Transformation Groups, Vol. ?, No. ?, ??, pp. 1–?? \begin{matrix}c \\ ◯ \end{matrix}Birkhäuser Boston (??)

TORIC DEGENERATIONS OF SCHUBERT VARIETIES

PH. CALDERO^{∗}

Institut Girard Desargues Université Claude Bernard – Lyon 1 69622 Villeurbanne cedex FRANCE caldero@desargues.univ-lyon1.fr

Abstract. Let G be a simply connected semi-simple complex algebraic group. We prove that every Schubert variety of G has a flat degeneration into a toric variety. This provides a generalization of results of [9], [7], [6]. Our basic tool is Lusztig’s canonical basis and the string parametrization of this basis.

0. Introduction

=restricted.

0.1. Let G be a simply connected semi-simple complex algebraic group. Fix a maximal torus T and a Borel subgroup B such that T ⊂ B ⊂ G. Let W the Weyl group of G relative to T. For any w in W, let X_{w} = \overline{BwB/B} denote the Schubert variety corresponding to w. This article is concerned with the following problem.

Degeneration Problem. Is there a flat family over Spec ℂ[t], such that the general fiber is X_{w} and the special fiber is a toric variety?

The existence of such a degeneration was obtained by N. Gonciulea and V. Lakshmibai for the flag variety G/B when G = SL_{n}, [9]. Their proof is based on the theory of standard monomials. The corner stone of their proof is the following : fundamental weights are minuscule weights, hence, a basis of every fundamental representation is endowed with a structure of distributive lattice.

A complete study of the degeneration problem was made in the case when G has rank two, [6]. Note also that the A_{n} case was studied by R. Dehy and R. Yu for a class of elements w in the Weyl group, [7], [8]. The proofs rely on the theory of standard monomials as well. A natural question would be : is there a (flat) toric degeneration of the flag variety G/B which restricts to a toric degeneration of the Schubert varieties X_{w} for any w in the Weyl group? In [5], R. Chirivi gives a degeneration of the flag variety which restricts into semi-toric degenerations of the Schubert varieties, i.e. finite unions of irreducible toric varieties. An explanation of this fact was given to us by O. Mathieu: intersections of irreducible toric varieties are irreducible toric varieties, but intersection of Schubert varieties can be a union of several Schubert varieties. Hence, the answer to the previous question is negative. In [5], the Degeneration Problem is solved with toric replaced by semi-toric.

∗Supported in part by the EC TMR network “Algebraic Lie Representations”, contract no.

ERB FMTX-CT97-0100

Received December 11, 1895. Accepted February 29, 1876.

0.2. Our approach of the problem is based on the canonical/global base of Lusztig/Kashiwara and the so-called string parametrization of this base studied by A. Berenstein and A. Zelevinsky, [1], [2] and P. Littelmann, [12].

Fix w in W. Let P^{+} be the semigroup of dominant weights. For all λ in P^{+}, let L_{λ} be the line bundle on G/B corresponding to λ. Then, the direct sum of global sections R_{w} := ⨁_{λ∈P^{+}} H^{0}(X_{w}, L_{λ}) carries a natural structure of P^{+}-graded ℂ-algebra. Moreover, there exists a natural action of T on R_{w}. Our principal result can be stated as follows :

Theorem. Fix w in W. There exists a filtration (F_{m}^{w})_{m∈ℕ} of R_{w} such that (i) for all m in ℕ, F_{m}^{w} is compatible with the P^{+}-grading of R_{w}, (ii) for all m in ℕ, F_{m}^{w} is compatible with the action of T, (iii) the associated graded algebra is the ℂ-algebra of the semigroup of integral points in a rational convex polyhedral cone.

This cone (and the filtration) depends on the choice of a reduced decomposition w̃_{0} of the longest element w_{0} of the Weyl group. Explicit equations for the faces of this cone can be obtained from [12] for so-called nice decompositions w̃_{0}. More generally, those equations were obtained in [2] from w̃_{0}-trails in fundamental Weyl modules of the Langlands dual of G.

This theorem gives a positive answer to the Degeneration Problem. Indeed, let λ be a regular dominant weight, then the line bundle L_{λ} is ample and X_{w} is the projective spectrum of ⨁_{m∈ℕ} H^{0}(X_{w}, L_{mλ}). Moreover, the spectrum of a noetherian graded algebra associated to a filtration of a noetherian algebra R is a flat degeneration of Spec(R). This is proved as follows by a standard argument : let t be an indeteminate and consider the filtration (R_{n})_{n∈ℕ} of R. Then, the ℂ[t]-algebra R^{t} = ⊕_{n}R_{n}t^{n} is flat over ℂ[t] and it verifies R^{t}/(t − t_{0})R^{t} ≃ R for t_{0} ≠ 0 and R^{t}/tR^{t} ≃ GrR.

Let U be the maximal unipotent subgroup of B. Hence, R_{w_{0}} is the algebra ℂ[G/U] of regular functions on G/U. For any w, the algebras R_{w} are quotients of this algebra. Let I_{w} be the ideal such that R_{w} = R_{w_{0}}/I_{w}. If w̃_{0} is adapted to w in the sense of Definition 2.4, then the filtration (F_{m}^{w})_{m∈ℕ} is the quotient filtration of (F_{m}^{w_{0}})_{m∈ℕ} defined by F_{m}^{w} := \overline{F}_{m}^{w_{0}} = F_{m}^{w_{0}} + I_{w}. For a general reduced decomposition w̃_{0}, the filtration (\overline{F}_{m}^{w_{0}})_{m∈ℕ} of R_{w} provides a graded associated algebra whose spectrum is a semi-toric variety.

The proof of the theorem is based on two facts. Let U^{−} be the maximal unipotent subgroup of G which is opposite to U. Set B^{−} = T U^{−}. Then, the algebra ℂ[G/U] embeds in C[B^{−}]. Moreover, we can embed the (specialized) dual of the canonical base in the algebra ℂ[U^{−}]. We prove that this dual has good multiplicative properties inherited from the quantum case, see Theorem 2.3. This part of the article is inspired by [15]. But here, we don’t use the positivity arguments or the elaborate Hall algebra model, only true for the simply laced case. In a second step we show how to restrict from flag variety to Schubert varieties : this part relies on the compatibility of the canonical base with the Demazure modules, [11],

1. Notations and recollection on global basis.

1.1. Denote by G a semisimple simply connected complex Lie group. Fix a torus T of G and let B be a Borel subgroup such that T ⊂ B ⊂ G. Denote by U the unipotent radical of B. Let B^{−} be the opposite Borel subgroup and U^{−} be its unipotent radical. Let 𝔤, resp. 𝔥, 𝔫, 𝔟, 𝔫^{−}, 𝔟^{−}, be the Lie ℂ-algebra of G, resp. T, U, B, U^{−}, B^{−}. Let n be the rank of 𝔤. We have the triangular decomposition 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫. Let {α_{i}}_{i} be a basis of the root system Δ corresponding to this decomposition. Let P be the weight lattice generated by the fundamental weights ϖ_{i}, 1 ≤ i ≤ n, and let P^{+} := ∑_{i} ℕϖ_{i} be the semigroup of integral dominant weights. Let W be the Weyl group, generated by the reflections s_{α_{i}} corresponding to the simple roots α_{i}, and let w_{0} be the longest element of W. We denote by ( , ) the W -invariant form on P.

1.2. Let d be an integer such that (P, P) ⊂ (2/d)ℤ. Let q be a indeterminate and set 𝕂 = ℂ(q^{1/d}). Let U_{q}(𝔤) be the simply connected quantized enveloping algebra on 𝕂, as defined in [4], i.e. it is the standard quantized enveloping algebra on 𝕂 whose Cartan subalgebra is isomorphic to the group algebra 𝕂[P]. Set d_{i} = (α_{i}, α_{i})/2 and q_{i} = q^{d_{i}} for all i. Let U_{q}(𝔫), resp. U_{q}(𝔫^{−}), be the subalgebra generated by the canonical generators E_{i}, resp. F_{i}, of weight, α_{i}, resp. −α_{i}, and the quantum Serre relations. For all λ in P, let K_{λ} be the element corresponding to λ in the algebra U_{q}^{0} = 𝕂[P] of the Cartan subalgebra of U_{q}(𝔤). We have the triangular decomposition U_{q}(𝔤) = U_{q}(𝔫^{−})⊗U_{q}^{0}⊗U_{q}(𝔫). We set

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

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

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

If M is a U_{q}^{0}-module and γ ∈ P, we set M_{γ} := {m ∈ M, K_{λ}.m = q^{(λ,γ)}m}.

For n a non negative integer, we set : [n]_{i} = \frac{1−q_{i}^{n}}{1−q_{i}}, [n]_{i}! = [n]_{i}[n − 1]_{i} … [1]_{i}.

1.3. The dual U_{q}(𝔤)^{∗} is endowed with a structure of left, resp. right, U_{q}(𝔤)-module by u.c(a) = c(au), resp. c.u(a) = c(ua), u, a ∈ U_{q}(𝔤), c ∈ U_{q}(𝔤)^{∗}. If M is a finite dimensional left U_{q}(𝔤)-module, we endow the dual M^{∗} with a structure of left U_{q}(𝔤)- module by uξ(v) = ξ(S(u)v), u ∈ U_{q}(𝔤), ξ ∈ M^{∗}, v ∈ M.

For all λ in P^{+}, let V_{q}(λ) be the simple U_{q}(𝔤)-module with highest weight λ. We can embed V_{q}(λ)^{∗} ⊗ V_{q}(λ) in U_{q}(𝔤)^{∗} by setting ξ ⊗ v(u) = ξ(u.v), u ∈ U_{q}(𝔤), ξ ∈ V_{q}(λ)^{∗}, v ∈ V_{q}(λ). Let v_{λ} be a highest weight vector of V_{q}(λ). For all integral dominant weights λ, let C(λ), resp. C^{+}(λ), be the subspace of U_{q}(𝔤)^{∗} generated by the ξ ⊗v, resp. ξ ⊗v_{λ}, ξ ∈ V_{q}(λ)^{∗}, v ∈ V_{q}(λ). We set ℂ_{q}[G] = ⨁_{λ∈P^{+}} C(λ), ℂ_{q}[G/U] = ⨁_{λ∈P^{+}} C^{+}(λ). Then, ℂ_{q}[G] and ℂ_{q}[G/U] are subalgebras of the Hopf dual of U_{q}(𝔤). ℂ_{q}[G], resp. ℂ_{q}[G/U], is the algebra of quantum regular functions on G, resp. on the quotient G/U.

1.4. There exists a unique bilinear form ( , ) on U_{q}(𝔟) × U_{q}(𝔟^{−}), see [16], [17], [4], such that :

(1.4.1)  (u^{+}, u_{1}^{−}u_{2}^{−}) = (Δ(u^{+}), u_{1}^{−} ⊗ u_{2}^{−}) ,  u^{+} ∈ U_{q}(𝔟) ; u_{1}^{−}, u_{2}^{−} ∈ U_{q}(𝔟^{−})
(1.4.2)  (u_{1}^{+}u_{2}^{+}, u^{−}) = (u_{2}^{+} ⊗ u_{1}^{+}, Δ(u^{−})) ,  u^{−} ∈ U_{q}(𝔟^{−}) ; u_{1}^{+}, u_{2}^{+} ∈ U_{q}(𝔟)

(1.4.3)  (K_{λ}, K_{μ}) = q^{−(λ,μ)} ,  λ, μ ∈ P
(1.4.4)  (K_{λ}, F_{i}) = 0 ,  λ ∈ P , 1 ≤ i ≤ n
(1.4.5)  (E_{i}, K_{λ}) = 0 ,  λ ∈ P , 1 ≤ i ≤ n
(1.4.6)  (E_{i}, F_{i}) = δ_{ij}(1 − q_{i}^{2})^{−1} ,  1 ≤ i, j ≤ n.

where the bilinear form on (U_{q}(𝔟)⊗U_{q}(𝔟))×(U_{q}(𝔟^{−})⊗U_{q}(𝔟^{−})) is defined by (u⊗v, u′ ⊗ v′) := (u, u′)(v, v′), u, v ∈ U_{q}(𝔟), u′, v′ ∈ U_{q}(𝔟^{−}). For all β in Q^{+} := ∑_{i} ℕα_{i}, let U_{q}(𝔫)_{β}, resp. U_{q}(𝔫^{−})_{−β}, be the subspace of U_{q}(𝔫), resp. U_{q}(𝔫^{−}), with weight β, resp. −β. The form ( , ) is non degenerate on U_{q}(𝔫)_{β}×U_{q}(𝔫^{−})_{−β}, β ∈ Q^{+}. We have, by (1.4.1-1.4.5) :

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

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

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

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

There exists an algebra isomorphism from U_{q}(𝔫) to U_{q}(𝔫^{−}) which maps E_{i} on F_{i} for all i. Via this isomorphism, the restriction of the form ( , ) on U_{q}(𝔫) × U_{q}(𝔫^{−}) coincides with the one defined by Lusztig in [13, par 1].

1.5. Define the maps :

β : U_{q}(𝔟) → U_{q}(𝔟^{−})^{∗}, β(u)(v) = (u, v) ;
ζ : U_{q}(𝔤) → U_{q}(𝔤)^{∗}, ζ(u)(v) =< u, v > .

It follows from 1.4 that:

Lemma. With the previous notations, we have

(i) β, ζ are injective. (ii) β is an anti-homomorphism of algebras.

Denote by ρ the restriction homomorphism from U_{q}(𝔤)^{∗} onto U_{q}(𝔟^{−})^{∗}. We know, see [3, Proposition 3.4]:

Proposition. The restriction of ρ to ℂ_{q}[G/U] is injective. Moreover, for all λ in P^{+}, we have (i) For all e in U_{q}(𝔫), ρ(ζ(eK_{−2λ})) = β(eK_{−λ}). (ii) there exists a (unique) subspace E_{λ} of U_{q}(𝔫) such that ζ(E_{λ}K_{−2λ}) = C^{+}(λ). ♢ 1.6. Let u ↦ \overline{u} be the 𝕂-antihomomorphism of U_{q}(𝔤) such that \overline{E}_{i} = E_{i}, \overline{K}_{λ} = K_{−λ}, \overline{F}_{i} = F_{i}. It is easily seen that

< u, v >= (u, v) = (\overline{u}, \overline{v}) =< \overline{u}, \overline{v} >, u ∈ U_{q}(𝔫), v ∈ U_{q}(𝔫^{−}).

Let B be the Lusztig’s canonical basis of U_{q}(𝔫^{−}), [13], which coincides with Kashiwara’s global basis, [11]. Let B^{∗} ⊂ U_{q}(𝔫) be the dual basis in U_{q}(𝔫), i.e. (b^{∗}, b′) = δ_{b,b′}. Note that what is usually called ”dual canonical basis” lies in the dual space U_{q}(𝔫^{−})^{∗}. To be more precise, the common dual canonical basis is the image of β(B^{∗}) by the restriction morphism U_{q}(𝔟^{−})^{∗} → U_{q}(𝔫^{−})^{∗}. 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 assignement 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}.

Let L_{i}, 1 ≤ i ≤ n, be the adjoint of the left multiplication operator u ↦ F_{i}u for the form ( , ) on U_{q}(𝔫) × U_{q}(𝔫^{−}). Then, L_{i} is a quantum derivation of U_{q}(𝔫), [13, par 1]:

L_{i}(e_{α}u) = L_{i}(e_{α})u + q^{(α,α_{i})}e_{α}L_{i}(u), u ∈ U_{q}(𝔫), e_{α} ∈ U_{q}(𝔫)_{α}, α ∈ Q.

Set L_{i}^{(r)} = \frac{1}{[r]_{i}!}L_{i}^{r}.

The following is a recollection of results about the canonical basis and its dual. Assertions (i) and (ii) can be read in [13, 14.4.13,14.4.14]. Assertion (iii) is a standard consequence of [13, 14.3.2 (c)] by dualization.

Theorem 1. For b, b′ in B, we have: (i) bb′ ∈ ℤ[q, q^{−1}]B, (ii) b^{∗}b^{′∗} ∈ ℤ[q, q^{−1}]B^{∗}, (iii) L_{i}^{(ε_{i}(b))}(b^{∗}) = (ẽ_{i}^{ε_{i}(b)}(b))^{∗}, (iv) \overline{b} ∈ B. ♢

The following theorem states precisely the compatibility of the canonical basis with the finite dimensional highest weight modules. It can be found in [11, Proposition 8.2].

Theorem 2. Fix λ in P^{+}, λ = ∑_{i} λ_{i}ϖ_{i}. Set

B_{λ} := \{b ∈ B, ε_{i}(\overline{b}) ≤ λ_{i}, 1 ≤ i ≤ n\}.

Then B_{λ} = {b ∈ B, b.v_{λ} ≠ 0} and B_{λ}.v_{λ} is a basis of V_{q}(λ). Moreover, the correspondence π_{λ} : b ↦ bv_{λ} is injective on B_{λ}. ♢

When b belongs to B_{λ}, and if no confusion occurs, we shall use the same symbol for b and b.v_{λ}, i.e. we set b = π_{λ}(b) = b.v_{λ}.

1.7. Let A = ℂ[q, q^{−1}]. Let U_{A}(𝔫^{−}) be the A-submodule of U_{q}(𝔫^{−}) generated by B. Then, U_{A}(𝔫^{−}) is a free A-space and a A-algebra. Indeed, U_{A}(𝔫^{−}) is the A-algebra generated by the \frac{1}{[m]_{i}!}F_{i}^{m}, see [10, Theorem 11.10 (b)]. Let U_{A} be the sub-A-algebra of U_{q}(𝔤) generated by the \frac{1}{[m]_{i}!}F_{i}^{m}, and the \frac{1}{[m]_{i}!}E_{i}^{m}. Set V_{A}(λ) = U_{A}(𝔫^{−}).v_{λ} ⊂ V_{q}(λ). Then, V_{A}(λ) is the (free) A-module generated by B_{λ}. By [10, Theorem 11.19], we know that B is compatible with specialization :

ℂ ⊗_{A} U_{A}(𝔫^{−}) ≃ U(𝔫^{−}), ℂ ⊗_{A} V_{A}(λ) ≃ V (λ), λ ∈ P^{+},

where ℂ = A/(q − 1)A as a A-module, U(𝔫^{−}) is the (classical) enveloping ℂ-algebra of 𝔫^{−} and V (λ) is the classical Weyl module with highest weight λ.

Let V_{A}(λ)^{∗} be the A-dual of V_{A}(λ). Then, it has a natural U_{A}-module structure by u.v^{∗}(m) = v^{∗}(S(u).m), u ∈ U_{A}, v ∈ V_{A}(λ)^{∗}, m ∈ V_{A}(λ). The module V_{A}(λ)^{∗} specializes at q = 1 onto the dual 𝔤-module V (λ)^{∗}.

1.8. Fix λ in P^{+} and w in W. We know that V_{q}(λ) verifies the Weyl character formula ; we denote by v_{wλ} an extremal vector of weight wλ. Then, the U_{q}(𝔟)-module V_{q,w}(λ) := U_{q}(𝔫)v_{wλ} verifies the Demazure character formula. We know, [11, Theorem 12.4], [12, 5.3-5.4], that

Theorem. There exists a subset B_{w} of B such that V_{q,w}(λ) is spanned by B_{w}.v_{λ}. Moreover, if b is in B_{w} then Δ(b) ∈< B_{w} > ⊗ < K_{μ}.B_{w}, μ ∈ P >, where < E > denotes the subspace generated by the part E of U_{q}(𝔤). ♢

In particular, the orthogonal V_{q,w}(λ)^{⊥} in V_{q}(λ)^{∗} of the Demazure module V_{q,w}(λ) is generated as a space by (B_{λ} ∩ B\B_{w})^{∗} and the dual V_{q,w}(λ)^{∗} is generated by the image of (B_{λ} ∩ B_{w})^{∗} by the quotient morphism.

As in 1.7, this allows us to define A-forms for Demazure modules. We denote by V_{A,w}(λ) the A-module generated by B_{λ} ∩ B_{w}. It specializes for q = 1 to the classical Demazure module V_{w}(λ).

2. A multiplicative property and string parametrizations of the dual canonical basis.

2.1. For λ in P^{+} and b in B_{λ}, let π_{λ}(b)^{∗} be the element of V_{q}(λ)^{∗} such that π_{λ}(b)^{∗}(b′.v_{λ}) = δ_{b,b′}, b′ ∈ B_{λ}, where δ means the Kroenecker symbol.

Lemma. For all λ in P^{+} and b in B_{λ}, whe have ζ(b^{∗}K_{−2λ}) = π_{λ}(b)^{∗} ⊗v_{λ}, β(b^{∗}K_{−λ}) = ρ(π_{λ}(b)^{∗} ⊗ v_{λ}), where ζ, β, and ρ are defined as in 1.5.

Proof. By Lemma 1.5, we only need to prove that < b^{∗}K_{−2λ}, u >= π_{λ}(b)^{∗}(u.v_{λ}), u ∈ U_{q}(𝔤). As U_{q}(𝔤) = U_{q}(𝔟^{−})⊕(U_{q}(𝔟^{−})⊗U_{q}(𝔫)∩Ker(ε)), we only need to prove this formula for u ∈ U_{q}(𝔟^{−}). Remark that a basis of U_{q}(𝔟^{−}) can be given (b′K_{μ}, b′ ∈ B, μ ∈ P). By (1.4.7) and (1.4.8),

< b^{∗}K_{−2λ}, b′K_{μ} >= (b^{∗}, b′)q^{(λ,μ)} = δ_{b,b′}q^{(λ,μ)} = π_{λ}(b)^{∗}(b′K_{μ}.v_{λ}).

This implies the lemma. ♢

Remark. By Proposition 1.5, the lemma implies that E_{λ} is spanned by a part of B^{∗}, namely B_{λ}^{∗}.

Let A_{q}[G/U] be the sub-A-module of ℂ_{q}[G] generated by the π_{λ}(b)^{∗} ⊗ v_{λ}, λ ∈ P^{+} and b ∈ B_{λ}. Let d_{b,b′}^{b′′} be the coefficient of b^{′′∗} in the product b^{∗}b^{′∗}.

Proposition. We have

(i) 𝕂 ⊗_{A} A_{q}[G/U] = ℂ_{q}[G/U] (ii) A_{q}[G/U] is a A-algebra. (iii) if b ∈ B_{λ}, b′ ∈ B_{λ′} and d_{b′,b}^{b′′} ≠ 0, then b′′ ∈ B_{λ+λ′}, (iv) A_{q}[G/U]/(q − 1)A_{q}[G/U] ≃ ℂ[G/U], where ℂ[G/U] is the ℂ-algebra of regular functions on G/U.

Proof. The A-basis π_{λ}(b)^{∗} of V_{A}(λ)^{∗} is a 𝕂-basis of V_{q}(λ)^{∗}. Hence, (i) is clear. ¿From Lemma 1.5, Proposition 1.5, and the previous lemma, we have

β^{−1}[(π_{λ}(b)^{∗} ⊗ v_{λ}).(π_{λ′}(b′)^{∗} ⊗ v_{λ′})] = β^{−1}(π_{λ′}(b′)^{∗} ⊗ v_{λ′}).β^{−1}(π_{λ}(b)^{∗} ⊗ v_{λ})

= (b^{′∗}K_{−λ′}).(b^{∗}K_{−λ}) = q^{−(λ′,ω(b^{∗}))}b^{′∗}b^{∗}K_{−λ−λ′} = q^{−(λ′,ω(b^{∗}))}(∑ d_{b′,b}^{b′′}b′′^{∗})K_{−λ−λ′}.

We know that (π_{λ}(b)^{∗} ⊗ v_{λ}).(π_{λ′}(b′)^{∗} ⊗ v_{λ′}) belongs to C^{+}(λ + λ′). Hence, the previous formula and Remark 2.1 imply (iii). Moreover, applying β, we obtain :

(2.1.1) (π_{λ}(b)^{∗} ⊗ v_{λ}).(π_{λ′}(b′)^{∗} ⊗ v_{λ′}) = q^{−(λ′,ω(b^{∗}))} ∑ d_{b′,b}^{b′′}(π_{λ+λ′}(b′′)^{∗} ⊗ v_{λ+λ′}).

This gives (ii). Now, (iv) is clear by 1.7 and the fact that specialization commutes with tensor product. ♢

Corollary. Fix b, b′, b′′ in B with d_{b,b′}^{b′′} non zero. Then, for all i, 1 ≤ i ≤ n, we have ε_{i}(b′′) ≤ ε_{i}(b) + ε_{i}(b′).

Proof. By applying the antiautomorphism ^{−}, see 1.6, we obtain that d_{\overline{b}′,\overline{b}}^{\overline{b}′′} = d_{b,b′}^{b′′}. Set λ = ∑_{i} ε_{i}(b)ϖ_{i} and λ′ = ∑_{i} ε_{i}(b′)ϖ_{i}. As ^{−} is an involution, we deduce from [1.6, Theorem 2] that \overline{b} ∈ B_{λ} and \overline{b}′ ∈ B_{λ′}. This gives \overline{b}′′ ∈ B_{λ+λ′} by (iii). Hence, we obtain the corollary from [1.6, Theorem 2]. ♢

Remark. In the simply laced case, this corollary is easily obtained by the positivity property of the dual canonical basis, i.e. d_{b,b′}^{b′′} ∈ ℕ[q, q^{−1}] by [13]. For general 𝔤, we can conclude by a standard argument, transmitted by E. Vasserot. It is based on the realization of d_{b,b′}^{b′′} in terms of traces of an automorphism of a diagram on spaces arising from perverse sheaves, see [13, 14.4.14].

2.2. We introduce the string parametrization of the (dual) canonical basis. Fix a reduced decomposition of the longest element of the Weyl group w_{0} : w̃_{0} = s_{i_{1}} … s_{i_{N}}, where N = dim 𝔫. For all u in U_{q}(𝔫) and 1 ≤ i ≤ n, set

a_{i}(u) = Max{r, L_{i}^{r}(u) ≠ 0}, Λ_{i}(u) = L_{i}^{(a_{i}(u))}(u).

For all b in B, set

Aw̃_{0}(b) = (a_{i_{N}} (b^{∗}), a_{i_{N−1}}(Λ_{i_{N}} (b^{∗})), …, a_{i_{1}}(Λ_{i_{2}} … Λ_{i_{N}} (b^{∗}))) ∈ ℕ^{N}.

This parametrization can be found in [12, par 1]. It coincides with the parametrization in [2, 3.2] by [1.6, Theorem 1 (iii)]. We now present a theorem due to Littelmann, [12, par 1], see also [2, 3.10].

Theorem. The map Aw̃_{0} embeds B into ℕ^{N}. Let C be its image. Then, C is the set of integral points of a rational convex polyhedral cone of ℝ^{N}. Moreover, set Γw̃_{0} := {(λ, ψ) ∈ P^{+} × C, ψ ∈ Aw̃_{0}(B_{λ})}. Then, Γw̃_{0} is the set of integral points of a rational convex polyhedral cone of ℝ^{n+N}. ♢

Note that equations of this cone are given in [2, 3.10]. For all ψ in C, set b^{ψ} = bψ ; w̃_{0} =

A−1 ; w̃_{0}(ψ). We also set d_{ψ,ψ′}^{ψ′′} := d_{b^{ψ},b^{ψ′}}^{b^{ψ′′}}.

2.3. Let ≺ be the lexicographical ordering of ℕ^{N}. We have the following

Theorem. Fix a reduced decomposition w̃_{0} of the longest element of the Weyl group. Let b, b′, b′′ be in B, Aw̃_{0}(b) = ψ, Aw̃_{0}(b′) = ψ′ Then, d_{b,b′}^{b′′} non zero implies Aw̃_{0}(b′′) ≺

ψ + ψ′. Moreover, d_{ψ,ψ′}^{ψ+ψ′} is a power of q.

Proof. First remark that a_{i}(uv) = a_{i}(u) + a_{i}(v), u, v in U_{q}(𝔫), by the quantized Leibniz rule. Write

(∗)  b^{∗}b^{′∗} = ∑ d_{b,b′}^{b′′}b′′^{∗}

Let ψ′′ = Aw̃_{0}(b′′), for b′′ in the sum. Then, ψ′′_{1} ≤ ψ_{1} + ψ′_{1} by Corollary 2.1 and [1.6, Theorem 1 (iii)]. This gives the first step of the induction. Now, applying L_{i_{N}}^{ψ_{1}+ψ′_{1}} and the Leibniz rule gives ξ_{1}Λ_{i_{N}} (b^{∗})Λ_{i_{N}} (b^{′∗}) = ∑ d_{b,b′}^{b′′}Λ_{i_{N}} (b^{′′∗}), where the sum is taken over the elements b′′ such that ψ′′_{1} = ψ_{1} +ψ′_{1}, and where ξ_{1} is a power of q. Now remark that, by [1.6, Theorem 1 (iii)], Λ_{i_{N}} (b^{∗}), Λ_{i_{N}} (b^{′∗}), and the Λ_{i_{N}} (b^{′′∗}) all belong to the dual of the canonical basis.So, we can proceed as for the first step by induction. We then have the first assertion of the theorem. For conclusion, note that we obtain

ξ_{N}Λ_{i_{1}} … Λ_{i_{N}} (b^{∗})Λ_{i_{1}} … Λ_{i_{N}} (b^{′∗}) = ∑ d_{b,b′}^{b′′}Λ_{i_{1}} … Λ_{i_{N}} (b^{′′∗}),

where the sum is taken over the elements b′′ such that ψ′′_{k} = ψ_{k} +ψ′_{k}, for all k and where ξ_{N} is a power of q. Hence, by Theorem 2.2 there is at most one element in the sum and for this element we have Aw̃_{0}(b′′) = ψ + ψ′. By [12, par 1],

Lemma. For all elements B of the canonical basis, one has: Λ_{i_{1}} … Λ_{i_{N}} (B^{∗}) = 1. ♢

Hence the coefficient d_{ψ,ψ′}^{ψ+ψ′} is a power of q. This finishes the proof of the theorem.

Remark. The Theorem and the Lemma appeared for type A in [1].

2.4. By 1.8, the results of 2.1, 2.2 can be easily generalized to quotients of ℂ_{q}[G/U] which correspond to Demazure modules. Indeed, fix w in W and let B\overline{w} be the complement of B_{w} in B. Set

I_{A,w} := ⨁_{λ∈P^{+}, b∈B\overline{w}∩B(λ)} Ab^{∗} ⊗ v_{λ} = ⨁_{λ} V_{A,w}(λ)^{⊥} ⊗ v_{λ}.

Then, I_{A,w} is the orthogonal < B_{w} >^{⊥} of < B_{w} > in A_{q}[G/U]. By Theorem 1.8, I_{A,w} is an ideal of A_{q}[G/U]. We have the following decomposition for the quotient algebra:

A_{q}[G/U]/I_{A,w} = ⨁_{λ} V_{A,w}(λ)^{∗} ⊗ v_{λ}.

Definition 1. Fix an element w in the Weyl group. Let w̃ = s_{i_{1}} … s_{i_{p}} be a reduced decomposition of w. Then there exists a reduced decomposition of w_{0} such that w̃_{0} = s_{i_{N}} … s_{i_{p}} … s_{i_{1}}. Such a reduced decomposition of w̃_{0} will be called adapted for w.

¿From [12, par 1], we have :

Theorem. Fix w in W and let w̃_{0} be a reduced decomposition of w_{0} which is adapted to w. Then, Aw̃_{0}(B_{w}) = C ∩ (ℕ^{p} × {0}^{N−p}). ♢

3. Specialization.

3.1. Fix a reduced decomposition w̃_{0} of w_{0}. At this stage of the article, we can construct a ℕ^{N}-filtration of the algebra ℂ[G/U] such that the associated graded algebra is the algebra of the semigroup Γw̃_{0}. To be more precise, let b_{λ,ψ}^{∗} in ℂ[G/U] be the image of the element (b_{λ}^{ψ})^{∗} ⊗ v_{λ} by the morphism of specialization at q = 1, see Proposition 2.1 (iv). We have by Theorem 2.3 and (2.1.1):

Proposition. The spaces F_{ψ} :=< b_{λ,ϕ}^{∗}, (λ, ϕ) ∈ Γw̃_{0}, ϕ ≺ ψ >, ψ ∈ C define a filtration of the algebra ℂ[G/U]. The graded associated algebra is naturally isomorphic to the ℂ-algebra of the semigroup Γw̃_{0}. ♢

3.2. What results from Proposition 3.1 is that there exists a finite sequence of degenerations of the flag variety which ends into a toric variety but what we want is a ”degeneration in one step”. Hence, we need a linear form ℕ^{N} → ℕ, which satisfies some strict inequalities, and which transforms the ℕ^{N}-filtration of ℂ[G/U] into a ℕ-filtration. This is made possible because the cone Γw̃_{0} is convex polyhedral and hence has a finite presentation. We start by a lemma.

Lemma. Let S be a finite set of points in ℕ^{N}. Then there exists a linear form e : ℕ^{N} → ℕ such that e(ϕ) < e(ψ) whenever ϕ ≺ ψ in S.

Proof. Let a_{1}, …, a_{N} be the coordinate linear forms on ℕ^{N}. Let C be an upper bound for all coordinates of all points in S. Choose positive integer ε_{1}, …, ε_{N} such that ε_{k} > C(ε_{k+1} + … + ε_{N}) for 1 ≤ k < N. Then the linear form e = ∑_{k} ε_{k}a_{k} obviously has the desired property. ♢

As a consequence of [2], see also [12], Γw̃_{0} is the set of integral points of a polyhedral convex cone. Hence, the semi-group Γw̃_{0} has a (unique) minimal set of generators and a finite presentation. Let (λ_{i}, ψ_{i}), 1 ≤ i ≤ K, be the minimal set of generators of Γw̃_{0} and (∑_{i} n_{i}^{k}(λ_{i}, ψ_{i}) − ∑_{i} m_{i}^{k}(λ_{i}, ψ_{i})), n_{i}^{k}, m_{i}^{k} ∈ ℕ, be the (finite) minimal set of generators of the relations. Then, from Proposition 3.1, we obtain:

Proposition. The (commutative) algebra ℂ[G/U] is defined by generators b_{λ_{i},ψ_{i}}^{∗}, 1 ≤ i ≤ K, and relations

(3.2.1)  ∏ b_{λ_{i},ψ_{i}}^{∗n_{i}^{k}} = ∏ b_{λ_{i},ψ_{i}}^{∗m_{i}^{k}} + ∑ c_{μ,ϕ}^{k}b_{μ,ϕ}^{∗}, with c_{μ,ϕ}^{k} ∈ ℂ

and c_{μ,ϕ}^{k} ≠ 0 ⇒ ϕ ≺ ∑_{i} n_{i}^{k}(λ_{i}, ψ_{i}) = ∑_{i} m_{i}^{k}(λ_{i}, psi_{i}).

Proof. The algebra ℂ[G/U] has a natural P^{+}-grading defined by ℂ[G/U]_{λ} = V (λ)^{∗}⊗v_{λ}. By construction, F_{ψ} = ⊕_{λ}F_{ψ,λ}, where F_{ψ,λ} = F_{ψ} ∩ ℂ[G/U]_{λ}. Clearly, F_{ψ,λ} is finite dimensional. Hence, by induction, we obtain that generators of the C-graded algebra of Proposition 3.1 lift to generators of the algebra ℂ[G/U]. Hence, we have the first part of the proposition.

Now, Proposition 3.1 implies the relations (3.2.1). Let ℂ[X_{i}, 1 ≤ i ≤ K] be the polynomial algebra with K indeterminates. There exists a surjective morphism φ : ℂ[X_{i}, 1 ≤ i ≤ K]/J → ℂ[G/U], where J is the ideal generated by the relations resulting from (3.2.1). This morphism maps X_{i} to b_{λ_{i},ψ_{i}}^{∗} for all i. Endow ℂ[X_{i}, 1 ≤ i ≤ K]/J with the quotient filtration < ∏ X_{i}^{m_{i}} + J, ∑ m_{i}ψ_{i} ≺ ψ >_{ψ∈C}. Then, the associated

graded algebra is defined by generators GrX_{i}, 1 ≤ i ≤ K, and relations ∏ Gr X_{i}^{n_{i}^{k}} =

∏ Gr X_{i}^{m_{i}^{k}}. Now, endow ℂ[G/U] with its C-filtration, see 3.1. By Proposition 3.1, Grφ is an isomorphism from Grℂ[X_{i}, 1 ≤ i ≤ K]/J onto Grℂ[G/U]. This implies that φ is an isomorphism. This finishes the proof of the proposition. ♢

Let ψ^{k} := ∑_{i} n_{i}^{k}ψ_{i}, and for each k, let the ϕ^{k,l} be the elements ϕ occuring in (3.2.1). Fix a ℕ-form e as in Lemma 3.2 associated to the finite set the ψ^{k} and ϕ^{k,l}. Then,

Corollary. The spaces F_{m}(ℂ[G/U]) =< ∏ b_{λ_{i},ψ_{i}}^{s_{i}^{k}}, e(∑_{i} s_{i}^{k}ψ_{i}) ≤ m >, m ∈ ℕ, define a filtration of ℂ[G/U]. The graded associated algebra is ℂ[Γw̃_{0}]. ♢

3.3. We now give some analogous results for Demazure modules. Fix w in W with length p. Set Γw ; w̃_{0} := Aw̃_{0}(B_{w}). Then,

Lemma. For all reduced decompositions w̃_{0} of w_{0}, the set Γw ; w̃_{0} is a finite union of rational convex polyhedral cones. Moreover, if w̃_{0} is adapted to w, see Definition 2.4, then Γw ; w̃_{0} is a p-dimensional face of Γw̃_{0}. In particular, it is a rational convex polyhedral cone.

Proof. Fix w in W and fix a reduced decomposition w̃_{0} adapted to w. By Theorem 2.4, Γw ; w̃_{0} is a p-dimensional face of Γw̃_{0} and so it is a convex polyhedral cone. Let w̃′_{0} be another reduced decomposition of w_{0} which is not supposed adapted to w. Then, by [2, 3.3], Γ_{w̃′_{0}}^{w} is the image of Γw ; w̃_{0} by a continuous piecewise linear map l. To be more precise, there exists a finite set of (convex) cones C_{i} in ℝ^{n+N} such that ⋃ C_{i} = ℝ_{≥0}^{n+N} and such that l is linear on Γw ; w̃_{0} ∩ C_{i}. Hence, we have the lemma. ♢

We still denote by v_{λ} a highest weight vector of the classical Weyl module V (λ). The algebra A_{q}[G/U] specializes for q = 1 onto the algebra ℂ[G/U] = ⨁_{λ} V (λ)^{∗} ⊗ v_{λ}. The algebra A_{q}[G/U]/I_{A,w} specializes for q = 1 onto the quotient algebra ℂ[G/U]/I_{w} = ⨁_{λ} V_{w}(λ)^{∗} ⊗ v_{λ}, where I_{w} := ⨁_{λ∈P^{+}} V_{w}(λ)^{⊥} ⊗ v_{λ}.

Let w̃_{0} be adapted to w. By the previous lemma, Γw ; w̃_{0} is generated as a semigroup by a part of the minimal set of generators (λ_{i}, ψ_{i}) of Γw̃_{0} and with the corresponding relations. This implies:

Theorem. Choose a reduced decomposition w̃_{0} adapted to w. Then, the graded associated algebra of the quotient filtration of (F_{m})_{m∈ℕ} on ℂ[G/U]/I_{w} is the ℂ-algebra ℂ[Γw ; w̃_{0}] of the semigroup Γw ; w̃_{0}. ♢

Remark. Note that this filtration (and hence the corresponding degeneration) depends on the choice of the ℕ-form e of Lemma 3.2, and of the reduced decomposition w̃_{0} adapted to w. For a general reduced decomposition w̃′_{0} of w_{0}, Lemma 3.3 implies that the spectrum of the associated graded algebra is a union of irreducible components which are toric varieties.

Acknowledgements

We wish to thank R. Yu for a general survey of the Degeneration Problem. This work owes much to helpful conversations with M. Brion and O. Mathieu. We are grateful to P. Baumann for true criticisms and to J. Germoni for encouragements and fan constructions.

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