Positivity of some intersections in K_{0}(G/B) * Olivier Mathieu

This paper is respectfully dedicated to David Buchsbaum.

0. Introduction: Start with a few notations. Let G be a simply connected connected simple algebraic group over C, let B be a Borel subgroup, let H be a Cartan subgroup of B, let P be the weight lattice, let P^{+} be the set of dominant weights and let W be the Weyl group of G. For λ ∈ P, we denote by C(λ) the corresponding one-dimensional B-module and by L(λ) the sheaf of sections of the line bundle C(λ) ×_{B} G → G/B. With these definitions, we recall that Γ(G/B, L(λ)) ≠ 0 if and only if −λ ∈ P^{+}. For any w ∈ W, let S_{w} = \overline{BwB}/B be the Schubert variety, let O_{w} be its structural sheaf, and set L_{w}(λ) = L(λ) ⊗ O_{w}. For any coherent sheaf M on G/B we denote by [M] its image in the Grothendieck ring K(G/B) := K_{0}(G/B) of the category of coherent sheaves on G/B.

It follows from the cell decomposition of G/B that ([O_{w}])_{w∈W} is a Z-basis of K(G/B). Therefore, for any v, w ∈ W, μ ∈ P, we can define the integer s_{w}^{v}(μ) by the following identity in K(G/B):

[L_{w}(−μ)] = ∑_{v∈W} s_{w}^{v}(μ)[O_{v}].

Theorem 0.1: For any w, v ∈ W and λ ∈ P^{+}, the integer s_{w}^{v}(λ) is ≥ 0.

To explain the title of the paper, note that [L_{w}(−μ)] is the intersection product of [L(−μ)] with S_{w} in K(G/B). This result has been proved by Fulton and Lascoux [FL94] for the group G = SL(n), and their paper was the motivation of the present work. We will show that Theorem 0.1 is a very simple corollary of the main result of [M_{2}89]. Indeed our proof is slighty more precise. Let K_{H}(G/B) be the Grothendieck ring of the category of the H-equivariant coherent sheaves on G/B and for any H-equivariant coherent sheaf M on G/B we denote by [M]_{H} its image in K_{H}(G/B). By [KK87], ([O_{w}]_{H})_{w∈W} is a Z[P]-basis of K_{H}(G/B) (see also [Ar86]), and therefore we can define the characters σ_{w}^{v}(μ) by the following identities in K_{H}(G/B):

[L_{w}(−μ)]_{H} = ∑_{v∈W} σ_{w}^{v}(μ)[O_{v}]_{H}.

We prove that for any dominant weight λ, the characters σ_{w}^{v}(λ) are effective, hence their degrees s_{w}^{v}(λ) are non-negative integers. Also in [M_{3}89], we proved that for any λ ∈ P^{+}, the sheaf O_{w} ⊗C(−λ) has a B-equivariant filtration whose any subquotient is some L_{v}(μ). At the end of Section 5, we will explain why this result is closely related with the effectivity of σ_{w}^{v}(λ).

Using the same ideas, it easy to deduce two other posivity results. These results hold in K_{H}(G/B), but for simplicity we will state them in K(G/B). For w ∈ W, set ∂S_{w} = ∪_{v<w} S_{v}, let I_{w} ⊂ O_{w} be the ideal defining ∂S_{w} in S_{w} and set I_{w}(λ) = L(λ) ⊗ I_{w}. It is clear that ([I_{w}])_{w∈W} is another basis of K(G/B). Therefore, we can define some integers k_{w}^{v}(μ) for v, w ∈ W, μ ∈ P by the following identities in K(G/B):

* Research supported by UA1 du CNRS at Strasbourg.

[I_{w}(−μ)] = ∑_{v∈W} k_{w}^{v}(μ)[I_{v}].

Theorem 0.2: For any w, v ∈ W and λ ∈ P^{+}, the integer k_{w}^{v}(λ) is ≥ 0.

Similarly we define the mixed numbers m_{w}^{v}(μ) by the following identity in K(G/B):

[I_{w}(−μ)] = ∑_{v∈W} m_{w}^{v}(μ)[O_{v}].

For λ ∈ P^{+}, denote by W_{λ} its stabilizer in W. Recall that each coset w.W_{λ} is an ordered set which is isomorphic to W_{λ}. Therefore each W_{λ}-coset contains a unique maximal and a unique minimal representative.

Theorem 0.3: Let w, v ∈ W and λ ∈ P^{+}. If w is minimal in w.W_{λ}, the integer m_{w}^{v}(λ) is ≥ 0.

Indeed Theorem 0.3 is a corollary of [M_{2}89] only when λ is strictly regular. However it follows from a refinement of a result of [M_{2}89], whose the proof is similar (a detailed proof will be given in a subsequent paper). Also we will also introduce some refined tensor product multiplicities in order to explain the relationship of these positivity results with the PRV conjecture (independently proved by Kumar [K88] and the author [M_{1}89]) and the refined PRV conjecture (due to Kumar [K89]). At the end of the paper, we show some identities between the numbers s_{w}^{v}(μ), k_{w}^{v}(μ), m_{w}^{v}(μ), from which we deduce a generalization of Pieri’s identity.

Aknowledgments: We heartily thank W. Fulton for its encouragemnts, M. Duflo for a useful discussion about harmonic functions and their relations to Demazure’s operators and P. Littelmann for discussion about his recent work [L98]. We also thank G. M. Cattaneo and E. Strickland for the organization of the meeting in University of Rome Tor Vergata, May 17-23 1998.

Main notations: Throughout the whole paper, we will use the notations of the introduction, together with the following ones. We will denote by Δ^{+} the set of positive roots, by Π the subset of simple positive roots. We set ρ = 1/2 ∑_{α∈Δ^{+}} α and for α ∈ Π we denote by s_{α} the corresponding simple reflection. We will use that W is an ordered set relative to the Bruhat ordering and w_{0} denotes its maximal element. For w ∈ W, let ϵ(w) be its signature.

1. Demazure operators. Denote by H the algebra End_{C(H)^{W}} (C(H)) where C(H) denotes the field of rational functions on H. By Galois theory, any X ∈ H can be uniquely written as X = ∑_{w∈W} ϕ_{w}w and also as X = ∑_{w∈W} w ψ_{w}, where ϕ_{w}, ψ_{w} ∈ C(H). For X ∈ H and ϕ ∈ C(H), it is important to distinguish the operator X ϕ ∈ H from the character X(ϕ) ∈ C(H): the first expression is the composition of X by the multiplication by ϕ, and the second one is the value of X at ϕ.

Lemma 1.1: There is a unique anti-involution X ↦ X^{∗} of the algebra H such that (i) ϕ^{∗} = ϕ, for any ϕ ∈ C(H). (ii) w^{∗} = ϵ(w)e^{ρ−w^{−1}ρ} w^{−1}.

Proof: For any ϕ, ψ ∈ C(H) and any v, w ∈ W, we have:

((ϕw) ∘ (ψv))^{∗} = (ϕψ^{w}wv)^{∗}

= ϵ(wv) e^{ρ−(wv)^{−1}ρ} (wv)^{−1}ϕψ^{w}

and we also have:

(ψv)^{∗} ∘ (ϕw)^{∗} = (ϵ(v) e^{ρ−v^{−1}ρ} v^{−1}ψ)(ϵ(w) e^{ρ−w^{−1}ρ} w^{−1}ϕ)

= ϵ(vw) e^{ρ−v^{−1}ρ} e^{v^{−1}ρ−v^{−1}w^{−1}ρ} v^{−1}w^{−1}ψ^{w}ϕ
= ϵ(vw) e^{ρ−v^{−1}w^{−1}ρ} v^{−1}w^{−1}ψ^{w}ϕ.

Hence (XY)^{∗} = Y^{∗}X^{∗} for any X, Y ∈ H, and ∗ is anti-morphism of algebras. Moreover w^{∗∗} = (ϵ(w)e^{ρ−w^{−1}ρ}w^{−1})^{∗} = e^{ρ−wρ} w e^{ρ−w^{−1}ρ} = w, therefore ∗ is involutive. Q.E.D.

For any α ∈ Π, set ∇_{s_{α}} = \frac{1}{1−e^{α}} (1−e^{α}s). For any reduced decomposition w = σ_{1} … σ_{n} of an element w ∈ W, set ∇_{w} = ∇_{σ_{1}} … ∇_{σ_{n}}. As it proved by Demazure [D74], the element ∇_{w} ∈ H is independent on a choice of a reduced decomposition for w. It is called the Demazure operator. Moreover we have (∇_{s_{α}})^{2} = ∇_{s_{α}}. Therefore, for any v, w ∈ W, we have ∇_{v}∇_{w} = ∇_{v∗w}, where the element v ∗ w ∈ W is defined by \overline{BvB}.\overline{BwB} = \overline{B(v ∗ w)B}.

Lemma 1.2: For any ϕ ∈ Z[P] and for any w ∈ W, we have

ϕ∇_{w} = ∑_{v∈W} ∇_{v} S_{w}^{v}(ϕ),

for some S_{w}^{v}(ϕ) ∈ Z[P].

Proof: It is clear that any X ∈ H can be uniquely written as X = ∑_{v∈W} ∇_{v}ϕ_{v}, with ϕ_{v} ∈ C(H). Therefore we have ϕ∇_{w} = ∑_{v∈W} ∇_{v} S_{w}^{v}(ϕ), for some S_{w}^{v}(ϕ) ∈ C(X). We only have to prove that S_{w}^{v}(ϕ) belongs to Z[P]. Let α ∈ Π. We have ϕ∇_{s_{α}} = \frac{ϕ}{1−e^{α}} (1 − e^{α} s_{α}), therefore we get:

ϕ∇_{s_{α}} = \frac{ϕ−ϕ^{sα}}{1−e^{α}} + ∇_{s_{α}}ϕ^{s_{α}}.

It is clear that \frac{ϕ−ϕ^{sα}}{1−e^{α}} belongs to Z[P]. Then one proves the lemma by induction over w. For w = 1, the lemma is obvious. For w ≠ 1, we set w = s_{α}u, where α ∈ Π and u < w. It follows that ϕ∇_{w} = \frac{ϕ−ϕ^{sα}}{1−e^{α}} ∇_{u} + ∇_{s_{α}}ϕ^{s_{α}}∇_{u}, therefore we get:

ϕ∇_{w} = ∑_{v∈W} ∇_{v}S_{u}^{v}(\frac{ϕ−ϕ^{sα}}{1−e^{α}}) + ∑_{v∈W} ∇_{s_{α}∗v}S_{u}^{v}(ϕ^{s_{α}}),

what proves the lemma. Q.E.D.

Denotes by H_{int} the subring of H generated by Z[P] and the Demazure operators (∇_{w})_{w∈W}. The following statement follows from Lemma 1.2:

Corollary 1.3: Any X ∈ H_{int} can be uniquely written as X = ∑_{w∈W} ϕ_{w}∇_{w} and as X = ∑_{w∈W} ∇_{w}ψ_{w}, where the elements ϕ_{w}, ψ_{w} belong to Z[P].

For any α ∈ Π, set ∇s_{α}^{′} = \frac{e^{α}}{1−e^{α}} (1 − s_{α}). For any reduced decomposition w = σ_{1} … σ_{n} of an element w ∈ W, set ∇w^{′} = ∇σ_{1}^{′} … ∇σ_{n}^{′}. The element ∇w^{′} ∈ H is also well-defined. It is called the modified Demazure operator. Moreover we have (∇s_{α}^{′})^{2} = −∇s_{α}^{′}. Therefore,

for any v, w ∈ W, we have ∇v^{′}∇w^{′} = ϵ(vw)ϵ(v ∗ w)∇v∗w^{′}.

Lemma 1.4: For any ϕ ∈ Z[P] and any w ∈ W, we have:

ϕ∇w^{′} = ∑_{v∈W} ∇v^{′}K_{w}^{v}(ϕ), and

ϕ∇w^{′} = ∑_{v∈W} ∇_{v}M_{w}^{v}(ϕ),

for some K_{w}^{v}(ϕ), M_{w}^{v}(ϕ) ∈ Z[P].

Proof: We have ∇_{s_{α}} = 1 + ∇s_{α}^{′}. By induction we get:

∇_{w} = ∑_{v≤w} ∇v^{′}, and ∇w^{′} = ϵ(w) ∑_{v≤w} ϵ(v)∇_{v}.

Therefore (∇w^{′})_{w∈W} is another left and right basis of the Z[P]-module H_{int} and the lemma follows. Q.E.D.

Lemma 1.5: We have (∇_{w})^{∗} = ∇_{w^{−1}} and (∇w^{′})^{∗} = ∇w^{−1}^{′} for any w ∈ W.

Proof: For any α ∈ Π, we have s_{α}^{∗} = −e^{α} s_{α} and (∇_{s_{α}})^{∗} = \frac{1}{1−e^{α}} + e^{α} s_{α} \frac{e^{α}}{1−e^{α}}

= \frac{1}{1−e^{α}} + e^{α} \frac{e^{−α}}{1−e^{−α}} s_{α}
= \frac{1}{1−e^{α}} − \frac{e^{α}}{1−e^{α}} s_{α}.

Therefore (∇_{s_{α}})^{∗} = ∇_{s_{α}}. Choose a reduced decomposition σ_{1} … σ_{n} of w. We get (∇_{w})^{∗} = (∇_{σ_{n}})^{∗} … (∇_{σ_{1}})^{∗} = ∇_{w^{−1}}. The proof for ∇w^{′} is similar. Q.E.D.

Corollary 1.6: For any ϕ ∈ Z[P] and any w ∈ W, we have:

∇_{w}ϕ = ∑_{v∈W} S_{w^{−1}}^{v^{−1}}(ϕ)∇_{v},

∇w^{′}ϕ = ∑_{v∈W} K_{w^{−1}}^{v^{−1}}(ϕ)∇v^{′},

∇w^{′}ϕ = ∑_{v∈W} M_{w^{−1}}^{v^{−1}}(ϕ)∇_{v}.

Proof: By Lemma 1.2 and 1.5, we have

∇_{w}ϕ = (ϕ∇_{w^{−1}})^{∗} = (∑_{v∈W} ∇_{v^{−1}}S_{w^{−1}}^{v^{−1}}(ϕ))^{∗} = ∑_{v∈W} S_{w^{−1}}^{v^{−1}}(ϕ)∇_{v},

what proves the first equality. The last two equalities are similar (use Lemma 1.4 instead of Lemma 1.2). Q.E.D.

2. Filtrations of B-modules. For any w ∈ W, λ ∈ P^{+}, set J_{w}(λ) = Γ(S_{w}, L_{w}(−λ)) and K_{w}(λ) = Γ(S_{w}, I_{w}(−λ)). It is easy to prove that the B-modules J_{w}(λ) are never 0, and J_{w}(λ) = J_{v}(μ) if and only if wλ = vμ. It is also easy to prove that K_{w}(λ) ≠ 0 if and only if w is minimal in W_{λ}. Moreover two non-zero B-modules K_{w}(λ) and K_{v}(μ) are isomorphic if and only if w = v and λ = μ. For a B-equivariant coherent sheaf M on G/B, we set χ_{B}(M) = ∑_{k≥0} (−1)^{k}ch H^{k}(G/B, M).

Theorem 2.1 (Demazure character formulas) Let λ ∈ P and w ∈ W. We have:

(i) χ_{B}(L_{w}(λ)) = ∇_{w}(e^{λ}). Moreover if λ is dominant, we have ch J_{w}(λ) = ∇_{w}(e^{−λ}). (ii) χ_{B}(I_{w}(λ)) = ∇w^{′}(e^{λ}). Moreover if λ is dominant, we have ch K_{w}(λ) = ∇′_{w}(e^{−λ}).

The reference for the theorem is [MR85][An85]. For any dominant weight λ, Demazure formulas follow from the vanishing of H^{k}(G/B, L_{w}(λ)) and H^{k}(G/B, I_{w}(λ)) for any k > 0. Formula 2.1(i) first appears in [D74]. Unfortunately, V. Kac found a serious gap in the Demazure’s beautiful paper [D74]. This gap has been filled by the introduction of Frobenius splittings in [MR85] and by slighty different method in [A85].

We say that a B-module M has a Joseph filtration (respectively: a van der Kallen filtration) if it admits a filtration whose any subquotient is some J_{w}(λ) (respectively: some K_{w}(λ)).

Theorem 2.2 Let λ, μ ∈ P^{+} and w ∈ W. Then:

(i) J_{w}(μ) ⊗ C(−λ) has a Joseph filtration and

(ii) K_{w}(μ) ⊗ C(−λ) has a van der Kallen filtration.

The reference for this theorem is [M_{2}89]. The Joseph filtrations has been first considered in [J85] and Joseph conjectured Theorem 2.2 (i) for w = w_{0}. The first step of a Joseph filtration of J_{w}(μ) ⊗ C(−λ) is J_{u}(ν), where λ + wμ = vν: this fact, which is a crucial ingredient in the proof of PRV conjecture independently proved by S. Kumar and the author in [K88][M_{2}89], is an easy consequence of Joseph’s paper. Then for G = SL(n), Theorem 2.1 (i) has been first proved by Polo [P89] by an had hoc method (a refinement of Wang’s trick [W82]) based on the fact that any fundamental weight is minuscule. The van der Kallen filtrations has been introduced in [vdK89] and van der Kallen proved the remarkable cohomological criterion 7.1 for the existence of a Joseph filtration or a van der Kallen filtration. It follows that Assertions (i) and (ii) are equivalent. For an arbitrary group G, Theorem 2.1 was proved by the author [M_{2}89] and the proof is strongly based on van der Kallen criterion [vdK89] and the Frobenius splittings invented by Metha, Ramanan and Ramanathan (it was stated as an open question in [P89]). See also later works [M_{3}89][M90] (extension to finite characteristics), [P93] (a more elementary proof for some cases) and [vdK93] (a very nice introduction to the subject).

Theorem 2.3: Let w ∈ W, λ, μ ∈ P^{+}. Assume that w^{−1} is minimal in w^{−1}.W_{λ}. Then K_{w}(μ) ⊗ C(−λ) has a Joseph filtration.

The references for this theorem is as follows: it is proved in [M_{2}89] under the condition λ ∈ ρ + P^{+}. The proof of this refined version is similar, and the details will be given in a subsequent paper.

Remark: As a matter of terminology, the Joseph filtrations has been previously called strong or excellent, and the van der Kallen filtration has been called weak or relative Schubert. The terminology ”excellent filtrations” conflicts with the well-established terminology ”good filtrations” (not defined in the present paper). For any union of Schubert varieties S and any dominant weight λ, set J_{S}(λ) = Γ(S, L(−λ) ⊗ O_{S}). Following [P89], the modules J_{S}(λ) are called the Schubert modules, and we call a Polo filtration of a Bmodule any filtration whose any subquotient is a Schubert module. Any Joseph filtration is obviously a Polo filtration. Moreover any Polo filtration can be refined to a van der Kallen filtration (what justified the strong/weak terminlogy). For any two unions of Schubert varieties S_{1}, S_{2}, we have ch J_{S_{1}}(λ)+ch J_{S_{2}}(λ) = ch J_{S_{1}∪S_{2}}(λ)+ch J_{S_{1}∩S_{2}}(λ), therefore the multiplicity of a Schubert module in a Polo filtration depends on the filtration. However in [L98] there is a combinatorial description of the multiplicities of Schubert modules in a Polo filtration of J_{w}(μ) ⊗ C(−λ).

3. The characters S, K and M are effective. Let μ ∈ P, w, v ∈ W. Recall from Section 1 that we define the characters S_{w}^{v}(e^{μ}), K_{w}^{v}(e^{μ}), M_{w}^{v}(e^{μ}) ∈ Z[P] by the following equalities in H:

e^{μ}∇_{w} = ∑_{v∈W} ∇_{v}S_{w}^{v}(e^{μ}),

e^{μ}∇′_{w} = ∑_{v∈W} ∇′_{v}K_{w}^{v}(e^{μ}),
e^{μ}∇′_{w} = ∑_{v∈W} ∇_{v}M_{w}^{v}(e^{μ}).

Denote by N the set of non-negative integers.

Theorem 3.1: Let λ ∈ P^{+}, w, v ∈ W.
(i) S_{w}^{v}(e^{−λ}) and K_{w}^{v}(e^{−λ}) belongs to N[P].
(ii) If w^{−1} is minimal in w^{−1}.W_{λ}, then M_{w}^{v}(e^{−λ}) belongs to N[P].

Proof: Fix w ∈ W and λ ∈ P^{+}. Set S_{w}^{v}(e^{−λ}) = ∑_{ξ∈P} m_{v}^{ξ} e^{−ξ} and Ω = {ξ ∈ P |m_{v}^{ξ} ≠ 0 for some v}. Let μ ∈ P^{+} and assume that μ is far away from the walls. Then μ−ξ ∈ P^{++} for any weight ξ ∈ Ω, where P^{++} = ρ + P^{+}. By Demazure formula 2.1, we get:

e^{−λ}ch J_{w}(μ) = ∑_{v∈W} ∑_{ξ∈Ω} m_{v}^{ξ}ch J_{v}(μ + ξ).

Up to repetitions, the family of characters (ch J_{x}(ν))_{x∈W ν∈P^{+}} form a basis of Z[P]. Therefore the characters (ch J_{x}(ν))_{x∈W ν∈P^{++}} are linearly independent. By Theorem 2.2(i), J_{w}(λ) ⊗ C(−μ) has a Joseph filtration. Hence m_{v}^{ξ} is the multiplicity of J_{v}(μ + ξ) in a Joseph filtration of J_{w}(μ) ⊗ C(−λ), for any ξ ∈ Ω, v ∈ W, when μ ∈ P^{+} is far away from the walls. Therefore m_{v}^{ξ} ≥ 0 and S_{w}^{v}(e^{−λ}) belongs to N[P].

The proof for K_{w}^{v}(e^{−μ}) and M_{w}^{v}(e^{−μ}) is similar: use Theorem 2.2(ii) and Theorem 2.3 instead of Theorem 2.2(i).

4. Harmonic functions on 𝔥^{∗}. Let 𝔥 be the Lie algebra of H. Identify S 𝔥^{∗} with the space of translation-invariant differential operators on 𝔥^{∗}, and denote by S^{+} 𝔥^{∗} its maximal homogenous ideal (therefore the elements of S^{+} 𝔥^{∗} are differential operators without constant term). A polynomial function F on 𝔥^{∗} is called harmonic if and only if θF = 0 for any θ ∈ (S^{+} 𝔥^{∗})^{W}. We say that a complex valued function f defined on P or on P^{+} is harmonic if it is the restriction of a harmonic polynomial function F on 𝔥^{∗} (as P and P^{+} are Zariski dense in 𝔥^{∗}, F is uniquely determined by f). Denote by Har the space of harmonic functions on P. As S 𝔥 is a free (S 𝔥)^{W} -module, Har is a regular representation of W and its dimension is Card W. For any coherent sheaf M on G/B, denote by χ(M) = ∑_{k≥0} (−1)^{k}dimH^{k}(G/B, M) its Euler characteristic.

Theorem: 4.1

(i) For any coherent sheaf M on G/B, the map λ ∈ P ↦ χ(M⊗L(−λ)) is harmonic. (ii) The map C ⊗ K_{0}(G/B) → Har, [M] ↦ χ(M ⊗ L(−λ)) is an isomorphism.

The reference for this statement is [J85].

For any w ∈ W and λ ∈ P, set j_{w}(λ) = χ(L_{w}(−λ)) and k_{w}(λ) = χ(I_{w}(−λ)). It follows from Theorem 4.1 that (j_{w})_{w∈W} and (k_{w})_{w∈W} are two bases of Har. Therefore we have:

Corollary 4.2: Let w ∈ W and λ, μ ∈ P. We have

(i) j_{w}(λ + μ) = ∑_{v∈W} s_{w}^{v}(λ)j_{v}(μ), (ii) k_{w}(λ + μ) = ∑_{v∈W} k_{w}^{v}(λ)k_{v}(μ), (ii) k_{w}(λ + μ) = ∑_{v∈W} m_{w}^{v}(λ)j_{v}(μ).

Proof: The corollary follows from the previous theorem, Demazure character formulas 2.1 and the definitions of the numbers s_{w}^{v}(λ), k_{w}^{v}(λ) and m_{w}^{v}(λ). Q.E.D.

5. Proofs of the Theorems 0.1, 0.2, and 0.3 and a filtration of O_{w} ⊗ C(−λ). The degree of a character ϕ = ∑_{ξ∈P} n^{ξ}e^{ξ} ∈ Z[P] is the integer deg ϕ = ∑_{ξ∈P} n^{ξ}.

Lemma 5.1: For any w, v ∈ W and λ ∈ P, we have s_{w}^{v}(λ) = deg S_{w^{−1}}^{v^{−1}}(e^{−λ}), k_{w}^{v}(λ) = deg K_{w^{−1}}^{v^{−1}}(e^{−λ}), and m_{w}^{v}(λ) = deg M_{w^{−1}}^{v^{−1}}(e^{−λ}).

Proof: By Corollary 1.6, we have ∇_{w}e^{−λ} = ∑_{v∈W} S_{w^{−1}}^{v^{−1}}(e^{−λ})∇_{v}. Therefore we get for any μ ∈ P

j_{w}(λ + μ) = χ(L_{w}(−λ − μ))

= deg ∇_{w}(e^{−λ}e^{−μ})
= deg [∑_{v∈W} S_{w^{−1}}^{v^{−1}}(e^{−λ})∇_{v}(e^{−μ})]
= ∑_{v∈W} deg S_{w^{−1}}^{v^{−1}}(e^{−λ}) j_{v}(μ).

Therefore the equality s_{w}^{v}(λ) = deg S_{w^{−1}}^{v^{−1}}(e^{−λ}) follows from Corollary 4.2 and the fact that (j_{w})_{w∈W} is a basis of Har. The proof of the other two equalities is similar. Q.E.D.

Proofs of Theorems 0.1, 0.2, and 0.3. Under the hypotheses of the theorems (i.e. λ ∈ P^{+} for the first two, and λ ∈ P^{+} and w is minimal in w.W_{λ} for the third one) the characters S_{w^{−1}}^{v^{−1}}(e^{−λ}), K_{w^{−1}}^{v^{−1}}(e^{−λ}), and M_{w^{−1}}^{v^{−1}}(e^{−λ}) are effective by Theorem 3.1. Therefore their degrees are ≥ 0, and the theorems follows from Lemma 5.1. Q.E.D.

Relation with [M_{3}89]: By [KK87], K_{H}(G/B) is a free Z[P]-module with basis ([O_{w}]_{H})_{w∈W}. Theorem 0.1 can be reinforced as follows: in K_{H}(G/B), we have, [L_{w}(−λ)]_{H} = ∑_{v∈W} S_{w^{−1}}^{v^{−1}}(e^{−λ})[O_{v}]_{H} and for any λ ∈ P^{+} the character S_{w^{−1}}^{v^{−1}}(e^{−λ}) is effective. Note that ([O_{w} ⊗ C(μ)]_{H})_{w∈W,μ∈P} is a Z-basis of K_{H}(G/B). However there is another natural Z-basis, namely ([L_{w}(μ)])_{w∈W,μ∈P}. Define the integers n_{w}^{v,ξ}(μ) by the following identity in K_{H}(G/B):

[O_{w} ⊗ C(μ)]_{H} = ∑_{v∈W,ξ∈P} n_{w}^{v,ξ}(μ)[L_{v}(ξ)]_{H}.

In [M_{3}89], we proved that for any λ ∈ P^{+}, O_{w} ⊗ C(−λ) has a B-equivariant filtration whose any subquotient is some L_{v}(ξ). Therefore the integers n_{w}^{v,ξ}(λ) are non-negative for any λ ∈ P^{+}. Indeed it is easy to see that S_{w}^{v}(e^{−λ}) = ∑_{ξ∈P} n_{w}^{v,ξ}(λ) e^{ξ}.

6. Some refined tensor product multiplicities. For any λ ∈ P^{+}, denote by L(λ) the simple G-module with lowest weight −λ. For λ, μ, ν, define the tensor product multiplicities K_{λμ}^{ν} by the identity

L(λ) ⊗ L(μ) = ⊕_{ν∈P^{+}} L(ν)^{K_{λμ}^{ν}}.

The tensor product multiplicities K_{λμ}^{ν} have been explicitely computed by Steinberg [S61]. We will now define some refined tensor multiplicities K_{λμ}^{ν}(w, v) indexed by the additional indices w, v ∈ W by the following requirements:

(i) Assume that w^{−1} is minimal in w^{−1}W_{μ} and v is maximal in v.W_{ν}. Then by Theorem 2.3, K_{w}(λ) ⊗ C(−μ) has a Joseph filtration and by definition K_{λμ}^{ν}(w, v) is the multiplicity of J_{v}(ν) in a Joseph filtration of K_{w}(λ) ⊗ C(−μ).

(ii) Otherwise, set K_{λμ}^{ν}(w, v) = 0. By definition the integers K_{λμ}^{ν}(w, v) are non-negative.

Theorem 6.1: For any λ, μ, ν ∈ P^{+}, we have

K_{λμ}^{ν} = ∑_{v,w∈W} K_{λμ}^{ν}(w, v).

Proof: We have ch L(μ) = ∇(e^{−λ}), where ∇ = ∇_{w_{0}}. Therefore ch L(λ) ⊗ L(μ) = ∇(e^{−μ}ch L(λ)). Moreover ch L(λ) = ∑_{w∈W} ch K_{w}(λ). Hence we get:

ch L(λ) ⊗ L(μ) = ∑_{w∈W} ∇(e^{−μ}ch K_{w}(λ)).

If w in not minimal in w.W_{λ}, then K_{w}(λ) = 0. Therefore the in previous sum, we let w runs over the minimal representatives of W/W_{λ}. Fix such a w. If w^{−1} is not minimal in w^{−1}.W_{μ}, there is a root α ∈ Π such that μ(h_{α}) = 0 and s_{α}w ≤ w, therefore ∇_{s_{α}}(e^{−μ}ch K_{w}(λ)) = e^{−μ}∇_{s_{α}}(ch K_{w}(λ)) = e^{−μ}∇_{s_{α}}∇s_{α}^{′}(ch K_{s_{α}w}(λ)) = 0, and therefore ∇(e^{−μ}ch K_{w}(λ)) = 0. Otherwise w^{−1} is minimal in w^{−1}.W_{μ}, and by definition we have:

e^{−μ}ch K_{w}(λ) = ∑_{v∈W} K_{λμ}^{ν}(w, v)chJ_{v}(ν).

Therefore, using the identity ∇(ch J_{v}(ν)) = ch L(ν), we get:

ch L(λ) ⊗ L(μ) = ∑_{ν∈P^{+}v,w∈W} K_{λμ}^{ν}(w, v) ch L(ν),

from which we get the required equality. Q.E.D.

Lemma 6.2: Let λ, μ, ν ∈ P^{+}. Let w, v ∈ W such that w is minimal in w.W_{λ}, w^{−1} is minimal in w^{−1}.W_{μ}, v is maximal in v.W_{ν}, and μ + wλ = vν. Then K_{λμ}^{ν}(w, v) = 1.

Proof: By Theorem 2.3, the B-module K_{w}(λ) ⊗ C(−μ) has a Joseph filtration. However, H^{0}(U, K_{w}(λ) ⊗ C(−μ)) is a one dimensional H-module of weight −μ − wλ, where U is the unipotent radical of B. Hence the first step of a Joseph filtration of K_{w}(λ) ⊗ C(−μ) is J_{v}(ν), and J_{v}(ν) does not occur further. Therefore K_{λμ}^{ν}(w, v) equals one. Q.E.D.

Corollary 6.3: (refined PRV conjecture) Let λ, μ, ν ∈ P^{+}. Let X be the set of all w, ∈ W such that μ + wλ ∈ W.ν. Then K_{λμ}^{ν} ≥ W_{μ}\X/W_{λ}.

Proof: It is clear that X is a union of double W_{μ} × W_{λ}-cosets. Each such double coset contains a minimal representative w, and therefore w is minimal in w.W_{λ} and w^{−1} is minimal in w^{−1}.W_{μ}. For such a w choose v ∈ W such that μ + wλ = vν and v is maximal in v.W_{ν}. By Lemma 6.2, K_{λμ}^{ν}(w, v) = 1. Therefore by Theorem 6.1, K_{λμ}^{ν} ≥ W_{μ}\X/W_{λ}. Q.E.D.

7. A generalization of Pieri’s identity. For w ∈ W set \overline{w} = ww_{0}, for λ ∈ P set \overline{λ} = −w_{0}λ and denote by ϕ ∈ Z[P] ↦ \overline{ϕ} ∈ Z[P] the involution sending e^{μ} to e\overline{μ}. The following two theorems are due to van der Kallen [vdK89].

Theorem 7.1: Let M be a B-module.

(i) M has a Joseph filtration if and only if H^{k}(B, M ⊗ K_{w}(λ)) = 0 for any k > 0, λ ∈ P^{+} and w ∈ W,

(ii) M has a van der Kallen filtration if and only if H^{k}(B, M ⊗ J_{w}(λ)) = 0 for any k > 0, λ ∈ P^{+} and w ∈ W.

Theorem 7.2: As a B × B-module, C[B] has a filtration whose the subquotients are J_{w}(λ) ⊗ K\overline{w}(\overline{λ}) and each of them occurs exactly once.

Corollary 7.3: Let M be a B-module, let λ ∈ P^{+}, v, w ∈ W. Assume that v is minimal in v.W_{λ} and w is maximal in w.W_{λ}.

(i) If M has a Joseph filtration, then the multiplicity of J_{w}(λ) in a Joseph filtration of M is dim H^{0}(B, M ⊗ K\overline{w}(\overline{λ})).

(ii) If M has a van der Kallen filtration, then the multiplicity of K_{v}(λ) in a van der Kallen filtration of M is dim H^{0}(B, M ⊗ J\overline{v}(\overline{λ})).

Proof: We have M ≃ H^{0}(B, C[B] ⊗ M). Therefore the filtration of C[B] defined in Theorem 7.1 induces a filtration of M. Assume that M has a Joseph filtration. By Theorem 7.1, the subquotient of the filtration are J_{w}(λ) ⊗ H^{0}(B, K\overline{w}(\overline{λ}) ⊗ M). Therefore Assertion (i) is proved. The second assertion is similar. Q.E.D.

Denote by τ : Z[P] → Z[P] the involution sending e^{μ} to e^{w_{0}μ}.

Lemma 7.4: For any v, w ∈ W, ϕ ∈ Z[P], we have:

(i) S_{w}^{v}(ϕ) = τ K\overline{w} ; \overline{v}(\overline{ϕ}),

(ii) M_{w}^{v}(ϕ) = τM\overline{w} ; \overline{v}(\overline{ϕ}),

(iii) K_{λμ}^{ν}(w, v) = K\overline{λ} ; \overline{ν}μ(\overline{v}, \overline{w}).

Proof: Let λ be any dominant weight and set S_{w}^{v}(e^{−λ}) = ∑_{ξ∈P} n_{ξ} e^{−ξ}, K\overline{v} ; \overline{w}(e^{−\overline{λ}}) = ∑_{ξ∈P} nξ^{′} e^{−ξ}. Fix ξ ∈ P and choose a dominant weight μ which is far away from the walls. It follows from the proof of Theorem 3.1 that n_{ξ} is the mutiplicity of J_{v}(μ + ξ) in a Joseph filtration of J_{w}(μ) ⊗ C(−λ). By Corollary 7.3, we get n_{ξ} = dim H^{0}(B, J_{w}(μ) ⊗ K\overline{w}(\overline{μ} + \overline{ξ}) ⊗ C(−λ)). As μ is an arbitrary dominant weight far away from the walls, we also get n_{ξ} = dim H^{0}(B, J_{w}(μ − \overline{ξ}) ⊗ K\overline{w}(\overline{μ}) ⊗ C(−λ)). Using again Corollary 7.3, we get n_{ξ} = n−\overline{ξ}^{′}, what proves the first assertion for ϕ = e^{−λ}. However, the maps ϕ ↦ S_{w}^{v}(ϕ) and ϕ ↦ τ K\overline{w} ; \overline{v}(\overline{ϕ}) are element of the algebra H. Therefore they are uniquely determined on by their values on e^{−λ} for λ ∈ P^{+}, and Assertion (i) follows. The proof of the other two assertions is similar. Q.E.D.

For α ∈ Π, set Δ_{s_{α}} = \frac{1}{1−e^{−α}} (1−e^{−α} s_{α}) and Δs_{α}^{′} = \frac{e^{−α}}{1−e^{−α}} (1−s_{α}). For any reduced decomposition σ_{1} … σ_{n} of an element w ∈ W, set Δ_{w} = Δ_{σ_{1}} … Δ_{σ_{n}} and Δw^{′} = Δσ_{1}^{′} … Δσ_{n}^{′}. These operators do not depend on the reduced decomposition of w. Recall that for any λ ∈ P^{+}, the highest weight of L(λ) is \overline{λ}. For any w ∈ W, let J_{w}^{∗}(λ) be the B-submodule of L(λ) generated by a vector v_{w} of weight w\overline{λ}, and set K_{w}^{∗}(λ) = J_{w}^{∗}(λ)/ ∑_{v<w} J_{v}^{∗}(λ). By Demazure character formula 2.1, we have

ch J_{w}^{∗}(λ) = Δ_{w}(e\overline{λ}),

ch K_{w}^{∗}(λ) = Δw^{′}(e\overline{λ}).

Lemma 7.5: For any w ∈ W, we have S_{w_{0}}^{w^{−1}}(e^{−λ}) = Δ\overline{w}^{′}(e\overline{λ}).

Proof: By Corollay 1.6, we have ∇w^{′}ϕ = ∑_{v∈W} K_{w^{−1}}^{v^{−1}}(ϕ) ∇v^{′}. However ∇v^{′}(1) = 0 for any v ≠ 1. Hence we get ∇w^{′}(ϕ) = K_{w^{−1}}^{1}(ϕ). Therefore the lemma follows from Lemma 7.4. Q.E.D.

Corollary 7.6: Let λ ∈ P^{+}. In K(G/B), we have

[L(−λ)] = ∑_{w∈W} k\overline{w}(\overline{λ})[O_{w}].

The corollary follows from Corollary 4.2 and the Lemma 7.5. It provides a generalisation for any group of the classical Pieri formula. It should be noted that the Littelmann’ path model provides a nice combinatorial formula for k_{w}(λ). More precisely, in L94] Littelmann defined the Lakshmibai-Sheshadri path of shape λ: they are indexed by a pair (a, τ), with a is an increasing sequence of rational numbers a : 0 = a_{0} < a_{1} < … < a_{r} = 1 and a decreasing sequence of W_{λ}-cosets τ : τ_{1} > … > τ_{r} = 1 together with an integrality condition on a (this condition depends on λ). He showed that j_{w}(λ) is the number of Lakshmibai-Sheshadri paths (a, τ) with τ_{1} ≤ w.W_{λ}. As j_{w}(λ) = ∑_{v≤w} k_{v}(λ), it is easy to deduce that k_{w}(λ) is the number of Lakshmibai-Sheshadi paths (a, τ) with τ_{1} = w.W_{λ}, whenever w is minimal in w.W_{λ} (otherwise k_{w}(λ) = 0).

Using Lemma 7.5, we also get the following identity:

Corollary 7.7: Let λ, μ ∈ P^{+}. We have

ch L(λ + μ) = ∑_{w∈W} ch J_{w}(λ) ⊗ K∗ ; \overline{w}(μ).

Bibiography:

[An85] H. H. Andersen: Schubert varieties and Demazure’s character formula. Invent. Math. 79 (1985) 611-618.

[A86] A. Arabia: Cycles de Schubert et cohomologie équivariante de K/T. Invent. Math. 85 (1986) 39-52.

[D74] M. Demazure: Désingularisation des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. 7 (1974) 53-88.

[FL94] W. Fulton and A. Lascoux: A Pieri formula in the Grothendieck ring of a flag bundle. Duke Math. J. 76 (1994) 711-729.

[J85] A. Joseph: On the Demazure character formula. Ann. Sci. École Norm. Sup. 18 (1985) 389-419.

[KK87] B. Kostant and S. Kumar: T -equivariant K-theory of generalized flag varieties. Proc. Nat. Acad. Sci. U.S.A. 84 (1987) 4351-4354.

[K88] S. Kumar: Proof of the Parthasaraty-Ranga-Rao-Varadarajan conjecture. Invent. Math. 93 (1988) 117-130.

[K89] S. Kumar: A refinement of the PRV conjecture. Invent. Math. 97 (1989) 305-311.

[L94] P. Littelmann: A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras. Invent. Math. 116 (1994), no. 1-3, 329-346.

[L98] P. Littelmann: The path model, the quantum frobenius map and standard monomial theory. Algebraic groups and their representations, Nato ASI series C 157 (1998) 175-212.

[M_{1}89] O.Mathieu: Construction d’un groupe de Kac-Moody et applications. Compositio Math. 69 (1989) 37-60.

[M_{2}89] O. Mathieu: Filtrations of B-modules. Duke Math. J. 59 (1989) 421-442. [M_{3}89] O. Mathieu: Frobenius action on the B-cohomology. Infinite-dimensional Lie

algebras and groups (Luminy-Marseille). World Sci. Publishing, Adv. Ser. Math. Phys. 7 (1989) 39-51.

[M90] O. Mathieu: Filtrations of G-modules. Ann. Sci. Ècole Norm. Sup. 23 (1990) 625-644.

[MR85] V. Mehta and A. Ramanathan: Frobenius splitting and cohomology vanishing for Schubert varieties. Ann. of Math. 122 (1985) 27-40.

[P89] P. Polo: Variétés de Schubert et excellentes filtrations. Orbites unipotentes et représentations, III. Astérisque 173-174 (1989) 281-311.

[P93] P. Polo: Modules associés aux variétés de Schubert. Indo-French Conference on Geometry. Hindustan Book Agency (1993) 155-171.

[S61] R. Steinberg: A general Clebsch-Gordan theorem. Bull. Amer. Math. Soc. 67 (1961) 406-407.

[vdK89] W. van der Kallen: Longest weight vectors and excellent filtrations. Math. Z. 201 (1989) 19-31

[vdK93] W. van der Kallen: Lectures on Frobenius splittings and B-modules. Springer-Verlag, Tata Institute Lecture Notes (1993).

[W82] Wang Jian-Pian: Sheaf cohomology on G/B and tensor products of Weyl modules. J. of Algebra 77 (1982) 162-185.

Author Adresses: Université Louis Pasteur, IRMA, 7 rue René Deascarte, 67000 Strasbourg, FRANCE. email: mathieu@math.u-strasbg.fr Université Paris 7, 2 place Jussieu, 75005 Paris, FRANCE.