) module,

(ii) adU_{q}(𝔫)K_{−2λ} is generated as a space by {K_{−2λ}b^{∗}, b^{∗} ∈ B(λ)^{∗}}, where B(λ)^{∗} := \{b^{∗} ∈ B^{∗}, E(\overline{b}) ⪯ λ\}. ♢ 1.7. Recall that the enveloping algebra U := U(𝔤) is endowed with a canonical filtration {U_{k}, k ∈ ℕ} such that the associated graded algebra is commutative, [14, 2.3]. Moreover, this filtration is compatible with the adU(𝔤)-module structure of U(𝔤). A generalization of this filtration in the quantum case is the Joseph-Letzter P^{+}-filtration, [21]. As in [27], this will permit us to define the generalized (P^{+})-exponents of 𝔤. First, let’s present a few facts on the Joseph-Letzter filtration. For λ ∈ P^{+} set

F_{q}^{λ} := adU_{q}(𝔤).K_{−2λ}, F_{q,λ} := ⨁_{ν≤λ} F_{q}^{ν}.

Then, F_{q,λ} is an adU_{q}(𝔤)-module. By [18, 7.1.1], {F_{q,λ}, λ ∈ P^{+}} defines a P^{+}-filtration of F (U_{q}(𝔤)). Moreover, let F̂_{q}^{λ}, resp. F̂_{q}(U_{q}(𝔤)), be the associated graded space of F_{q}^{λ}, resp. F_{q}(U_{q}(𝔤)). As adU_{q}(𝔤)-modules, F̂_{q}^{λ} and F_{q}^{λ} are isomorphic. By [20], the Joseph-Letzter’s filtration extends to a filtration of U_{q}(𝔤) such that Û_{q}(𝔤) is the algebra with the canonical generators, quantum Serre relations, weight relations, and [Ê_{i}, F̂_{j}] =

−δ_{ij} \frac{K̂_{−α_{i}}}{q−q^{−1}} . Explicitly this filtration is F_{λ}(U_{q}(𝔤)) = ⨁_{α≥−2λ} U_{q}(𝔫)⊗K_{α} ⊗V_{q}(𝔫^{−}) where V_{q}(𝔫^{−}) is the ℂ(q)-algebra generated by K_{α_{j}}F_{j}, 1 ≤ j ≤ n. In general, if E denotes an element, resp. subspace, of U_{q}(𝔤), Ê will denote the associated graded element, resp. space. Clearly, the algebra F̂(U_{q}(𝔤)) ⊂ Û_{q}(𝔤) admits a Ringel filtration as in 1.3 and the associated graded algebra will be GrF̂_{q}(U_{q}(𝔤)). In F̂_{q}(U_{q}(𝔤)) we have, [18, 7.1.7 (iv)] :

(1.7.1)  F̂_{q}^{λ}F̂_{q}^{λ′} = F̂_{q}^{λ+λ′},  λ, λ′ ∈ P^{+}.

Hence, ⨁_{i} F̂_{q}^{ϖ_{i}} generates the graded associated algebra.

1.8. For λ ∈ P^{+}, let z_{λ} be the unique adU_{q}(𝔤)-invariant element in adU_{q}(𝔤).K_{−2λ}. By Theorem 1.4, the center Z_{q} of U_{q}(𝔤) is generated (as a space) by the z_{λ}, λ ∈ P^{+}. Moreover, Z_{q} is generated as an algebra by the z_{i} := z_{ϖ_{i}}, [18, 7.1.17].

Definition 1. Let J_{q} be the ideal of F (U_{q}(𝔤)) generated by the z_{i}−ε(z_{i}), 1 ≤ i ≤ n. Let ℍ̂_{q} be a complementary adU_{q}(𝔤)-module of Ĵ_{q} such that ℍ̂_{q} = ⨁_{λ} ℍ̂_{q} ∩ F̂_{q}^{λ}. Identifying in a canonical way the U_{q}(𝔤)-modules F̂_{q}^{λ} and F_{q}^{λ}, we obtain a sub-adU_{q}(𝔤)-module ℍ_{q} of F (U_{q}(𝔤)) such that ℍ_{q} = ⨁_{λ} ℍ_{q} ∩ F_{q}^{λ}. For all λ in P^{+}, set ℍ_{q}^{λ} = ℍ_{q} ∩ F_{q}^{λ}, and let ℍ_{q}^{λ}(μ), resp. F_{q}^{λ}(μ), be the isotypical component of ℍ_{q}^{λ}, resp. F_{q}^{λ}, of type V_{q}(μ).

Definition 2. Let μ ∈ P^{+}. In the algebra ℂ[P^{+}] generated by the e^{λ}, λ ∈ P^{+}, we define the element Q_{μ}(e) = Q_{μ} := ∑_{λ}[ℍ_{q}^{λ} : V_{q}(μ)]e^{λ}. The dominant weights occurring in this sum with multiplicities will be called P^{+}-exponents of μ. As in the classical case, we have, [18, Lemma 8.1.5] :

(1.8.1)  [ℍ_{q} : V_{q}(μ)] = dim V_{q}(μ)_{0}.

Hence, by specializing e on 1, we obtain Q_{μ}(1) = dim V_{q}(μ)_{0}. We now give an explicit expression of Q_{μ} in terms of the canonical basis. This is nothing but another formulation of a result of Baumann [1, 3.4]. We give a proof for completion.

Proposition. For μ in P^{+}, let B(μ), resp. B(μ)_{0}, be the crystal basis, resp. the set of 0-weight elements in the crystal basis, of V_{q}(μ). Then, Q_{μ} = ∑_{b∈B(μ)_{0}} e^{E(b)}.

Proof. For λ, μ ∈ P^{+}, consider the following subsets of B(μ)_{0}.

N_{λ,μ} = {b ∈ B(μ)_{0}, E(b) ⪯ λ}, N_{λ,μ}^{0} = {b ∈ B(μ)_{0}, E(b) = λ}, N_{λ,μ}^{+} = N_{λ,μ}\N_{λ,μ}^{0},

with cardinals, respectively, n_{λ,μ}, n_{λ,μ}^{0} and n_{λ,μ}^{+}. Now, let’s work in the Joseph-Letzter graded algebra. The theorem of separation of variables, [21], gives F̂_{q} = ℍ̂_{q} ⊗ Ẑ_{q}. Moreover, by semi-simplicity and from the basic properties of the crystal basis, [18, 6.3.18] :

(1.8.2) [F̂_{q}^{λ} : V_{q}(μ)] = [V_{q}(λ)^{∗} ⊗ V_{q}(λ) : V_{q}(μ)] = [V_{q}(λ) ⊗ V_{q}(μ) : V_{q}(λ)] = n_{λ,μ}.

We claim that

(1.8.3)  F̂_{q}^{λ}(μ) ≃ n_{λ,μ}^{0}V_{q}(μ) ⊕ Ĵ_{q} ∩ F̂_{q}^{λ}(μ).

Indeed, we can prove this fact by induction on the |λ| := ∑_{i} λ_{i}, with λ = ∑_{i} λ_{i}ϖ_{i} in the following way. Suppose this is true for all λ′ ∈ P^{+}, |λ′| < |λ|. Then, we have [ℍ̂_{q}(μ) ∩ F̂_{q}^{λ′} : V_{q}(μ)] = n_{λ′,μ}^{0}, for such λ′. Note that Ĵ_{q} is the ideal generated by the ẑ_{ϖ_{i}}, [18, 7.3.5]. By the separation of variables theorem,

∑_{λ′≺λ} ẑ_{λ−λ′}(ℍ̂_{q} ∩ F̂_{q}^{λ′}(μ)) = Ĵ_{q} ∩ F̂_{q}^{λ}(μ).

By induction hypothesis, the left hand term is isomorphic to ∑_{λ′≺λ} n_{λ′,μ}^{0}V_{q}(μ) = n_{λ,μ}^{+}V_{q}(μ). By (1.8.2), this gives (1.8.3) and the proposition is proved. ♢

Remark. When 𝔤 is of type A_{n}, then the function Q_{μ} is the multivariable Kostka-Foulkes polynomial 𝕂_{μ,0} defined in [29, 6.2] by Lascoux-Leclerc and Thibon. In fact, the previous proposition materializes their definition in the following sense : 𝕂_{μ0} as defined in [Loc. cit.] is the Hilbert polynomial of the isotypical component of ℍ_{q} of type V_{q}(μ) in the Joseph-Letzter P^{+}-filtration.

1.9. We present a result which can be seen as a quantum version of Dixmier’s antihomomorphism [14, 8.4.1]. We consider the Joseph-Letzter associated graded algebra. Let π̂ be the natural projection from F̂_{q}(U_{q}(𝔤)) ≃ ⊕_{λ}V_{q}(λ)^{∗} ⊗ V_{q}(λ) onto ⊕adU_{q}(𝔫) K̂_{−2λ} ≃ ⊕V_{q}(λ)^{∗} ⊗ v_{λ}. Then, by [22, Lemma 7.2], Lemma. The restriction of π̂ to F̂_{q}(U_{q}(𝔤))^{U_{q}(𝔫)} is an injective algebra q-antihomo-morphism. To be more precise, let a_{μ} ∈ F̂_{q}(U_{q}(𝔤))^{U_{q}(𝔫)} , b_{ν} ∈ (F̂_{q}^{λ})^{U_{q}(𝔫)} of weights, respectively, μ and ν. Then, π̂(a_{μ}b_{ν}) = q^{(μ,2ν−λ)}π̂(b_{ν})π̂(a_{μ}). Moreover π̂(ẑ_{λ}) = K̂_{−2λ}. ♢

For λ in P^{+}, let π̂_{λ} be the λ component of π̂_{λ}. By say [8, Proposition 3.4], it is clear that π̂_{λ} is the restriction of the natural projection on Û_{q}(𝔫) ⊗ K̂_{−2λ}. Hence, π̂ is compatible with the Ringel filtration.

2. A-form and specialization.

2.1. The following results can be deduced from [12, 1.5]. Let A be the local algebra ℂ[q]_{(q−1)}. We define the A-form U_{A} generated by E_{i}, F_{i}, \frac{K_{ϖ_{i}} −K_{−ϖ_{i}}}{q−q^{−1}} , K_{λ}, where λ ∈ P, 1 ≤ i ≤ n. Remark that U_{A} is a sub-ad U_{A}-module of U_{q}(𝔤). Moreover, U_{A} is A-free and U_{q}(𝔤) = ℂ(q) ⊗_{A} U_{A}.

Claim. Let ℂ[P/2P] be the algebra of the group P/2P, generated as a space by the elements e\overline{λ}, \overline{λ} ∈ P/2P. There exists a natural isomorphism of bialgebras between U_{A}/(q−1)U_{A} and ℂ[P/2P]⊗U(𝔤) such that the canonical surjection ϕ : U_{A} → U_{A}/(q− 1)U_{A} ≃ ℂ[P/2P] ⊗ U(𝔤) sends K_{λ} to e\overline{λ}, λ ∈ P. ♢ Let U_{A}^{+}, resp. U_{A}^{0}, resp. U_{A}^{−}, be the positive, resp. Cartan, resp. negative, part of U_{A}. As in 1.2, the triangular decomposition holds for A-forms. We define F (U_{A}) = U_{A} ∩F (U_{q}(𝔤)) and Z_{A} := Z ∩U_{A}. From the definitions, the algebra Z_{A} is the center of U_{A}. Let J_{q} be the ideal of F (U_{q}(𝔤)) generated by the z_{i} − ε(z_{i}), 1 ≤ i ≤ n and set J_{A} := J_{q} ∩ F (U_{A}). Then, J_{A} is a adU_{A}-module. Moreover, the quotient F (U_{A})/J_{A} ⊂ F (U_{q}(𝔤))/J has no A-torsion. Proposition. The morphism ϕ sends

(i) F (U_{A}) onto U(𝔤), realized as the subalgebra 1 ⊗ U(𝔤) in ℂ[P/2P] ⊗ U(𝔤), with kernel (q − 1)F (U_{A}).

(ii) J_{A} onto the minimal primitive ideal U(𝔤)Kerε_{1}, where ε_{1} is the augmentation re-

stricted to Z(𝔤). Proof. By Theorem 1.4 (ii), F (U_{A}) contains K_{−2ϖ_{i}}E_{α_{i}}, for all i. Hence, by the Claim, ϕ(F (U_{A})) contains U(𝔫). In the same way, it contains U(𝔫^{−}), and both algebras generate U(𝔤). Hence, U(𝔤) ⊂ ϕ(F (U_{A})). Now, by [20, Theorem 6.4], F (U_{A}) ⊂ U_{A}^{−} ⊗ Û^{0} ⊗ U_{A}^{+}, with (2.1.1)

Û^{0} = ℂ(q)[K_{2ϖ_{i}}, K_{−2ϖ_{i}}]∩A[K_{ϖ_{i}}, K_{−ϖ_{i}}, \frac{K_{ϖ_{i}} − K_{−ϖ_{i}}}{q − q^{−1}} ] = A[K_{2ϖ_{i}}, K_{−2ϖ_{i}}, \frac{1 − K_{−2ϖ_{i}}}{q − q^{−1}} ].

Hence, ϕ(F (U_{A})) ⊂ U(𝔤). This gives (i). Let’s prove that ϕ(J_{A}) = U(𝔤)Kerε_{1}. U(𝔤)Kerε_{1} ⊂ ϕ(J_{A}) : Let J′_{A} = F (U_{A})(Kerε ∩ Z_{A}). Then, J′_{A} ⊂ J_{A}. Now, by the surjectivity of ϕ : Z_{A} → Z(𝔤), see Theorem 2.2 below, this inclusion implies U(𝔤)Kerε_{1} = ϕ(J′_{A}) ⊂ ϕ(J_{A}). ϕ(J_{A}) ⊂ U(𝔤)Kerε_{1} : Let M_{q}(0) be the quantum Verma module with highest weight 0, as defined in [18, 3.4.9]. Then, M_{q}(0) is a cyclic U_{q}(𝔫^{−})-free module generated by a highest weight vector v_{0} of weight 0. Let M_{A}(0) be the sub-U_{A}-module of M_{q}(0) generated by v_{0}. Then, M_{A}(0)/(q − 1)M_{A}(0) has a natural structure of U(𝔤)-module by (i). It is easily seen that this is a cyclic U(𝔤)-module with highest weight 0 and with same formula character than the classical Verma module M(0). Hence, M_{A}(0)/(q − 1)M_{A}(0) ≃ M(0) as U(𝔤)-modules. Now, J_{A} annihilates M_{A}(0) ⊂ M_{q}(0). Hence, ϕ(J_{A}) ≃ J_{A}/(q − 1)J_{A} annihilates M(0). By Duflo’s theorem [14, 8.4.3], we have the desired inclusion. Hence, (ii) holds. ♢ 2.2. Let U_{q}(𝔫)^{+}, resp. U_{q}(𝔫^{−})^{+}, be the augmentation ideal of U_{q}(𝔫), resp. U_{q}(𝔫^{−}). Let φ be the Harish-Chandra map of U_{q}(𝔤), i.e. the projection U_{q}(𝔤) → U_{q}^{0} with kernel U_{q}(𝔤)U_{q}(𝔫)^{+} + U_{q}(𝔫^{−})^{+}U_{q}(𝔤). Recall the so-called twisted action of W on U_{q}^{0}, see [12, par 2]. Moreover, by [12, 2.2] :

Proposition. Let Ũ_{A}^{0} be the intersection of U_{A} with the ℂ(q)-algebra generated by the K_{2λ}, λ ∈ P. Then, Ũ_{A}^{0} is a sub-W -module of U^{0} and the Harish-Chandra homomorphism φ restricts into an algebra isomorphism Z_{A} ≃ (Ũ_{A}^{0})^{W}. ♢

Remark that the inverse isomorphism can be given explicitly by the Tolstoi projector introduced in [26]. This is implicit in the proof of [12, Proposition 2.2]. The proposition above is the quantum integral analogue of the following classical result, see [14, 7.4] : the Harish-Chandra map establishes an algebra isomorphism between the center Z(𝔤) of U(𝔤) and the W -invariant subalgebra of the Cartan part U(𝔥) of U(𝔤).

Theorem. The A-algebra Z_{A} is a polynomial algebra over A. It specializes for q = 1, onto the center Z(𝔤) of the enveloping algebra U(𝔤), realized as the subalgebra 1 ⊗ U(𝔤) in ℂ[P/2P] ⊗ U(𝔤). Proof. From Proposition 2.1 (i), Z_{A} specializes in the subalgebra 1 ⊗ U(𝔤). Clearly, specialization sends an element of the center of U_{A} to an element of Z(𝔤). Moreover, Ũ_{A}^{0} specializes onto U(𝔥) and specialization commutes with W -action. As W is a finite group, (Ũ_{A}^{0})^{W} specializes onto U(𝔥)^{W}. The last assertion of the theorem follows because specialization commutes with the Harish-Chandra homomorphisms. Let’s prove that Z_{A} is polynomial over A. Let’s identify the center Z of U(𝔤) with S(𝔥)^{W} and the symmetric algebra S(𝔥) with the algebra of restrictions to Q of the regular functions on 𝔥^{∗} = ℂ ⊗ Q restricted to Q. In the same way, let’s identify Z_{q} with (U_{q}^{0})^{W}, and U_{q}^{0} with the ℂ(q)-algebra generated by the functions K_{ϖ_{i}}( ) = q^{(ϖ_{i}, )} on Q. This enables us to give formal expansions at q = 1 from elements of Z_{q}, as in [8, 7.1] and [21, 6.13], i.e. we can embed Z_{q} in the ring of Laurent series S(𝔥)^{W} ((q−1)) ≃ Z((q−1)). Let’s consider the augmentation ideal Z^{+} of Z. Fix a set of homogeneous elements (C_{m_{j}}), (with some incorrectness of notations if multiplicities occur) 1 ≤ j ≤ n, of degree 1 < m_{1} ≤ … ≤ m_{n}, which is a basis of Z^{+} modulo (Z^{+})^{2}. Let z_{i} be the quantum central element as in 1.8 and z′_{i} = z_{i} − ε(z_{i}). From the formula in the proof of [21, Lemme 6.15], we have the claim

Claim. The coefficient of (q − 1)^{m} of the image of z′_{i} in Z((q − 1)) is an homogeneous element of degree m in Z. ♢

Now, the image of z′_{i} in Z^{+}/(Z^{+})^{2}((q − 1)) is ∑ a_{ij}C_{m_{j}}(q − 1)^{m_{j}}, a_{ij} ∈ ℂ, 1 ≤ i, j ≤ n. Moreover the products z′_{i}z′_{j} vanish in this quotient. Recall that the z′_{i} generate the ℂ(q)- algebra Z_{q} and recall that, by the previous arguments, Z_{A} specializes onto Z at q = 1. This implies that there exists a polynomial combination of the z′_{i} (and of (q − 1)^{−1}) which specializes on C_{m_{j}} at q = 1. This combination is linear modulo (Z^{+})^{2} and this implies that the matrix (a_{ij}) is invertible. Hence, we can change the indexation of the z_{i} in order to obtain non zero minors (a_{ij})_{1≤i,j≤k}, 1 ≤ k ≤ n. Then, by induction on k, we obtain elements C_{m_{k}}^{q} in Z_{A} which are linear combination of z_{k} and products of z_{i} for i < k and which specializes on C_{m_{k}} (at least up to an automorphism of the graded vector space spanned by the C_{m_{j}}, 1 ≤ j ≤ n ). Now, let’s prove that Z_{A} is the A polynomial algebra generated by the C_{m_{k}}^{q}. Remark that, by construction

1) These elements specialize on a polynomial basis of Z.

2) They generate Z_{q} as a ℂ(q)-algebra. It is clear that the C_{m_{k}}^{q} are algebraically independant by 1). Moreover, let z be in Z_{A}, then, by 2), there exists an integer k such that (q − 1)^{k}z belongs to the A-algebra generated by the C_{m_{k}}^{q}. Assume k minimal and non zero. By specialization, 1) and Proposition 2.1 (i) yield a contradiction with the minimality of k. Hence, k = 0, this proves the theorem. ♢

2.3. Set F_{A,λ} = U_{A} ∩ F_{q,λ}.

Proposition. We have the following

(i) The A-module F_{A} is free, (ii) there exists an adU_{A}-modules graduation (F_{A}^{λ})_{λ∈P^{+}} of F (U_{A}) such that

F (U_{A}) = ⨁_{λ∈P^{+}} F_{A}^{λ}, ⨁_{λ≤μ} F_{A}^{λ} = F_{A,μ}.

Proof. The module U_{A} is A-free. The submodule F_{A,λ} is therefore also free. Since U_{A} is an A-form of U_{q}(𝔤) and F_{q,λ} is a finte dimensional subspace of U_{q}(𝔤), the A-module F_{A,λ} is of finite rank. Fix λ ≤ μ in P^{+}. Then, F_{A,μ}/F_{A,λ} embeds in F_{q,μ}/F_{q,λ}, so it has no A-torsion. Hence, the A-module F_{A,λ} is a direct summand in F_{A,μ}. Remark that F (U_{A}) = ⋃_{μ} F_{A,μ} to obtain (i), (ii). ♢

Remark. F (U_{A}) strictly contains ⊕_{λ}(U_{A} ∩ F_{q}^{λ}). For example, set 𝔤 = 𝔰𝔩_{2} and let z_{ϖ} ∈ F_{q}^{ϖ} be the Casimir element corresponding to the quantum trace of the fundamental module V_{q}(ϖ). Then, [8, 5.3], ϕ(z_{ϖ}) = 2, \frac{z_{ϖ}−2}{q−q^{−1}} belongs to F (U_{A}) but not to ⊕_{λ}U_{A}∩F_{q}^{λ}.

2.4. We prove in this section that the separation theorem at the quantum level, i.e. on ℂ(q), and on the classical level implies the separation theorem for the A-forms. The proof relies on general properties of torsion-free modules over A-polynomial algebras. These properties are given in the Appendix. Theorem. The separation theorem holds for the A-forms : let ℍ_{q} be a quantum space of harmonics as in Definition 1. Putting H_{A} = U_{A}∩ℍ_{q}, the multiplication of U_{A} affords an isomorphism

Z_{A} ⊗ H_{A} ≃ F (U_{A}),

for an adU_{A}-module H_{A}. Moreover, H_{A} can be obtained as U_{A} ∩ ℍ_{q}, where ℍ_{q} is as in 1.8, Definition 1. Proof. Remark that F (U_{A}) is the direct sum of its ”isotypical components F_{A}(ν) := F_{A}∩ F_{q}(ν), ν ∈ P^{+}. Thus, it is sufficient to prove the theorem on the isotypical component, i.e F_{A}(ν) = Z_{A}⊗H_{A}(ν), ν ∈ P^{+}. Now, from Theorem 2.2, the center Z_{A} is a polynomial ring over A. We are in the framework of [Appendix, Proposition A4] with T = q −1 and R = Z_{A}. From the quantum and classical theorems of separation of variables, [21] and [27], the hypothesis of Proposition A4 are verified with r =dimV (ν)dimV (ν)_{0}. Hence, F_{A}(ν) is a free Z_{A}-module with rank r.

Fix a basis of this module. The theorem is a consequence of the following facts :

(i) H_{A}(ν) generates the Z_{q} module F_{q}(ν). (ii) H_{A}(ν) is a A-direct summand in F_{A}(ν). Indeed, fix a A-basis of H_{A}(ν). As ℍ_{q}(ν) = ℂ(q) ⊗_{A} H_{A}(ν), we obtain that this basis generates the ℂ(q)-space ℍ_{q}(ν). Hence, it also generates the Z_{q}-module F_{q}(λ) by [18, 7.3.8]. This implies (i). Moreover, by contruction H_{A}(ν) is a free A-module with finite rank and F (U_{A})(ν)/H_{A}(ν) has no (q − 1)-torsion. So, (ii) holds. Now, fix a A-basis of H_{A}(ν) and let d be the determinant of the matrix of this basis in the fixed basis of the Z_{A}-module F_{A}(ν). Then, (i) asserts that (q − 1)^{−m}d is a unit of Z_{A}, hence of A, for some positive integer m. By (ii), we obtain m = 0. This ends the proof. ♢

Let ϕ be the morphism of specialization as in 2.1. We have :

Corollary. The morphism ϕ sends

(i) H_{A} onto a complementary ad𝔤-module H of U(𝔤)Kerε_{1} in U(𝔤), (ii) The μ isotypical component of H_{A} onto the μ isotypical component of H. Proof. For all element z in Z_{A}, (and more generally for all element in U_{A}, ε(z) belongs to A. Hence, Z_{A} = (Z_{A}∩Kerε)⊕A as A-modules. By the previous theorem, this implies that F (U_{A}) = (Z_{A}∩Kerε)H_{A}⊕H_{A} = (Z_{A}∩Kerε)Z_{A}H_{A}⊕H_{A} = (Z_{A}∩Kerε)F (U_{A})⊕H_{A}. Moreover, by Proposition 2.1 (i) and Theorem 2.2, this implies that the specialization of the augmentation ideal of Z_{A} is the augmentation ideal of Z(𝔤). Hence, Proposition 2.1 (i) gives the corollary. ♢

2.5. Fix H_{A} as in Theorem 2.4. Then, as in Proposition 2.3, we can construct an adU_{A}-modules graduation (H_{A}^{λ})_{λ∈P^{+}}, resp. (Z_{A}^{λ})_{λ∈P^{+}}, such that H_{A} = ⨁_{λ∈P^{+}} H_{A}^{λ}, resp. Z_{A} = ⨁_{λ∈P^{+}} Z_{A}^{λ}, and ⨁_{λ≤μ} H_{A}^{λ} = H_{A} ∩ F_{q,μ}, resp. ⨁_{λ≤μ} Z_{A}^{λ} = Z_{A} ∩ F_{q,μ}.

Lemma. Fix λ ∈ P^{+}. We have

(2.5.1)  ⨁_{μ′⪯μ≤λ} Z_{A}^{μ−μ′} ⊗ H_{A}^{μ′} = F_{A,λ}.

Proof. By 1.7, multiplication affords a map ⨁_{μ′⪯μ≤λ} Z_{A}^{μ−μ′} ⊗H_{A}^{μ′} → F_{A,λ}. By Theorem 2.4 and again 1.7, this map is bijective. ♢ From this lemma, we can construct a graduation (F_{A}^{λ})_{λ∈P^{+}} which verifies the properties of Proposition 2.3 by setting F_{A}^{λ} = ⨁_{μ⪯λ} Z_{A}^{λ−μ} ⊗ H_{A}^{μ}. This graduation also verifies H_{A} = ⨁ H_{A} ∩ F_{A}^{λ} and Z_{A} = ⨁ Z_{A} ∩ F_{A}^{λ}. Let Z_{A}^{λ} and H_{A}^{λ} be the graded component of Z_{A} and H_{A}.

For all λ ∈ P^{+}, set F_{λ} = F_{λ}(U(𝔤)) = ϕ(F_{A,λ}) ∈ U(𝔤). This is an adU(𝔤)-submodule of U(𝔤). In general, we omit the index A to note the image by ϕ, i.e. we note E := ϕ(E_{A}). We want to prove that F_{λ} is a filtration of U(𝔤) which is analogue to the Joseph-Letzter filtration in the quantum case. Indeed we have : Proposition. The algebra U(𝔤) is filtered by the adU(𝔤)-modules (F_{λ})_{λ∈P^{+}}. Moreover, (i) F_{λ} ≃ ⨁_{μ≤λ} F^{μ} (ii) F^{μ} ≃ V (μ)^{∗} ⊗ V (μ) (iii) ⨁_{μ′⪯μ≤λ} Z^{μ−μ′} ⊗ H^{μ′} = F_{λ}. Proof. The adU_{A} modules (F_{A,λ})_{λ∈P^{+}} give a filtration of F (U_{A}). Hence, by specialization and by Proposition 2.3, we have the first assertion. (i) and (ii) are obtained by specialization, using the fact that the A-spaces F_{A,λ} and F_{A}^{λ} are A-summands in F (U_{A}). (iii) is obtained from (2.5.1). ♢

3. Gröbner basis and harmonic elements.

3.1. Recall the definition of roots packages, see 1.5. In the graded algebra GrU_{q}(𝔤), the generators Gr z_{i} of Gr Z_{q} are given by :

Proposition. For 1 ≤ i ≤ n, we have, up to a multiplicative scalar

Gr z_{i} = ( ∏_{β∈Δ_{i}^{+}} Gr E_{β}Gr F_{β})Gr K_{−2ϖ_{i}}.

The Gr z_{i}, 1 ≤ i ≤ n, generate the algebra Gr Z_{q}. Proof. Recall that z_{i} corresponds to the quantum trace on V_{q}(ϖ_{i}), [8, 5.1]. The formula is a direct consequence of [8, Proposition 3.4], and [11, Lemma 4.1 (iii)]. From 1.5, {Δ_{i}^{+}, i} is a partition of Δ^{+}, so the Gr ∏_{i} z_{i}^{a_{i}}, a_{i} ∈ ℕ are linearly independant. Hence, they generate Gr Z_{q} as a space. This proves the proposition. ♢

3.2. Definition. Let M be the set of monomials M in U_{q}(𝔤) such that

(i) M = ∏_{j,m} E_{α_{j,m}}^{a_{j,m}}K_{λ}F_{α_{j,m}}^{b_{j,m}}, 0 ≤ a_{j,m}, b_{j,m}, λ ∈ P, (ii) For all j, ∏_{m} a_{j,m}b_{j,m} = 0. Then, the following separation theorem holds.

Theorem. Let J_{q}^{0} be the ideal of U_{q}(𝔤) generated by the z_{i}, 1 ≤ i ≤ n. Fix a reduced decomposition of w_{0}. Then :

(i) The set {z_{i}, 1 ≤ i ≤ n} is a Gröbner basis of J_{q}^{0} for the lexicographic ordering of the PBW-basis, i.e. u ∈ J_{q}^{0} =< z_{i} >⇒ Gr u ∈< Gr z_{i} >.

(ii) The algebra Gr U_{q}(𝔤) is free over Gr Z_{q} with basis Gr M. (iii) The algebra U_{q}(𝔤) is free over Z_{q} with basis M. Proof. First remark that, by construction and by Proposition 3.1, the image of Gr M in Gr U_{q}(𝔤)/ < Gr z_{i}, 1 ≤ i ≤ n > is a basis of this quotient. Let Gr z_{j}, resp. Gr m_{j}, be monomials in Gr Z_{q}, resp. Gr M, j = 1, 2. Then, {Δ_{i}^{+}, i} being a partition of Δ^{+}, we obtain by Proposition 3.1 :

(3.2.1)  Gr z_{1}Gr m_{1} = Gr z_{2}Gr m_{2} ⇒ Gr z_{1} = Gr z_{2} and Gr m_{1} = Gr m_{2}.

Now, considering Gr U_{q}(𝔤) as a ℕ^{N} × P × ℕ^{N}-graded algebra, see 1.3, we remark that each graded component of Gr U_{q}(𝔤) has dimension 0 or 1. Hence, (3.2.1) implies that Gr U_{q}(𝔤) is free over Gr Z_{q}, with basis Gr M. This implies (ii) and then (iii).

Let Z_{q}^{+} be the ideal of Z_{q} generated by the z_{i}. Then, by (iii), each element u of J_{q}^{0} can be decomposed in a unique way into u = ∑_{k} P_{k}m_{k}, with P_{k} ∈ Z_{q}^{+} and m_{k} ∈ M. Now, by (3.2.1) and the remark above, we obtain that this sum contains a unique term of highest degree. Hence, Gru =GrP_{s}m_{s}, for some integer s. This gives (i). ♢ Remark 1. It is possible to generalize the theory of Gröbner basis in some noncommutative contexts, [23]. (i) can also be proved in this context by using the first Buchsberger criterion [2]. Indeed, Proposition 3.1 asserts that the initial monomials of the z_{i} are relatively prime. Remark 2. Unfortunately, the specialization of the center is not compatible with the associated graded algebra. Indeed, the initial terms of the generators z_{i} may vanish at q = 1. The basis given in Theorem 3.2 (iii) has no analogue in the classical case.

3.3. This section will be of no use in the sequel. The results of 1.3 generalize when q = ε is a root of one. We define the quantum algebra U_{ε}(𝔤) as in [12, 1.5]. By the description of the center Z_{ε} of quantum groups at roots of one, see [13], we obtain :

Theorem. When q = ε is a root of one, the algebra GrU_{ε}(𝔤) is free over the graded algebra Gr Z_{ε}. This implies that U_{ε}(𝔤) is free over its center. ♢ For example, when ε is a lth root of one, with l odd, then a basis of U_{ε}(𝔤) over Z_{ε} is given by M^{l} := {M ∈ M, M = ∏_{i,j,m} E_{α_{j,m}}^{a_{j,m}}K_{ϖ_{i}}^{c_{i}}F_{α_{j,m}}^{b_{j,m}}, 0 ≤ a_{j,m}, b_{j,m}, c_{i} < l}.

3.4. Let π the canonical lifting of π̂, see 1.9. The next proposition gives a basis of π(ℍ_{q}^{U_{q}(𝔫)}) in terms of the dual of the canonical basis of U_{q}(𝔫^{−}), (see 1.6). When u is a weight vector in a U_{q}(𝔤)-module, then we note Ω(u) the weight of u.

For λ in P^{+}, set

B_{H} := \{b, b ∈ B, E(b) ⪯ −Ω(b)\}, B_{H}^{λ} := \{b, b ∈ B_{H}, E(\overline{b}) = λ\}.

Proposition. There exists an harmonic space ℍ_{q}, in the sense of 0.4, such that a basis of π(ℍ_{q}^{U_{q}(𝔫)}) is given by (K_{−2λ}b^{∗}, λ ∈ P^{+}, \overline{b} ∈ B_{H}^{λ}). Proof. Fix μ = ∑ μ_{i}ϖ_{i} ∈ P^{+}. Recall the notations of Proposition 1.8. we have

(3.4.1)  dim (ℍ_{q}^{λ})(μ)^{U_{q}(𝔫)} = [ℍ_{q}^{λ}(μ) : V_{q}(μ)] = n_{λ,μ}^{0}.

Via the natural embedding of B(μ) in the canonical basis, we have by [24, Proposition 8.2] N_{λ,μ}^{0} = \{b, b ∈ B, Ω(b) = −μ, E(\overline{b}) ⪯ μ, E(b) = λ\}. In the same way, dim(F̂_{q}^{λ})^{U_{q}(𝔫)}(μ) = n_{λ,μ} and N_{λ,μ} = \{b, b ∈ B, Ω(b) = −μ, E(\overline{b}) ⪯ μ, E(b) ⪯ λ\}.

Claim. A basis of π(F_{q}^{λ}(μ)^{U_{q}(𝔫)}) is given by (K_{−2λ}b^{∗}), where b runs over the set \overline{N}_{λ,μ}, i.e. the image of N_{λ,μ} by the involution described in 1.6. Proof of the claim. Fix λ, μ, ξ in P^{+}, λ = ∑ λ_{i}ϖ_{i}, μ = ∑ μ_{i}ϖ_{i}. By say, [18, 6.3.20] the image of the highest weight space of weight μ in V_{q}(ξ) ⊗ V_{q}(λ) by the natural projection on V_{q}(ξ) ⊗ v_{λ} is (V_{q}(ξ))_{μ−λ}^{λ} ⊗ v_{λ}, where

(V_{q}(ξ))_{μ−λ}^{λ} := {v ∈ V_{q}(ξ), Ω(v) = μ − λ, F_{i}^{μ_{i}+1}v = 0, ∀i}

(3.4.2)  = {v ∈ V_{q}(ξ), Ω(v) = μ − λ, E_{i}^{λ_{i}+1}v = 0, ∀i}.

We have the succession of space isomorphisms (V_{q}(λ)^{∗})_{μ−λ}^{λ} =< b^{∗} ∈ B(λ)^{∗}, Ω(b) = −μ, E(b) ⪯ μ >=< b^{∗} ∈ B^{∗}, Ω(b) = −μ, E(\overline{b}) ⪯ λ, E(b) ⪯ μ >= \overline{N}_{λ,μ}. Indeed, the first equality comes from [30, Theorem 4.4 (c), 4.3] and (3.4.2), the second one is deduced from [24, Proposition 8.2]. The claim results from Theorem 1.4 (ii) and Theorem 1.6. ♢ end of proof of the proposition. Suppose ν in P^{+}, ν ⪯ λ, ν ≠ λ. In the graded algebra F̂_{q}(U_{q}(𝔤)), < K_{−2λ}b^{∗}, b ∈ B(μ)^{∗}, Ω(b) = −μ, E(b) ⪯ λ >⊂ π̂(Ĵ_{q}) by the claim and Lemma 1.9. Then, the proposition results from the claim by (3.4.1). ♢

Remark. What is behind the proof is a natural twist τ : B(λ) ⊗ B(μ) → B(μ) ⊗ B(λ) defined to be the unique crystal map such that the image of the ẽ_{i}-invariant elements, 1 ≤ i ≤ n, is given by τ(v_{λ} ⊗ bv_{μ}) = v_{μ} ⊗ \overline{b}v_{λ}, for E(b) ⪯ λ, E(\overline{b}) ⪯ μ. It would be interesting to understand the braiding of this twist on B(ϖ)^{⊗k}, where ϖ is a basic weight.

3.5. We suppose in this section that 𝔤 is of type A-D-E and that the decomposition of w_{0} corresponds to an orientation of the Coxeter graph of 𝔤. In this case, from Ringel’s Hall algebra approach, [33], the canonical basis can be labelled by the Kostant partitions. We present a few results which can be found in [31], [32, par 2 and 3]. Recall that {E_{ψ}, ψ ∈ ℕ^{N}} is a basis of U_{q}(𝔫). There exists a parametrization B = {b_{ψ}, ψ ∈ ℕ^{N}} of the canonical basis such that :

(3.5.1)  Gr b_{ψ}^{∗} = Gr E_{ψ}, up to a multiplicative scalar in ℤ[q],

(3.5.2)  Gr b_{ψ}^{∗}b_{ϕ}^{∗} = Gr q^{[ψ,ϕ]}b_{ψ+ϕ}^{∗}, for a power [ψ, ϕ] of q,

(3.5.3)  E(b_{ψ+ϕ}) ⪯ E(b_{ψ}) + E(b_{ϕ}).

The following definition flows from Proposition 3.4 :

Definition. The harmonic cone is defined by C_{0} := {ψ, b_{ψ} ∈ B_{H}} ⊂ ℕ^{N}.

Theorem. The ℂ-algebra ℂ[C_{0}] of the semigroup C_{0} is normal, see [16, par. 4]. Via the anti-homomorphism Gr π̂, see 1.9, the algebra Gr ((F̂_{q}(U_{q}(𝔤))/Ĵ)^{U_{q}(𝔫)}) is isomorphic to the following algebra : a basis is a_{ψ}, ψ ∈ C_{0} with the multiplication given by

a_{ψ}a_{ϕ} = \{ \begin{matrix}0 & if E(ψ + ϕ) ≺ E(ψ) + E(ϕ), \\ a_{ψ+ϕ} & if E(ψ + ϕ) = E(ψ) + E(ϕ), \end{matrix}

up to a power of q. Proof. The cone C_{0} is {ψ ∈ ℕ^{N}, E(b_{ψ}) ⪯ Ω(ψ)}. Ω is clearly additive on ℕ^{N} and so, C_{0} is a semigroup by (3.5.3). For all ψ in ℕ^{N}, set a_{ψ} := Gr π̂(Gr (K̂_{−2E(ψ)}Ê^{ψ})). The set {a_{ψ}, ψ ∈ C_{0}} is a basis of Gr π̂(Gr (F̂_{q}(U_{q}(𝔤))/Ĵ)^{U_{q}(𝔫)}) by Proposition 3.4 and the previous formulas. The multiplication rule is given by Lemma 1.9 and (1.3.1). By [5, Theorem 3.8], we obtain that E(kψ) = kE(ψ), ψ ∈ ℕ^{N}. This gives kψ ∈ C_{0}, k ∈ ℕ^{∗}, ψ ∈ ℕ^{N} ⇒ ψ ∈ C_{0}. Hence, ℂ[C_{0}] is normal. ♢

Remark. Calculating the power of q in the formula does not seem to be an absolute requirement, but the demanding reader can easily obtain this power by using (1.3.1) and Lemma 1.9.

4. The A_{n} case.

4.1. Let 𝔤 of type A_{n}. Set F_{q} = F_{q}^{ϖ_{1}}. For all λ = ∑ λ_{i}ϖ_{i} ∈ P, we define h(λ) = ∑_{i} iλ_{i}. Define the total order ≤ on P^{+} such that

1) h(λ) < h(μ) ⇒ λ ≤ μ and

2) Fix λ = ∑_{i} λ_{i}ϖ_{i}, μ = ∑_{i} μ_{i}ϖ_{i}, with h(λ) = h(μ). Then, λ ≥ μ iff (λ_{n}, …, λ_{1}) is lower than (μ_{n}, …, μ_{1}) for the lexicographical order of ℕ^{n}. Example. For 𝔤 = 𝔰𝔩_{4}, 0 ≤ ϖ_{1} ≤ ϖ_{2} ≤ 2ϖ_{1} ≤ ϖ_{3} ≤ ϖ_{1} + ϖ_{2} ≤ 3ϖ_{1} ≤ ϖ_{1} + ϖ_{3} ≤ 2ϖ_{2} ≤ 4ϖ_{1} ≤ … It is easily seen that this ordering verifies the hypothesis of 1.1. We have :

Lemma. For all k in ℕ, ∑_{0≤m≤k} F_{q}^{m} = ⨁_{h(λ)≤k} F_{q}^{λ}.

Proof. It is well known that the minimal integer k such that V_{q}(λ) ⊂ V_{q}(ϖ_{1})^{⊗k} is h(λ). Hence the lemma results from [19, Corollary 3.10]. ♢

4.2. This section is devoted to the specialization of the center of U_{q}(𝔰𝔩_{n+1}). Recall that the center Z_{q} of U_{q}(𝔰𝔩_{n+1}) is generated as a space by the z_{λ}, λ ∈ P^{+}, defined as in 1.8 and as an algebra by the z_{i} ∈ F_{q}^{ϖ_{i}}, 1 ≤ i ≤ n. Recall the following theorem of [8, Proposition 6.3, Théorème 7.3].

Theorem. There exists an n × n matrix A with coefficients in A and a column matrix C_{q} = (C_{q,i})_{1≤i≤n} with C_{q,i} ∈ Z_{A} such that

(i) AC_{q} = (z_{i} − C_{n+1}^{i})_{1≤i≤n}, (ii) ϕ(C_{q,i}), 1 ≤ i ≤ n, has degree i + 1 in Z and they generate the polynomial algebra

Z, (iii) A = (a_{i,j}(q − q^{−1})^{j+1}), with a_{i,j} = 1^{j}C_{n+1}^{i−1} − 2^{j}C_{n+1}^{i−2} + … + (−1)^{i−1}i^{j}C_{n+1}^{0} ♢

4.3. In the following, we set U := U(𝔰𝔩_{n+1}). The aim of this section is to describe the P^{+}-filtration (F_{λ})_{λ∈P^{+}} of U(𝔰𝔩_{n+1}) in terms of the natural ℕ-filtration (U_{k})_{k∈ℕ} of U(𝔰𝔩_{n+1}). For λ = ∑_{i} λ_{i}ϖ_{i} ∈ P^{+}, set z(λ) = ∑(i + 1)λ_{i}. Set J := ϕ(J_{A}). Recall that, by Proposition 2.1 (ii), J is a minimal primitive ideal. By Lemma 4.1, we can choose ℍ_{q} in Theorem 2.4 such that

(∗)  ℍ_{q} ∩ ⨁_{h(λ)≤k} F_{q}^{λ} ⊂ ∑_{0≤m≤k}(ℍ_{q}^{ϖ_{1}})^{m},

for all k. Set H_{A} = ℍ_{q} ∩ U_{A}, as in Theorem 2.4. Set H_{A}^{λ} and Z_{A}^{λ} as in 2.5. By construction, H_{A} ∩ (∑_{h(λ)≤k} F_{q,λ}) = ⨁_{h(λ)≤k} H_{A}^{λ} and Z_{A} ∩ (∑_{z(λ)≤k} F_{q,λ}) = ⨁_{z(λ)≤k} Z_{A}^{λ}.

Proposition. Set U = U(𝔰𝔩_{n+1}), H = ϕ(H_{A}), H_{k} = H ∩ U_{k}, Z_{k} = Z ∩ U_{k}, k ∈ ℕ, we have

(i) Z ⊗ H = U via multiplication, (ii) ϕ(∑_{h(λ)≤k} H_{A}^{λ}) = H_{k}, (iii) ϕ(∑_{z(λ)≤k} Z_{A}^{λ}) = Z_{k},

Proof. (i) follows from Theorem 2.4. Let’s prove (ii). For k in ℕ, set F_{q,k} := ∑_{h(λ)≤k} F_{q,λ} and F_{A,k} = F (U_{A}) ∩ F_{q,k}. In the sequel, the symbol \overline{E}, where E is a space or an element, will denote the image of E by the quotient morphism modulo J_{q}, J_{A}, or J, if no confusion occurs. Let \overline{ϕ} be the quotient morphism \overline{ϕ} : F (U_{A})/J_{A} → U/J. Then, \overline{ϕ} is surjective. We want to show that \overline{ϕ}(\overline{F_{A,k}}) is the image \overline{U}_{k} of U_{k} by the canonical surjection. First of all, F_{q,ϖ_{1}} = V_{q}(0)⊕V_{q}(ϖ_{1}+ϖ_{n}) modulo J_{q} as U_{q}(𝔤)-modules. Moreover, ϕ(\overline{F_{A,ϖ_{1}}}) contains \overline{E}_{1}. Hence, \overline{ϕ}(\overline{F_{A,ϖ_{1}}}) = ℂ ⊕ 𝔰𝔩_{n+1} and \overline{ϕ}(∑_{m≤k} \overline{(F_{A,ϖ_{1}}})^{m}) = \overline{U_{k}}. Lemma 4.1 gives

(4.3.1)  \overline{U_{k}} ⊂ \overline{ϕ}(\overline{F}_{A,k}).

For μ in P^{+} and m in ℕ, let c_{μ}^{m} be the coefficient of q^{m} in the Kostka-Foulkes fonction K_{μ,0}(q). For λ ∈ P^{+}, let c_{μ}^{λ} = n_{λ,μ}^{0}, be the coefficient of e^{λ} in Q_{μ}. Then, by the Hesselink formula and the Lascoux-Leclerc-Thibon theorem, see 0.3 :

dim\overline{U_{k}} = ∑_{μ∈P^{+}} ∑_{m≤k} c_{μ}^{m}dimV (μ) = ∑_{μ∈P^{+}} ∑_{m≤k} ∑_{h(λ)=m} c_{μ}^{λ}dimV_{q}(μ).

So, by 1.8, Proposition 2.3 and Theorem 2.4, dim\overline{U_{k}} = dim\overline{F_{q,k}} = dim(ϕ(\overline{F_{A,k}})). Hence, we have equality in (4.3.1). This gives (ii) by the hypothesis (*) on ℍ_{q}. Let’s prove (iii). Fix s, 1 ≤ s ≤ n. Let Δ_{s} be the determinant of the matrix (a_{i,j})_{1≤i,j≤s}, see 4.2. It is an easy exercise to use the Van der Monde determinant to obtain

Δ_{s} = (−1)^{\frac{s(s−1)}{2}} ∏_{m=1}^{s} m! ≠ 0.

From Theorem 4.2, we obtain that C_{q,s} = D_{q,s} modulo (q − 1)Z_{A}, for all 1 ≤ s ≤ n, with D_{q,s} ∈ (q − q^{−1})^{−(s+1)} ∑_{i=1}^{s} Q(z_{i} − C_{n+1}^{i}). The elements ϕ(D_{q,s}) = ϕ(C_{q,s}), 1 ≤ s ≤ n, generate the algebra Z(𝔤). So, by induction on the P^{+}-filtration of Z_{A}, D_{q,s}, 1 ≤ s ≤ n, generate the algebra Z_{A} by using Nakayama’s lemma on the finite rank A-modules Z_{A} ∩ (∑_{z(λ)≤m} F_{q,λ}), m ∈ ℕ. The left hand term in (iii) is generated by ∏_{s} D_{q,s}^{n_{s}}, with ∑(s + 1)n_{s} ≤ k. This implies (iii). ♢

Set H^{λ} = ϕ(H_{A}^{λ}), resp. Z^{λ} = ϕ(Z_{A}^{λ}). This gives the following theorem : Theorem. The ℕ-natural filtration and the P^{+}-filtration of U, see 2.4, can be compared as follow :

U_{k} = ⨁_{h(λ)+z(μ)≤k} H^{λ} ⊗ Z^{μ}.

♢

4.4. In this section, we illustrate, in the case 𝔰𝔩_{n+1}, some of the main results and techniques of this article. The reduced decomposition of w_{0} is given by

w_{0} = s_{1} … s_{n}s_{1} … s_{n−1} … s_{1}s_{2}s_{1}.

Let α_{i,j} := α_{i} + … + α_{j}, 1 ≤ i ≤ j ≤ n, be the set of positive roots ordered by the rule given in 1.3. In particular, the ordering of the positive roots is given by

α_{1,1} > α_{1,2} > … > α_{1,n} > α_{2,2} > α_{2,3} > … < α_{2,n} > α_{3,3} > … > α_{n,n}.

See [25] for the bijection between the canonical basis and semi-standard Young tableaux and for the calculation of E(b), where b is associated to a given tableau. The correspondence between Lusztig’s parametrization of the canonical basis (for this choice of reduced decomposition of w_{0}) and semi-standard Young Tableaux can be described as follows : to a semi-standard Young tableau T, we associate the monomial of the PBW-basis ∏ E_{α_{i,j}}^{m_{i,j}} such that m_{i,j} is the number of times j + 1 occurs on the ith line of T.

See [4, Theorem 4.1] for inequations in Example 3. Examples 1-2-3 are devoted to the cone C_{0}. Example 1. For 𝔤 = 𝔰𝔩_{2}, the cone C_{0} is ℕ, where n ∈ ℕ corresponds in the canonical basis to the semi-standard tableau of shape 2nϖ and of weight 0.

| 1 | . . . . . . | 1 | 2 | . . . . . . | 2 |

Example 2. For 𝔤 = 𝔰𝔩_{3}, and w_{0} = s_{1}s_{2}s_{1}, the cone C_{0} is generated as a semi-group by (1, 0, 1), (0, 1, 0), (1, 1, 0), (1, 0, 2), corresponding to the following tableaux of weight 0.

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

In the ℂ(q) algebra of C_{0}, the multiplication corresponds to concatenation, see [11]. We have the relation :

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

In the degenerated algebra :

| 1 | 2 | 3 | ∗ \begin{matrix}| 1 & | 1 & | 2 & | \\ | 2 & | 3 & | 3 & | \end{matrix} = 0

This gives, for the case 𝔰𝔩_{3}, an interpretation of a result of Joseph-Letzter, [22, 7.6].

Example 3. For 𝔤 = 𝔰𝔩_{n+1} and w_{0} = s_{1} … s_{n}s_{1} … s_{n−1} … s_{1}s_{2}s_{1}. The cone C_{0} is the set of points (n_{i,j}, 1 ≤ i ≤ j ≤ n) such that

∑_{k=j}^{n} n_{i,k} − ∑_{k=j+1}^{n} n_{i+1,k} ≤ ( ∑_{k≤k′} n_{k,k′}α_{k,k′}, α_{i}), 1 ≤ i ≤ j ≤ n.

In Example 4, we calculate P^{+}-exponents of type μ, μ ∈ P^{+}, in a particular case. Then, we deduce the ℕ-exponents of μ. Example 4. For 𝔤 of type A_{3}, let’s calculate the ℕ-exponents of μ = ϖ_{1} + 2ϖ_{2} + ϖ_{3}. By [loc. cit., 3.4.2 (i)], B(μ)_{0} is parametrized by the semi-standard Young tableaux of shape μ with entries (1, 1, 2, 2, 3, 3, 4, 4). These tableaux are given by

\begin{matrix}| 1 & | 1 & | 2 & | 2 & | \\ | 3 & | 3 & | 4 & | \\ | 4 & | \end{matrix} \begin{matrix}| 1 & | 1 & | 2 & | 3 & | \\ | 2 & | 3 & | 4 & | \\ | 4 & | \end{matrix} \begin{matrix}| 1 & | 1 & | 2 & | 3 & | \\ | 2 & | 4 & | 4 & | \\ | 3 & | \end{matrix}

\begin{matrix}| 1 & | 1 & | 2 & | 4 & | \\ | 2 & | 3 & | 3 & | \\ | 4 & | \end{matrix} \begin{matrix}| 1 & | 1 & | 2 & | 4 & | \\ | 2 & | 3 & | 4 & | \\ | 3 & | \end{matrix} \begin{matrix}| 1 & | 1 & | 3 & | 3 & | \\ | 2 & | 2 & | 4 & | \\ | 4 & | \end{matrix}
| 1 | 1 | 3 | 4 |
| 2 | 2 | 4 |
| 3 |

By [loc. cit., 3.4.2 (ii)] and Proposition 1.8, the P^{+}-exponents of μ are respectively 2ϖ_{1} + ϖ_{3}, ϖ_{1} + ϖ_{2}, ϖ_{1} + ϖ_{2} + ϖ_{3}, ϖ_{1} + ϖ_{2} + ϖ_{3}, ϖ_{1} + 2ϖ_{3}, 2ϖ_{2}, ϖ_{2} + ϖ_{3}. After reordering, the exponents of μ are (3, 4, 5, 5, 6, 6, 7). from the second theorem of 0.3. Example 5 is devoted to the decription of the Joseph-Letzter filtration for 𝔰𝔩_{2}. Example 5. For 𝔤 = 𝔰𝔩_{2}, we have F_{ϖ} = ℂ ⊕ 𝔤 ⊕ ℂc, where c is the quadratic Casimir element. F_{nϖ} = F_{ϖ}^{n}. F_{nϖ} can be described by its 𝔫^{+}-invariant component ⨁_{k≤n}(F^{kϖ})^{𝔫^{+}} = ⨁_{0≤i≤k≤n} ℂE^{k−i}c^{i}. Moreover, U_{n} = ⨁_{0≤h+2z≤n} H^{hϖ} ⊗ Z^{zϖ}.

APPENDIX

Lemma A1. Let M be a torsion-free module on a discrete valuation ring R, 𝔪 be its maximal ideal, K be its fraction field and k be its residue field. Suppose dim_{K}K ⊗_{R} M =dim_{k}k ⊗_{R} M = r. Then, M is a free R-module with rank r.

Proof. Let (b_{i}), 1 ≤ i ≤ r, be a part of M such that (1 ⊗ b_{i}) is a basis of the k-espace k ⊗_{R} M. Then, (b) is A-free. By cardinality, it is a basis of the K-space K ⊗_{R} M. Let N be the A-module generated by (b). Then, M/N is 𝔪-torsion. Let’s prove that M/N has no non zero 𝔪-torsion elements. Let m in M be the lift of a 𝔪-torsion element in M/N. For all m in M, there exists r such that 𝔪^{r}m ∈ N. Choose r minimal non zero. By tensoring by k, and using the fact that each element of R − 𝔪 is invertible, we obtain that (1 ⊗ b) is not free, which is absurd. Hence, r = 0 and this gives the Lemma. ♢

Lemma A2. Let K be a field and let R be a localization of the K-algebra K[T] by a multiplicative part S such that T ∉ S. Let M be a torsion free R-module. Suppose that

1) M[T^{−1}] is R[T^{−1}]-free with rank r,

2) M/T M is a r-dimensional R/T R-vector space. Then, M is a free R-module with rank r.

Proof. Set k = R/T R. Let’s prove the following assertion.

Assertion. There exist e_{1}, …, e_{r} in M such that N = ⊕Re_{i} is free with rank r, {1⊗e_{i}} is a basis of M/T M and N[T^{−1}] = M[T^{−1}].

We proceed by noetherian induction. Let e_{1}^{0}, …, e_{r}^{0} in M such that (1 ⊗ e_{i}^{0}) is a basis of the k-vector space k ⊗_{R} M. Set N_{0} = ∑ Re_{i}^{0} ⊂ M. By Krull intersection theorem, [16, Corollary 5.4], (e_{i}^{0}) is a base of the R-module N_{0}. If N_{0}[T^{−1}] = M[T^{−1}], there is nothing to prove. If not, then there exists x_{1} in M such that T^{k}x_{1} ∉ N_{0} for all k in ℤ. Set N_{1} = N_{0} + Rx_{1}. Then, N_{1} is a finitely generated torsion free R-module, so, it is a free R-module. Localizing by T, we obtain that rkN_{1} = r. There exists a basis e_{1}^{1}, …, e_{r}^{1} of the R module N_{1} and, as N_{1} contains N_{0}, (k ⊗ e_{i}^{1}) is a basis of the k-space k ⊗_{R} M. Using induction and the fact that M[T^{−1}] is a noetherian R[T^{−1}]-module gives the assertion. We want to prove that the module N of the assertion is equal to M. It is sufficient to prove that N_{𝔪} = M_{𝔪} for all maximal ideal 𝔪 of R. If T ∉ 𝔪, the equality results from N[T^{−1}] = M[T^{−1}]. If T ∈ 𝔪, then M_{𝔪}/N_{𝔪} is T -torsion by the previous formula. Moreover, as in the proof of the previous lemma, M_{𝔪}/N_{𝔪} has no nonzero T -torsion elements. This ends the proof. ♢

Proposition A4. Define the polynomial ring R_{1} = ℂ[X_{1}, .., X_{n}] and the local ring R_{2} = ℂ[T]_{(T)}. Let K_{1} = ℂ(X_{1}, .., X_{n}) et K_{2} = ℂ(T) be their fraction fields. Set R = R_{1} ⊗_{ℂ} R_{2} endowed with its natural structure of ring. Let M be a torsion-free R-module such that

(i) M/T M, is a free R/T R-module with rank r, (ii) K_{2}M = M_{T} is a free R_{1} ⊗_{ℂ} K_{2}-module with rank r.

Then, M is a free R-module with rank r.

Proof. From Lemma A2, we deduce that K_{1}M is a free K_{1}R-module with rank r. We now have to prove that M is finitely generated over R. This is the aim of the next lemma :

Lemma A3. Let M be a torsion-free R-module such that K_{1}M, resp. K_{2}M = M_{T}, is a free K_{1} ⊗ R_{2}-module with rank r, resp. R_{1} ⊗ K_{2}-module with rank r. Then, M is finitely generated as a R-module.

Proof. First remark that R is a UFD domain. Recall that for all multiplicative part S of R, we have S^{−1}(⋀^{r} M) = ⋀^{r}(S^{−1}M), as S^{−1}M-modules. Let F = ⋀^{r} M. From the hypothesis, we have that K_{1}F and K_{2}F are free with rank 1. Let e_{1}^{s}, …, e_{r}^{s} ∈ M such that e_{i}^{s} is a basis of the K_{s}R-module K_{s}M, s = 1 or 2. Set e^{s} := e_{1}^{s} ∧ … ∧ e_{r}^{s} ∈ F. The element e^{s} is a basis of the K_{s}R module K_{s}F. Let x be in F. We have x = (α_{s}/a_{s})e^{s} and e^{2} = (γ/d)e^{1} with α_{s}, γ ∈ R, a_{s} ∈ R_{s}, d ∈ R_{2}, GCD(α_{s}, a_{s}) =GCD(γ, d) = 1. Hence, we obtain α_{1}/a_{1} = (α_{2}γ)/(a_{2}d) and then, r := α_{1}d/a_{1} = α_{2}γ/a_{2} ∈ K_{1}R ∩ K_{2}R = R. In conclusion, F is a subset of (1/d)e^{1}. Fix an element y in M, y = ∑_{i}(μ_{i}/p_{i})e_{i}^{1}, μ_{i} ∈ R, p_{i} ∈ R_{1}, GCD(μ_{i}, p_{i}) = 1. By the previous assertion, there exists s_{i} such that

e_{1}^{1} ∧ … ∧ e_{i−1}^{1} ∧ y ∧ e_{i+1}^{1} ∧ … ∧ e_{r}^{1} = (μ_{i}/p_{i})e^{1} = (s_{i}/d)e^{1}.

We deduce that μ_{i}/p_{i} = s_{i}/d and so y ∈ (1/d) ∑ Re_{i}^{1}. We obtain that M is finitely generated as a R-module. ♢

end of proof of proposition. In order to show the proposition, it suffices, by the Quillen-Suslin theorem, cf. [28, Introduction], to prove that M is a projective, i.e. locally free, R-module. Let 𝔪 be a maximal ideal of R. Let’s prove that M_{𝔪} is free. If T does not belong to 𝔪, then M_{𝔪} is a localization of the module M[T^{−1}] = K_{2}M and so, it is a free module. If T belongs to 𝔪, it is sufficient to prove that M_{𝔪} is a flat R_{𝔪}-module. This results from the hypothesis and [16, Corollary 6.9]. ♢

ACKNOWLEDGEMENTS

I thank R. Berger and M. Cretin for helpful discussions on Gröbner basis, M. Reineke for e-mail conversations on Hall algebras and T. Levasseur for precious remarks about the appendix. I am grateful to B. Leclerc and S. Zelikson for several informations about crystal basis. I also wish to thank J. Germoni for comments and encouragements. Most of all, I am in debt to P. Baumann for his remarks and corrections. And last, but not least, the article owes a lot to the careful remarks of the referee.

BIBLIOGRAPHY

1. P. BAUMANN. Another proof of Joseph and Letzter’s separation of variables theorem for quantum groups, Transform. Groups, 5, (2000), 3-20.

2. T. BECKER and V. WEISPFENNING. Gröbner bases : a computational approach to commutative algebra, Graduate texts in Mathematics, 141, New-York, Springer-Verlag, (1993).

3. D.J. BENSON. Representations and cohomology I, Basic representation theory of finite groups and associative algebras, Cambridge studies in Advanced Mathematics, 30, Cambridge University Press.

4. A. BERENSTEIN and A. ZELEVINSKY. Canonical bases for the quantum group of type A_{r} and piecewise-linear combinatorics., Duke Math. J., 82, (1996), no. 3, 473-502.

5. A. BERENSTEIN and A. ZELEVINSKY. Tensor product multiplicitie, Canonical bases and Totally positive varieties, ArXiv:Math.RT/9912012.

6. N. BOURBAKI. Groupes et Algèbres de Lie, Chap. VI, Masson, Paris, (1981).

7. R.K. BRYLINSKI. Limits of weight spaces, Lusztig’s q-analogs, and fiberings of adjoint orbits, J. Amer. Math. Soc., 2, (1989), 517-533.

8. P. CALDERO. Générateurs du centre de U_{q}(sl(N + 1)), Bull. Sci. Math., 118, (1994), 177-208.

9. P. CALDERO. Elements ad-finis de certains groupes quantiques, C. R. Acad. Sci. Paris, Serie I, t. 316, (1993), 327-329.

10. P. CALDERO. Etude des q-commutations dans l’algèbre U_{q}(𝔫), J. Algebra, 178, (1995), 444-457.

11. P. CALDERO. On the q-commutations in U_{q}(𝔫) at roots of one, J. Algebra, 210, (1998), 557-576.

12. C. DE CONCINI and V. G. KAC. Representations of quantum groups at roots of 1, Colloque Dixmier, Progress in Math., 92, (1990), 471-506.

13. C. DE CONCINI, V. G. KAC, C. PROCESI. Quantum coadjoint action, J. Amer. Math. Soc., 5, (1992), 151-189.

14. J. DIXMIER. Algèbres enveloppantes, Cahiers scientifiques n^{∘} 37, Gauthiers-Villars, Paris, 1974.

15. V. G. DRINFELD. On almost cocommutative Hopf algebras, Leningrad Math. J., I, (1990), 321-342.

16. D. EISENBUD. Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Berlin: Springer-Verlag, (1995).

17. W.H. HESSELINK. Characters of the Nullcone, Math. Ann., 252, (1980), 179-182.

18. A. JOSEPH. Quantum groups and their primitive ideals, Springer-Verlag, 29, Ergebnisse der Mathematik und ihrer Grenzgebiete, (1995).

19. A. JOSEPH. On the mock Peter-Weyl theorem and the Drinfeld double of a double, Reine Angew. Math., 507 ,(1999), 37-56.

20. A. JOSEPH and G. LETZTER. Local finiteness of the adjoint action for quantized envelopping algebras, J. Algebra, 153, (1992), 289-318.

21. A. JOSEPH and G. LETZTER. Separation of variables for quantized enveloping algebras, Amer. J. Math., 116, (1994), 127-177.

22. A. JOSEPH and G. LETZTER. Verma modules annihilators for quantized enveloping algebras, Ann. Sci. Ec. Norm. Sup., 28, (1995), 493-526.

23. A. KANDRI-RODY and V. WEISPFENNING. Non-commutative Gröbner bases in Algebras of solvable type, J. Symb. Comp., 9, (1990), 1-26.

24. M. KASHIWARA. On Crystal Bases, Canad. Math. Soc., Conference Proceed., 16, (1995), 155-195.

25. M. KASHIWARA and T. NAKASHIMA. Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Alg., 165, (1994), 295-345.

26. S.M. KHOROSHKIN and V.N. TOLSTOI. The uniqueness theorem for the Universal R-matrix, Lett. Math. Phys., 24, (1992), 231-244.

27. B. KOSTANT. Lie groups representations on polynomial rings, Amer. J. Math., 85, (1963), 327-404.

28. T.Y. LAM. Serre’s conjecture, Lecture Notes in Mathematics, 635, Berlin - Heidelberg - New York: Springer-Verlag, (1978).

29. A. LASCOUX, B. LECLERC and J-Y. THIBON. Crystal graphs and q-analogues of Weight Multiplicities for the Root System A_{n}, Lett. Math. Phys., 35, (1995), 359-374.

30. G. LUSZTIG. Canonical bases arising from quantized enveloping algebras. II, Prog. Theor. Phys. Suppl., 102, (1990), 175-201.

31. G. LUSZTIG. Introduction to quantum groups, Progress in Mathematics, 110, Birkauser, (1993).

32. M. REINEKE. Multiplicative Properties of Dual Canonical Bases of Quantum Groups, J. Alg., 211, (1999), 134-149.

33. C.M. RINGEL. Hall algebras and quantum groups, Invent. Math., 101, (1990), 583-592.

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

35. T. TANISAKI. Killing forms, Harish-Chandra isomorphisms, and universal Rmatrices for quantum algebras, Int. J. Mod. Phys. A, Vol. 7, Suppl. 1B, (1992), 941-961. Institut Girard Desargues, UPRES-A-5028 Université Claude Bernard Lyon I, Bat 101 69622 Villeurbanne Cedex, France e-mail : caldero@desargues.univ-lyon1.fr

Keywords : Quantum groups, enveloping algebras, canonical basis, harmonic space.