ON THE RELATION BETWEEN INTERSECTION COHOMOLOGY AND REPRESENTATION THEORY IN

POSITIVE CHARACTERISTIC

WOLFGANG SOERGEL

Abstract. In this article we investigate the relation between the rational representation theory of semisimple algebraic groups over a field k of finite characteristic p and intersection cohomology with coefficients in k of (complex) Schubert varieties.

Contents

1. Introduction 2 2. Arguments from representation theory 4 2.1. Description of some endomorphism rings 4 2.2. Quotient categories 5 2.3. The regular subquotient category 6 2.4. The semiregular subquotient categories 7 2.5. Translation 8 2.6. Homomorphisms between projective objects 9 2.7. Decomposition with and without grading 11 2.8. Definition of the modules D_{x} 11 2.9. Properties of the modules D_{x} 12 2.10. Grothendieck groups 14 2.11. Proof of the first theorem from the introduction 15 3. Arguments from topology 16 3.1. Notations 16 3.2. The category K 17 3.3. Perverse sheaves on G/B and G/P_{s} 21 4. Topology and representation theory 23 4.1. Preliminary remarks 23 4.2. Equivalence of K and C 24 References 26

partially supported by the EC TMR network “Algebraic Lie Representations”, EC-contract no ERB FMRX-CT97-0100.

1. Introduction

Let k be a field of positive characteristic p and let G ⊃ B ⊃ T be a semisimple split simply connected algebraic group over k with a Borel and a maximal torus. A weight on the torus λ ∈ X(T) leads to an induced representation ind_{B}^{G}k_{λ} = H^{0}(G/B, O(λ)) = H^{0}(λ). If this induced representation is nonzero, it admits a unique simple subrepresentation L(λ) ⊂ H^{0}(λ), and every simple G-module is isomorphic to precisely one such L(λ), see e.g. [Jan87].

We would like to compute the multiplicities [H^{0}(λ) : L(μ)] ∈ ℕ of L(μ) occuring in a composition series of H^{0}(λ) for weights λ, μ such that H^{0}(λ) ≠ 0 ≠ H^{0}(μ). For these multiplicities Lusztig [Lus80] has put forward a conjecture. It is proven up to now [AJS94] that these conjectures are valid for every given root system in sufficiently high characteristic. If however the root system is none of A_{1}, A_{2}, A_{3}, B_{2}, G_{2}, one doesn’t know for a single characteristic whether it is sufficiently high.

The goal of this article is to forward this problem to the topologists or geometers. Certainly we know that intersection cohomology of (complex) Schubert varieties with coefficients in a field of sufficiently high charcteristic behaves as if the characteristic would be zero. But again it is not known for any characteristic whether it actually is sufficiently high. We want to explain how a solution of this topologico-geometrical problem would at least solve part of our representation theoretic problem.

To state our results more precisely, we need more notations. Let X(T) ⊃ R ⊃ R^{+} be the weight lattice, the root system R = R(G, T), and the system of positive roots R^{+} = −R(B, T). In particular we have H^{0}(λ) ≠ 0 if and only if λ is dominant. Let h denote the Coxeter number of our root system. Let ρ ∈ X(T) be the half sum of positive roots and (W, S) the Coxeter system of G ⊃ B. In particular W is the (finite) Weyl group.

Let on the other hand G ⊃ B be a complex semisimple algebraic group corresponding to the dual root system R^{∨} with a Borel, and let us identify its Coxeter system with (W, S) in the obvious way. Let st = (p − 1)ρ ∈ X(T) denote the “Steinberg weight”. In the following we equip complex varieties always with their natural “metric” topology, never with the Zariski-topology. In section 2.11 we will show:

Theorem 1.1. Let the characteristic of our field k be bigger than the Coxeter number of our root system. For fixed y ∈ W the following are equivalent:

… INTERSECTION COHOMOLOGY AND REPRESENTATION THEORY … 3

1. [H^{0}(st + xρ) : L(st + yρ)] = 1 for all x ∈ W such that x ≤ y in the Bruhat order.

2. The cohomology ring H^{•}(\overline{ByB/B}; k) with coefficients in k of the complex Schubert variety \overline{ByB/B} satisfies Poincaré duality.

Remarks 1.0.1. Let us call a (complex) variety X “k-smooth” if and only if H^{•}(X; k) satisfies Poincaré duality. For root systems of type ADE every ℚ-smooth Schubert variety is actually smooth by unpublished results of Peterson [Car94] and hence is k-smooth for all k. In general this is not true. It would be interesting to know wether every ℚ-smooth Schubert variety is at least k-smooth for all k with chark > Coxeter number.

Consider now the bounded below derived category D^{+}Sh_{k}(G/B) of the category Sh_{k}(G/B) of all sheaves of k-vector spaces on the complex variety G/B. Let K ⊂ D^{+}Sh_{k}(G/B) denote the full subcategory consisting of all objects of the form Rπ_{∗}(k_{BS}) along with their direct sums and summands and their [ ]-translates, where k_{BS} denotes the constant sheaf k on a Bott-Samelson variety BS and π : BS → G/B is the projection. Put into formulas we take all varieties BS of the form

BS = P_{s} ×^{B} P_{t} ×^{B} … ×^{B} P_{r} ×^{B} B/B

for any sequence s, t, …, r ∈ S of simple reflections with corresponding minimal parabolics B ⊂ P_{s}, P_{t}, …, P_{r} ⊂ G.

Theorem 1.2. Suppose p > h. Then we have:

1. For every y ∈ W there exists (up to isomorphism) a unique indecomposable object L_{y} ∈ K such that L_{y}[−l(y)]|_{ByB/B} is the constant sheaf k on the Bruhat cell ByB/B and suppL_{y} ⊂ \overline{ByB/B}.

2. The dimensions of the stalks of the cohomology sheaves of the L_{y} give us multiplicities, more precisely we have for all x, y ∈ W the formula

[H^{0}(st + xρ) : L(st + yρ)] = ∑_{i} dim_{k}(H^{i}L_{y})_{x}.

Remarks 1.0.2. 1. The theorem implies that L_{y} is selfdual under Verdier duality. For almost all p (I hope, for all p > h) our L_{y} is just the intersection cohomology complex IC(\overline{ByB/B}; k) with coefficients in k of the Schubert variety \overline{ByB/B}, extended by zero to all of G/B.

2. We will show that any indecomposable object of K is isomorphic to one of the L_{y}, up to a [ ]-translation.

Corollary 1.0.3. For p > h the following are equivalent:

1. For all x, y ∈ W the multiplicities [H^{0}(st + xρ) : L(st + yρ)] are as conjectured by Lusztig.

2. For every simple reflection s ∈ S and any x ∈ W with xs > x the derived direct image Rπ_{s∗}IC(\overline{BxB/B}; k) of the intersection cohomology complex of the Schubert variety under the projection π_{s} : G/B → G/P_{s} is a semisimple perverse sheaf on G/P_{s}.

The article represents my efforts to translate the arguments of [Soe90] into finite characteristic. This works well most of the time, but for the decomposition theorem there are problems. This is the meaning the above theorems.

2. Arguments from representation theory

In this section 2 we will assume that the characteristic p of k is bigger than the Coxeter number h.

2.1. Description of some endomorphism rings. We describe the endomorphism rings of the G-modules one gets by translating the Steinberg module St = H^{0}(st) out from some walls.

Let λ ↦ wλ denote the action of the Weyl group W on X(T). Let λ ↦ w · λ = w(λ + ρ) − ρ denote the W -action translated to have fixed point −ρ. Let W_{p} ⊂ AffX(T) denote the subgroup generated by all (w·) for w ∈ W and the translations (+μ) with μ ∈ pℤR inside the group of all affine transformations of X(T). We call W_{p} the affine Weyl group and denote its action on X(T) also λ ↦ w · λ, since it extends the dot-action of W. By [Jan87] the category G-Mod of all rational G-modules decomposes as

G-Mod = ∏_{Ω} G-Mod_{Ω}

where Ω runs through all W_{p}-orbits in X(T) and G-Mod_{Ω} consists of all G-modules having only simple subquotients L(λ) with λ ∈ Ω. For any two W_{p}-orbits Ω, Γ ⊂ X(T) we define as in [Jan87] the translation functor T_{Ω}^{Γ} : G-Mod_{Ω} → G-Mod_{Γ}.

Now we take the integral weights in the (suitably translated) lower closure of the fundamental dominant alcove, more precisely the set

I = {λ ∈ X(T) | 0 ≤ ⟨λ, α̌⟩ < p ∀α ∈ R^{+}}

as a convenient parameter set for part of the W_{p}-orbits, and denote for λ, μ ∈ I by T_{λ}^{μ} the translation functor from the W_{p}-orbit of st + λ to the W_{p}-orbit of st + μ.

Put 𝔥 = Lie(T), U = U(𝔥) the symmetric algebra over 𝔥, U^{+} ⊂ U the ideal generated by 𝔥 and C = C(W, 𝔥) = U/(U^{+})^{W} U the algebra of coinvariants. For λ ∈ X(T) we denote W_{λ} its stabilizer in the Weyl group. We thus have W_{0} = W. In [AJS94], 19.8 we show:

Theorem 2.1.1. [AJS94] For any λ ∈ I there exists an isomorphism can_{λ} : End_{G}T_{0}^{λ}St → C^{W_{λ}} of our endomorphism ring with the ring of W_{λ}-invariants in the coinvariant algebra C.

Let more precisely λ, μ ∈ I be given such that W_{λ} ⊃ W_{μ}. Then by [Jan87] there is an isomorphism iso : T_{λ}^{μ}T_{0}^{λ}St → T_{0}^{μ}St, and since the endomorphism rings of our modules are commutative, the map End_{G}T_{0}^{λ}St → End_{G}T_{0}^{μ}St, f ↦ iso ∘ (T_{λ}^{μ}f) ∘ iso^{−1} does not depend on the choice of iso. We abbreviate our notation and denote this map by

T_{λ}^{μ} : End_{G}T_{0}^{λ}St → End_{G}T_{0}^{μ}St.

More precisely than stated in the theorem we actually construct in [AJS94], 19.8 explicit isomorphisms can_{λ} : End_{G}T_{0}^{λ}St → C^{W_{λ}} and our construction shows:

Theorem 2.1.2. [AJS94] For λ, μ ∈ I with W_{λ} ⊃ W_{μ} the diagram

End_{G}T_{0}^{λ}St ⟶^{can_{λ}} C^{W_{λ}}
↓  ↓
End_{G}T_{0}^{μ}St ⟶^{can_{μ}} C^{W_{μ}}

commutes, where the left vertical is our map T_{λ}^{μ} and the right vertical the obvious inclusion.

2.2. Quotient categories. In this section we recall some definitions and results concerning quotient categories. See [Gab62] for a more detailed discussion.

Definition 2.2.1. A full subcategory N of an abelian category A is called a “Serre subcategory” if and only if for any short exact sequence M′ ↪ M→ →M′′ in A we have M ∈ N ⇔ M′, M′′ ∈ N . In other words, N is a full subcategory and is stable under subquotients and extensions.

Let A be an abelian category and N ⊂ A a Serre subcategory. One defines the quotient category A/N as follows: The objects of A/N are just the objects of A, the morphisms however are given as

Hom_{A/N} (M, N) = lim_{U,I} Hom_{A}(U, I),

where the direct limit goes over all subobjects i : U ↪ M and all quotients p : N → I such that coker(i) and ker(p) respectively belong to N .

One shows that A/N is an abelian category, that the obvious functor Q : A → A/N is exact, and that QM ≅ 0 for all M ∈ N . If on the other hand B is an abelian category and F : A → B an exact functor such that F M ≅ 0 for all M ∈ N , then F factorizes uniquely over an exact functor F′ : A/N → B into F = F′ ∘ Q.

Definition 2.2.2. An artinian category is an abelian category all of whose objects are of finite length.

To give a Serre subcategory of an artinian category we only have to say what should be its simple objects.

Let now A be an artinian category with a projective generator. In particular A has only finitely many simple isomorphism classes. Let {L_{x}}_{x∈W} be a system of representatives for the simple isomorphism classes. Any L_{x} has a projective cover P_{x}. For any subset T ⊂ W let A_{T} ⊂ A be the Serre subcategory of all objects having “support in T ”, i.e. with simples \{L_{x}\}_{x∈T} . Let M ↦ \overline{M} denote the quotient functor A → A/A_{T} . For all x ∈ W \ T and M ∈ A we have Hom_{A}(P_{x}, M) = Hom_{A/A_{T}} (\overline{P_{x}}, \overline{M}). In particular for all x ∈ W \ T the \overline{P_{x}} are projective covers of the \overline{L_{x}}.

For a ring R let Mod^{fl}-R denote the category of all finite length right modules. Is A an artinian category and P ∈ A a projective generator, then it is known [Bas68] that Hom_{A}(P) : A → Mod^{fl}-End_{A}P is an equivalence of categories. For T ⊂ W let us put P_{T} = ⨁_{x∈T} P_{x}. In the commutative diagram given by the universal property of A/A_{T}

A = A
↓  ↓ Hom_{A}(P_{W \T} , )
A/A_{T} → Mod^{fl}-End_{A}P_{W \T}

by our above remarks the lower horizontal is an equivalence of the categories.

Later we will use the following terminology:

Definition 2.2.3. An exact functor F : A → B is called a “quotient functor” if and only if it induces an equivalence A/ ker F → B.

Remark 2.2.4. Let F : A → B be an exact functor between abelian categories. Let N_{A} ⊂ A, N_{B} ⊂ B be Serre subcategories and suppose F (N_{A}) ⊂ N_{B}. Then F induces a functor \overline{F} : A/N_{A} → B/N_{B}.

Let us now assume in addition, that F admits a left adjoint G such that G is exact and G(N_{B}) ⊂ N_{A}. Then also \overline{G} is left adjoint to \overline{F} .

2.3. The regular subquotient category. We put G-Mof = {M ∈ G-Mod | dim_{k}M < ∞}. There is a contravariant involutive functor d : G-Mof → G-Mof such that dL ≅ L for any simple object L. More precisely take as dM the contravariant representation M^{∗} twisted by a Chevalley-involution of G. We put V (λ) = dH^{0}(λ). We consider the partial order ↑ on X defined in [Jan87], II.6.4 and put

A = {M ∈ G-Mof | [M : L(λ)] ≠ 0 ⇒ λ ↑ st + ρ}.

Inside we consider the subcategory

N = {M ∈ A | [M : L(λ)] = 0 ∀λ ∈ st + W ρ}

and define the “regular subquotient arround the Steinberg point” O = A/N . Let M ↦ \overline{M} denote the quotient functor A → O. We put

M_{x} = \overline{V (st + xρ)} ∈ O.

Now we list some properties of O and the M_{x}.

1. Every object of O has finite length. All spaces of morphisms are finite dimensional.

2. Every M_{x} has a unique simple quotient L_{x}. The {L_{x}}_{x∈W} form a system of representatives for the simple isomorphism classes of O.

3. There exists a contravariant involution d : O → O such that dL_{x} ≅ L_{x} for all x ∈ W.

4. There exists a partial order ≤ on W such that [M_{x} : L_{y}] ≠ 0 ⇔ y ≤ x. Furthermore [M_{x} : L_{x}] = 1 for all x ∈ W. (Certainly this is just the Bruhat order, but we don’t care.)

5. M_{e} is the projective cover of L_{e} in O. This follows from the fact that V (st + ρ) is the projective cover of L(st + ρ) in A.

6. \overline{T_{0}^{ρ}St} = P_{w_{∘}} is the projective cover of L_{w_{∘}} in O. This P_{w_{∘}} admits a filtration, in which any M_{x} with x ∈ W appears exactly once as a subquotient. Furthermore can_{ρ} gives a canonical isomorphism can : End_{O}P_{w_{∘}} → C.

2.4. The semiregular subquotient categories. For any simple reflection s ∈ S let α ∈ R^{+} be the corresponding simple root and ϖ_{α} the fundamental weight. We consider μ = μ(s) = ρ − ϖ_{α} ∈ I. This weight lies on the s-wall, W_{μ} = ⟨s⟩. As before we take

A^{s} = {M ∈ G-Mof | [M : L(λ)] ≠ 0 ⇒ λ ↑ st + μ},

consider in there the subcategory

N^{s} = {M ∈ A^{s} | [M : L(λ)] = 0 ∀λ ∈ st + W μ}

and form the quotient O^{s} = A^{s}/N^{s}. In there we have the standard objects

M_{x}^{s} = \overline{V (st + xμ)}.

Certainly M_{x}^{s} = M_{xs}^{s} for x ∈ W. Any M_{x}^{s} has a unique simple quotient L_{x}^{s}.

We will need in particular that \overline{T_{0}^{μ}St} = P_{w_{∘}}^{s} is the projective cover of L_{w_{∘}}^{s} in O^{s}. Again P_{w_{∘}}^{s} admits a filtration such that every M_{x}^{s} with x ∈ W/⟨s⟩ occurs exactely once as a subquotient. Furthermore can_{μ} gives a canonical isomorphism

can^{s} : End_{O^{s}}P_{w_{∘}}^{s} → C^{s}.

2.5. Translation. Let s ∈ S, μ = μ(s) be as before. The translation functors onto the wall and out from the wall T_{ρ}^{μ} and T_{μ}^{ρ} induce functors between our subquotient categories, which we denote T^{s} : O → O^{s} and T_{s} : O^{s} → O. Known properties of T_{ρ}^{μ}, T_{μ}^{ρ} imply lots of properties of T^{s}, T_{s}.

1. The functors T^{s}, T_{s} are exact and right- as well as leftadjoint to one another.

2. We have T^{s}M_{x} ≅ M_{x}^{s} and if y < ys in the Bruhat order there is a short exact sequence

M_{y} ↪ T_{s}M_{y}^{s}→ →M_{ys}.

3. We have T_{s}P_{w_{∘}}^{s} ≅ P_{w_{∘}}, one gets independently of of such an isomorphism a map T_{s} : End_{O^{s}}P_{w_{∘}}^{s} → End_{O}P_{w_{∘}} and the diagram

End_{O^{s}}P_{w_{∘}}^{s} ⟶^{can^{s}} C^{s}
T_{s} ↓  ↓
End_{O}P_{w_{∘}}  ⟶^{can} C

commutes with the inclusion as right vertical.

4. The translation through the wall

θ_{s} = T_{s}T^{s} : O → O

is an exact selfadjoint functor. If y ∈ W is given such that y < ys, we have a short exact sequence

M_{y} ↪ θ_{s}M_{y}→ →M_{ys}.

5. There are enough projectives in O, and each one of them is the direct summand of an object of the form θ_{s} … θ_{t}M_{e}. Indeed, M_{e} is projective by 2.3.5 and the θ_{s} are exact and selfadjoint, hence all θ_{s} … θ_{t}M_{e} are projective. Now if x = t … s is a reduced decomposition, then by the preceeding point we have a surjection θ_{s} … θ_{t}M_{e} ↠ M_{x} ↠ L_{x}. This shows what we need.

6. Let P_{x}→ →L_{x} be the projective cover in O. Every P_{x} admits a filtration with subquotients of the form M_{y}. If (P_{x} : M_{y}) denotes the multiplicity of M_{y} as a subquotient, we have the reciprocity formula (P_{x} : M_{y}) = [M_{y} : L_{x}].

7. For all s ∈ S, M ∈ O we have [θ_{s}M : L_{w_{∘}}] = 2[M : L_{w_{∘}}]. Indeed we know by 2.3.6 that (P_{w_{∘}} : M_{x}) = 1 ∀x ∈ W. From 2.5.4 we get comparing the characters θ_{s}P_{w_{∘}} ≅ P_{w_{∘}} ⊕ P_{w_{∘}}. Now we have

[θ_{s}M : L_{w_{∘}}] = dimHom_{O}(P_{w_{∘}}, θ_{s}M)
= dimHom_{O}(θ_{s}P_{w_{∘}}, M)
= 2dimHom_{O}(P_{w_{∘}}, M)
= 2[M : L_{w_{∘}}].

8. We have [M_{x} : L_{w_{∘}}] = (P_{w_{∘}} : M_{x}) = 1 for all x ∈ W. More precisely we even have socM_{x} = L_{w_{∘}} for all x ∈ W. Indeed it is sufficient to show: Hom_{O}(L_{y}, M_{x}) ≠ 0 ⇒ y = w_{∘}. This follows by induction from above over the length of x. For x = w_{∘} the statement is clear. Suppose xs < x. Then we have a short exact sequence M_{xs} ↪ θ_{s}M_{x}→ →M_{x} and we deduce:

Hom_{O}(L_{y}, M_{xs}) ≠ 0 ⇒ Hom_{O}(L_{y}, θ_{s}M_{x}) ≠ 0
⇒ Hom_{O}(θ_{s}L_{y}, M_{x}) ≠ 0
⇒ [θ_{s}L_{y} : L_{w_{∘}}] ≠ 0
⇒ y = w_{∘}.

2.6. Homomorphisms between projective objects. Our canonical isomorphism can : End_{O}P_{w_{∘}} → C gives us an exact functor

𝕍 = Hom_{O}(P_{w_{∘}}, ) : O → C-Mod^{fl}.

We are going to show

Theorem 2.6.1. For M, Q ∈ O with Q projective the functor 𝕍 induces an isomorphism

Hom_{O}(M, Q) → Hom_{C}(𝕍M, 𝕍Q).

Proof. We proceed in steps.

Q = P_{w_{∘}}: In this case we have Q = \overline{T_{0}^{ρ}St} also is the injective hull of M_{w_{∘}} = L_{w_{∘}}. For M ∈ C-Mod^{fl} also the dual space M^{∗} = Hom_{k}(M, k) is a C-Modul. By our restriction on the characteristic C is the cohomology ring of the smooth manifold C = H^{•}(G/B; k), see [Dem73], and Poincaré duality gives us an isomorphism C ≅ C^{∗} in C-Mod^{fl}. Therefore C is not only the projective cover, but also the injective hull of k ∈ C-Mod^{fl}.

Since 𝕍Q = C in this case, both sides of the hoped-for isomorphism Hom_{O}(M, Q) → Hom_{C}(𝕍M, 𝕍Q) are exact in M. By the five lemma it will be sufficient to show this isomorphism for simple M. This is left to the reader.

Q = M_{e}: Let us consider once more our isomorphism from the preceding step and take on both sides the part annihilated by the maximal ideal C^{+} ⊂ C. We get an isomorphism Hom_{O}(M, N) → Hom_{C}(𝕍M, k) where N ⊂ P_{w_{∘}} is the biggest subobject killed by C^{+} and we use k = socC on the right. We only have to show N = M_{e}. Clearly we have a short exact sequence M_{e} ↪ P_{w_{∘}}→ →coker and coker admits a filtration with subquotients M_{x}.

Since socM_{x} = L_{w_{∘}} for all x we deduce for any nonzero subobject U ⊂ coker, that [U : L_{w_{∘}}] ≠ 0. Now certainly we have M_{e} ⊂ N. In case M_{e} ≠ N, the image of N in coker would be nonzero, thus we would have [N : L_{w_{∘}}] ≥ 2. But Hom_{O}(P_{w_{∘}}, N) ⊂ End_{O}P_{w_{∘}} is an ideal killed by C^{+}, thus we have 1 ≥ dimHom_{O}(P_{w_{∘}}, N) = [N : L_{w_{∘}}]. This contradiction shows that indeed M_{e} = N.

Q arbitrary: Any projective Q ∈ O is a direct summand of an object of the form θ_{s} … θ_{t}M_{e}, thus we can assume Q = θ_{s} … θ_{t}M_{e}. Now we deduce with the last paragraph of 2.2 for any M ∈ O :

Hom_{O}(M, θ_{s} … θ_{t}M_{e}) = Hom_{O}(θ_{t} … θ_{s}M, M_{e})
= Hom_{C}(𝕍θ_{t} … θ_{s}M, 𝕍M_{e})
= Hom_{C}(\overline{θ_{t}} … \overline{θ_{s}}𝕍M, 𝕍M_{e})
= Hom_{C}(𝕍M, \overline{θ_{s}} … \overline{θ_{t}}𝕍M_{e})
= Hom_{C}(𝕍M, 𝕍θ_{s} … θ_{t}M_{e})

Next we proof:

Theorem 2.6.2. i.) For all x ∈ W we have 𝕍M_{x} = k.

ii.) For all M ∈ O, s ∈ S we have 𝕍θ_{s}M ≅ C ⊗_{C^{s}} 𝕍M.

Proof. i.) is clear by 2.5.8. For ii.) we consider the adjoint pair (T_{s}, T^{s}) of functors between O and O^{s}. Let us consider also the quotient functor

𝕍^{s} = Hom_{O^{s}}(P_{w_{∘}}^{s}, ) : O^{s} → C^{s}-Mod^{fl}.

Since T_{s}P_{w_{∘}}^{s} ≅ P_{w_{∘}} we have [T^{s}M : L_{w_{∘}}^{s}] = [M : L_{w_{∘}}] for M ∈ O. Furthermore we have [T_{s}M : L_{w_{∘}}] = 2[M : L_{w_{∘}}^{s}] for M ∈ O^{s}, for example since this holds for all standard objects M = M_{x}^{s}.

Therefore the adjoint pair (T_{s}, T^{s}) gives us an adjoint pair (\overline{T_{s}}, \overline{T^{s}}) of functors between the quotient categories C-Mod^{fl} and C^{s}-Mod^{fl}. By 2.5.3 we deduce that \overline{T^{s}} = res : C-Mod^{fl} → C^{s}-Mod^{fl} is just the restriction functor. Therefore we necessarily have \overline{T_{s}} = C⊗_{C^{s}} and we deduce 𝕍θ_{s}M ≅ 𝕍T_{s}T^{s}M ≅ 𝕍\overline{T_{s}} \overline{T^{s}}M ≅ C ⊗_{C^{s}} 𝕍M.

2.7. Decomposition with and without grading. Let C be a finite dimensional graded k-algebra – we even permit homogeneous components of negative degree. Let C-mod^{fl} be the category of all graded finite dimensional C-modules, denote by hom_{C} the morphisms in this category, and let v : C-mod^{fl} → C-Mod^{fl} be the functor forgetting the grading. For M = ⨁ M^{n} ∈ C-mod^{fl} let us define the grading shifted modules M⟨ν⟩ ∈ C-mod^{fl} by (M⟨ν⟩)^{n} = M^{n−ν}. Thus we have v(M⟨ν⟩) = vM and Hom_{C}(vM, vN) = ⨁_{ν} hom_{C}(M, N⟨ν⟩).

Definition 2.7.1. Take M ∈ C-Mod^{fl}. We say “M lifts” if and only if there exists M̃ ∈ C-mod^{fl} such that vM̃ ≅ M.

Proposition 2.7.2. Let M, N ∈ C-Mod^{fl} be given such that (1) N, M ⊕ N lift and (2) N is indecomposable. Then also M lifts. More precisely, for any lift { \atop M ⊕ N} of M ⊕ N we find a lift M̃ of M, which is a direct summand of { \atop M ⊕ N}.

Proof. The inclusion i : N ↪ M ⊕N and the projection p : M ⊕N→ →N decompose on lifts of these objects into homogeneous components i = ∑ i_{ν}, p = ∑ p_{ν}. Since we assumed N indecomposable we have End_{C}N local. Since id = p ∘ i = ∑ p_{ν} ∘ i_{−ν} is an isomorphism, necessarily for some ν the map p_{ν}∘i_{−ν} has to be an isomorphism. Let u ∈ end_{C} Ñ be its inverse. Then i_{−ν} is a splitting of the projection u∘p_{ν} : { \atop M ⊕ N} → Ñ⟨ν⟩ and ker(u ∘ p_{ν}) is a lift of M.

On the other hand we have:

Lemma 2.7.3. Any two lifts of an indecomposable C-module of finite dimension M are isomorphic up to a shift of the grading

Proof. Let M̃, M̃′ be two lifts of M. The identity on M decomposes into homogeneous components in Hom_{C}(M̃, M̃′). Since M is indecomposable, one of these homogeneous components has to be an isomorphism.

2.8. Definition of the modules D_{x}. Let now C be again the coinvariant algebra with its even grading C = ⨁ C^{2ν}. Let us consider the additive subcategory C ⊂ C-mod^{fl} consisting of all C ⊗_{C^{s}} C …⊗_{C^{t}} k⟨ν⟩ with s, …, t ∈ S, ν ∈ ℤ and their direct sums and summands.

Theorem 2.8.1. 1. There exists a family {D_{x}}_{x∈W} of indecomposable objects in C-mod^{fl} such that for any reduced decomposition x = t … s (s, …, t ∈ S) we have

C ⊗_{C^{s}} C … ⊗_{C^{t}} k⟨−l(x)⟩ ≅ D_{x} ⊕ Rest,

where Rest is a direct sum of D_{y}⟨ν⟩ with y < x, ν ∈ ℤ.

2. If {D′_{x}}_{x∈W} is another such family, then we have D′_{x} ≅ D_{x} ∀x ∈ W.

3. The D_{x} stay indecomposable, when we forget their grading.

4. {D_{x}⟨ν⟩}_{x∈W,ν∈ℤ} is a system of representatives for the isomorphism classes of indecomposable objects in C.

We prove this theorem along with the following theorem which makes its meaning clear:

Theorem 2.8.2. We have 𝕍P_{x} ≅ vD_{x} for all x ∈ W.

Proof of both theorems. First we prove the first part of the first theorem along with the second theorem by induction on the length of x. For x = e everything is trivial. For x = t … s (reduced decomposition) we have by 2.5, 4, 6 and 2.3, 4

θ_{s} … θ_{t}P_{e} ≅ P_{x} ⊕ Rest

where Rest is a sum of P_{y} with y < x. We deduce

\begin{aligned}C ⊗_{C^{s}} C … ⊗_{C^{t}} k & = & 𝕍θ_{s} … θ_{t}P_{e} \\ & ≅ & 𝕍(P_{x}) ⊕ 𝕍(Rest). \end{aligned}

By induction 𝕍(Rest) is a sum of vD_{y} with y < x. Every one of these vD_{y} lifts by the inductive assumption. By Proposition 2.7.2 also 𝕍P_{x} lifts, and even lifts to a direct summand of C ⊗_{C^{s}} C … ⊗_{C^{t}} k⟨−l(x)⟩. This proves the first part of the first theorem and the second theorem.

We still have to prove the rest of the first theorem. 2 follows easily by Krull-Schmid. 3 follows since End_{C}D_{x} = End_{O}P_{x} is a local ring. Part 4 follows by Krull-Schmidt and Lemma 2.7.3, once we know, that in C-Mod the subcategory consisting of all direct sums of D_{x} is stable under C⊗_{C^{s}} for all s ∈ S. This follows since C ⊗_{C^{s}} D_{x} = 𝕍θ_{s}P_{x} and since θ_{s}P_{x} is projective, hence a direct sum of indecomposable projectives.

2.9. Properties of the modules D_{x}. By [Dem73] our coinvariant algebra C = H^{•}(G/B; k) is just the cohomology ring of the flag manifold G/B. For the homology the Bruhat cells BxB/B give a canonical basis, and we denote the dual basis in cohomology with {p_{x}}_{x∈W} . Any s ∈ S with corresponding simple root α gives a Demazure operator ∂_{s} : S → S, f ↦ (f −sf)/α̌. This induces a map ∂_{s} : C → C on the coinvariants, and it is known for example from [BGG73], 3.14(i), that

∂_{s}p_{x} = \{ \begin{matrix}p_{xs} & xs < x; \\ 0 & else. \end{matrix}

The restriction H^{•}(G/B; k) → H^{•}(\overline{BxB/B}; k) has kernel

I_{x} = ∑_{y≰x} kp_{y} ⊂ C.

We want to show:

Proposition 2.9.1. For all x ∈ W we have dimD_{x}^{−l(x)} = dimD_{x}^{l(x)} = 1 and Ann_{C}D_{x} = I_{x}.

Proof. First we claim:

f ∈ Ann_{C}(C ⊗_{C^{s}} M) ⇔ f, ∂_{s}f ∈ Ann_{C}M.

Indeed any element of C ⊗_{C^{s}} M can be written uniquely in the form 1⊗m_{1}+α̌⊗m_{2}, in addition 2f = (f+sf)+(α̌(∂_{s}f)) is the decomposition of 2f ∈ C along C = C^{s} ⊕ α̌C^{s}, and thus we have

\begin{aligned}2f(1 ⊗ m_{1} + α̌ ⊗ m_{2}) = & 1 ⊗ ((f + sf)m_{1} + (α̌)^{2}(∂_{s}f)m_{2}) \\ & +α̌ ⊗ ((∂_{s}f)m_{1} + (f + sf)m_{2}). \end{aligned}

The claim follows immediately. The same calculation also shows f ∈ Ann_{C}(1 ⊗ m) ⇔ f, ∂_{s}f ∈ Ann_{C}(m).

Now let x = t … s be a reduced decomposition. With induction we deduce Ann_{C}(C ⊗_{C^{s}} C … ⊗_{C^{t}} k) = I_{x} and even Ann_{C}(1 ⊗ 1 … ⊗ 1) = I_{x}. Now the two extremal components of degrees l(x) and −l(x) of C ⊗_{C^{s}} C …⊗_{C^{t}} k⟨−l(x)⟩ are one dimensional, and since (C/I_{x})^{2l(x)} ≠ 0 the highest homogeneous component lies in the submodule generated by the lowest one. The proposition follows by induction over the length of x.

Next we prove:

Lemma 2.9.2. Let M ∈ C-mod^{fl}, s ∈ S. Then we have (C ⊗_{C^{s}} M)^{∗} ≅ C ⊗_{C^{s}} (M^{∗})⟨−2⟩.

Proof. Again any element of C ⊗_{C^{s}} M can be written uniquely in the form 1 ⊗ m_{1} + α̌ ⊗ m_{2}. Now we define the pairing

(C ⊗_{C^{s}} M) × (C ⊗_{C^{s}} (M^{∗})) → k, (v, w) ↦ ⟨v, w⟩

by ⟨(1 ⊗ m + α̌ ⊗ n), (1 ⊗ f + α̌ ⊗ g)⟩ = f(n) + g(m) for all m, n ∈ M, f, g ∈ M^{∗}. It is not degenerated and we check that ⟨cv, w⟩ = ⟨v, cw⟩ for c ∈ C^{s} and c = α̌, hence for all c ∈ C.

Proposition 2.9.3. The modules D_{x} are selfdual, D_{x} ≅ D_{x}^{∗}.

Proof. By now we know: If x = t … s is a reduced decomposition and M ⊂ C ⊗_{C^{s}} C … ⊗_{C^{t}} k⟨−l(x)⟩ is a direct summand with M^{l(x)} = k or M^{−l(x)} = k, then we have M ≅ D_{x}. But by the preceding lemma C ⊗_{C^{s}} C … ⊗_{C^{t}} k⟨−l(x)⟩ is selfdual.

Theorem 2.9.4. For x ∈ W are equivalent:

1. D_{x} ≅ (C/I_{x})⟨−l(x)⟩.

2. The cohomology ring H^{•}(\overline{BxB/B}; k) = C/I_{x} with coefficients in k the Schubert variety \overline{BxB/B} satisfies Poincaré duality.

Proof. 1 ⇒ 2. Certainly C/I_{x} has exactly one simple quotient and in the light of the preceding proposition 1 implies that C/I_{x} also has a unique simple submodule, soc(C/I_{x}) = k. This immediately implies 2.

2 ⇒ 1. We consider the inclusion

(C/I_{x})⟨−l(x)⟩ ↪ C ⊗_{C^{s}} C … ⊗_{C^{t}} k⟨−l(x)⟩

with 1 ↦ 1 ⊗ 1 … ⊗ 1. By assumption resp. Lemma 2.9.2 both sides are selfdual, therefore we get a surjection

C ⊗_{C^{s}} C … ⊗_{C^{t}} k⟨−l(x)⟩→ →(C/I_{x})⟨−l(x)⟩.

Since I_{x} is already the annihilator of the left hand side, this surjection has to split. This implies immediately D_{x} ≅ (C/I_{x})⟨−l(x)⟩.

To prove the first theorem from the introduction, we still need some preparations.

2.10. Grothendieck groups. To every abelian category A we associate its Grothendieck group [A], and every object A ∈ A determines an element [A] ∈ [A]. On the other hand we associate to every additive category A its split Grothendieck group ⟨A⟩ as follows: ⟨A⟩ is the free abelian group generated by the objects of A with the relations A′ + A′′ = A whenever there is a split short exact sequence A′ ↪ A ↠ A′′. Every object A ∈ A gives us an element ⟨A⟩ ∈ ⟨A⟩. Every (co- or contravariant) exact resp. additive functor F : A → B gives us a group homomorphism F : [A] → [B] resp. F ⟨A⟩ → ⟨B⟩.

Let us now consider our category O. Certainly {[L_{x}]}_{x∈W} is a ℤbasis of [O]. For M ∈ O the coefficients of [M] in this basis are just the Jordan-Hölder multiplicities [M] = ∑_{x}[M : L_{x}][L_{x}]. Now the transition matrix [M_{y} : L_{x}] is upper triangular with ones on the diagonal, hence also {[M_{x}]}_{x∈W} is a ℤ basis of [O]. We write the coefficients of [M] in this basis (M : M_{x}), i.e. [M] = ∑_{x}(M : M_{x})[M_{x}]. If M has a Verma flag (i.e. a filtration such that all subquotients are among the M_{x}), then (M : M_{x}) is precisely the multiplicity of M_{x} as a subquotient.

Elements of the group ring ℤ[W] will be written in the form ∑_{x} n_{x}x with n_{x} ∈ ℤ. We define an isomorphism

\begin{matrix}A : [O] & → & ℤ[W] \\ [M] & ↦ & ∑(M : M_{x})x. \end{matrix}

This is an isomorphism of abelian groups, and from 2.5 we can even deduce A[θ_{s}M] = A[M](1 + s) for all M ∈ O, s ∈ S. Finally we consider the additive category pO of all projective objects of O. From 2.5 we deduce that the obvious map ⟨pO⟩ → [O], ⟨P ⟩ ↦ [P] is an isomorphism. The composition of this isomorphism with A will be denoted A : ⟨pO⟩ → ℤ[W]. The functors θ_{s} map projective objects to projective objects. We deduce that also A⟨θ_{s}P ⟩ = A⟨P ⟩(1 + s) for all P ∈ pO, s ∈ S.

2.11. Proof of the first theorem from the introduction. We need the following commutative diagram:

⟨pO⟩ →^{𝕍} ⟨vC⟩ →^{dim} ℤ
A ↓ B ↓ ||
ℤ[W] = ℤ[W] →^{tr} ℤ

The map B is defined by the condition that the first square is commutative. The map dim is given by dim⟨M⟩ = dim_{k} M for all M ∈ vC. The map tr is given by tr(∑ n_{x}x) = ∑ n_{x}. We leave it to the reader to check that also the second square commutes. Furthermore from 2.6.2 we deduce the formula B⟨C ⊗_{C^{s}} M⟩ = B⟨M⟩(1+s) for s ∈ S, M ∈ vC. Theorem 1.1 follows using 2.9.4 and 2.5.6 from

Theorem 2.11.1. Take y ∈ W. The following are equivalent: 1. (P_{y} : M_{x}) = 1 for all x ∈ W with x ≤ y in the Bruhat order.

2. D_{y} ≅ (C/I_{y})⟨−l(y)⟩.

Proof. From 1 we deduce ∑_{x≤y} x = A⟨P_{y}⟩ = B⟨vD_{y}⟩, therefore ♯{x ∈ W | x ≤ y} = dim_{k} D_{y}, and then comparing dimensions the map (C/I_{y})⟨−l(y)⟩ ↪ D_{y}, 1 ↦ 1 ⊗ 1 … 1 has to be an isomorphism.

If on the other hand we have (P_{y} : M_{x}) > 1 for some x, we deduce dim_{k}(C/I_{y}) < dim_{k} D_{y} by the same argument.

For later applications we add a formula for the dimension of Homspaces between objects of C. Let us denote by

⟨ , ⟩ : ℤ[W] × ℤ[W] → ℤ

the bilinear form for which the group elements form an orthonormal basis, ⟨x, y⟩ = δ_{x,y} for x, y ∈ W.

Lemma 2.11.2. For all D, D′ ∈ S we have the formula

dim_{k} Hom_{C}(D, D′) = ⟨B(D), B(D′)⟩.

Proof. Certainly it is sufficient to show for P, P′ ∈ pO the corresponding formula dim_{k} Hom_{O}(P, P′) = ⟨A(P), A(P′)⟩. Without restriction of generality we may assume P = P_{x}, P′ = P_{y} and then we have indeed

dim_{k} Hom_{O}(P_{x}, P_{y}) = [P_{y} : L_{x}]
= ∑_{z}(P_{y} : M_{z})[M_{z} : L_{x}]
= ∑_{z}(P_{y} : M_{z})(P_{x} : M_{z}).

3. Arguments from topology

In this part 3 let k be an arbitrary field.

3.1. Notations. As possible sources for the following generalities we cite [Bor84, KS90]. For every topological space X denote by ShX = Sh_{k}X the category of all sheaves of k-vectorspaces on X. We consider the bounded below derived category D^{+}ShX and the cohomology sheaves H : D^{+}ShX → ShX. Morphisms in D^{+}ShX will be denoted Hom_{D}. Any continous map f : X → Y induces functors f_{∗}, f_{!} : ShX → ShY and f^{−1} : ShY → ShX. All these have derived functors. We change our notations and put f_{∗} = R^{+}f_{∗}, f_{!} = R^{+}f_{!} and f^{∗} = R^{+}f^{−1}. Thus we have triangulated functors f_{∗}, f_{!} : D^{+}ShX → D^{+}ShY and f^{∗} : D^{+}ShY → D^{+}ShX and an adjunction (f^{∗}, f_{∗}). We also need Verdier duality. This doesn’t exist in full generality, therefore we restrict to a comfortable situation.

Let X be a complex algebraic variety and X(ℂ) the corresponding analytic space (with its “metric” topology). We consider the category of “algebraically constructible sheaves” Sh_{ac}X(ℂ) = {F ∈ ShX(ℂ) | There is a decomposition X = ⋃_{α} X_{α} into (finitely many) local closed subvarieties such that j_{α}^{∗}F ∈ ShX_{α}(ℂ) is locally constant of finite rank, for all inclusions j_{α} : X_{α}(ℂ) ↪ X(ℂ)}. Here we also could have written j_{α}^{−1}F, but since we already renamed j_{α}^{−1} into j_{α}^{∗} we stay with it. Finally we associate to any complex algebraic variety X the triangulated kcategory

D(X) = \{F ∈ D^{+}ShX(ℂ) | \begin{matrix}H^{ν}F ∈ Sh_{ac}X(ℂ) for all ν, \\ H^{ν}F = 0 for |ν| ≫ 0 \end{matrix} \}

For F, E ∈ D(X) we can form RHom(F, E) ∈ D(X). If f : X → Y is a morphism of complex varieties, then f_{∗}, f_{!}, f^{∗} map our D(X), D(Y) into one another. Furthermore the functor f_{!} : D(X) → D(Y) admits a right adjoint f^{!} : D(Y) → D(X). I will in addition use the following nonstandard notation: Let \underline{pt} ∈ Sh_{ac}(pt) ⊂ D(pt) be the constant sheaf k on a point, and \underline{X} ∈ D(X) the constant sheaf \underline{X} = c^{∗}\underline{pt}, where c :

X → pt is the constant map. Furthermore let 𝔻F = RHom(F, c^{!}\underline{pt}) be the Verdier dual of F. We have 𝔻^{2} = id and 𝔻f^{!} = f^{∗}𝔻, 𝔻f_{!} = f_{∗}𝔻.

3.2. The category K. In the introduction we introduced K as the full subcategory K ⊂ D(G/B) consisting of all objects π_{∗}\underline{BS} along with their direct sums and summands and the [ ]-translates of those, for all projections

π : BS ⟶ G/B

of a Bott-Samelson variety to a flag manifold. For our purposes it is better to use the following more technical description of K. We could have taken the following lemma also as the definition of K, and then wouldn’t have to prove it. With the proof we pay for greater elegance in the introduction.

Lemma 3.2.1. K is the smallest full subcategory of D(G/B) satisfying all of the following:

1. K contains with an object all isomorphic objects.

2. K contains the skyscraper sheaf on the one point Bruhat cell, j_{e∗}\underline{B/B} ∈ K for j_{e} : B/B ↪ G/B.

3. F ∈ K ⇒ π_{s}^{∗}π_{s∗}F ∈ K ∀s ∈ S, for π_{s} : G/B → G/P_{s}.

4. F ∈ K ⇒ F[n] ∈ K ∀n ∈ ℤ.

5. K is stable under direct sums and direct summands, i.e. if F, E ∈ D(G/B) are given then we have: F ⊕ E ∈ K ⇔ F, E ∈ K.

Proof. The lemma easily follows once we know that

π_{∗}\underline{BS} ≅ π_{t}^{∗}π_{t∗} … π_{s}^{∗}π_{s∗}j_{∗}\underline{B/B}

for t, …, s ∈ S and π : BS = P_{s} ×^{B} … P_{t} ×^{B} B/B → G/B the multiplication. To show this we rename t, …, s into s_{r}, …, s_{1} and abbreviate P_{s_{i}} = P_{i}. In these notations the maps

\begin{matrix}f_{i} : Y = G ×^{B} P_{1} ×^{B} P_{2} … ×^{B} P_{r}/B → G/B \\ f_{i} : [(g, p_{1}, …, p_{r})] ↦ gp_{1} … p_{i}B (i = 0, …, r) \end{matrix}

give us an isomorphism

Y ⟶^{∼} \{(x_{0}, …, x_{r}) ∈ (G/B)^{r+1} | \begin{matrix}x_{i−1} and x_{i} have \\ the same image in G/P_{i} \\ for i = 1, …, r \end{matrix} \}

If we write Y = Y (1, 2, …, r), then we have in hopefully selfexplaining notations a commutative diagram of cartesian squares that for r = 3 has the following form:

Y (1, 2, 3) → Y (2, 3) → Y (3) → G/B
↓ ↓ ↓ ↓ π_{t}
Y (1, 2) → Y (2) → G/B →_{π_{t}} G/P_{3}
↓  ↓  ↓
Y (1) → G/B → G/P_{2}
↓  ↓ π_{s}
G/B →_{π_{s}} G/P_{1}

Now proper base change implies

f_{r∗}f_{0}^{∗} = π_{t}^{∗}π_{t∗} … π_{s}^{∗}π_{s∗}.

Finally we remark that the left vertical of our diagram is part of a cartesian square

BS ↪ Y (1, …, r)
↓  ↓
B/B ↪  G/B.

The claim follows.

Remark 3.2.2. F ∈ K ⇒ 𝔻F ∈ K. Indeed we have 𝔻π_{s∗} = 𝔻π_{s!} = π_{s∗}𝔻 since π_{s} is proper, and 𝔻π_{s}^{∗} = π_{s}^{!}𝔻 = π_{s}^{∗}[2]𝔻 since π_{s} is smooth with fibres of real dimension 2.

To study K we need some auxiliary categories. We put D_{B}(G/B) = {F ∈ D(G/B) | H^{ν}j_{x}^{∗}F is a constant sheaf on BxB/B, for all x ∈ W, ν ∈ ℤ} and

D(G/B)^{ev} = {F ∈ D(G/B) | H^{ν}F ≠ 0 ⇒ ν is even }

and consider

D_{B}(G/B)^{ev} = D_{B}(G/B) ∩ D(G/B)^{ev}.

In addition let H = H(W, S) = ⨁_{x∈W} ℤ[t, t^{−1}]T_{x} be the Hecke algebra as in [KL79] or [Soe97], with t = q^{1/2} = v^{−1}. We define h : D_{B}(G/B) → H by

h(F) = ∑_{ν,x} rkH^{ν}(j_{x}^{∗}F)t^{ν}T_{x}.

Clearly we have h(F[1]) = t^{−1}h(F). Less clear is the following

Lemma 3.2.3. Take F ∈ D_{B}(G/B)^{ev} and s ∈ S. Then we also have π_{s}^{∗}π_{s∗}F ∈ D_{B}(G/B)^{ev} and h(π_{s}^{∗}π_{s∗}F) = h(F)(T_{s} + 1).

Proof. We repeat a proof of [Spr82]. Springer assigns it to MacPherson. Consider the cartesian digaram

BxP_{s}/B →^{j} G/B
π ↓  ↓ π_{s}
BxP_{s}/P_{s} →^{j} G/P_{s}

and assume that xs > x. Then we have a decomposition of the upper left corner

BxsB/B ↪^{u} BxP_{s}/B ↩^{i} BxB/B

into an open and a closed subset with inclusions u and i. For any F ∈ D(G/B) proper base change gives j_{x}^{∗}(π_{s}^{∗}π_{s∗}F) = i^{∗}π^{∗}π_{∗}j^{∗}F and j_{xs}^{∗}(π_{s}^{∗}π_{s∗}F) = u^{∗}π^{∗}π_{∗}j^{∗}F.

Now the distinguished triangle π^{∗}π_{∗}(u_{!}u^{!}, id, i_{∗}i^{∗})j^{∗}F leads under i^{∗}, u^{∗} to two distinguished triangles which can be written after some standard transformations as

\begin{matrix}(π ∘ i)^{∗}(π ∘ u)_{!}j_{xs}^{∗}F → j_{x}^{∗}(π_{s}^{∗}π_{s∗}F) → (π ∘ i)^{∗}(π ∘ i)_{∗}j_{x}^{∗}F →^{[1]} \\ (π ∘ u)^{∗}(π ∘ u)_{!}j_{xs}^{∗}F → j_{xs}^{∗}(π_{s}^{∗}π_{s∗}F) → (π ∘ u)^{∗}(π ∘ i)_{∗}j_{x}^{∗}F →^{[1]} . \end{matrix}

But (π∘i) is an isomorphism and (π∘u) is a projection of the affine space BxsB/B to the affine space BxP_{s}/P_{s} with fibre ℂ. Therefore we have (π ∘ u)_{!}\underline{BxsB/B} ≅ \underline{BxP_{s}/P_{s}}[−2]. If we now assume F ∈ D_{B}(G/B)^{ev}, then with our two distinguished triangles we have π_{s}^{∗}π_{s∗}F ∈ D_{B}(G/B)^{ev} and

rkH^{ν−2}j_{xs}^{∗}F + rkH^{ν}j_{x}^{∗}F = rkH^{ν}j_{x}^{∗}(π_{s}^{∗}π_{s∗}F) = rkH^{ν}j_{xs}^{∗}(π_{s}^{∗}π_{s∗}F).

These equations imply by an easy calculation in the Hecke algebra h(π_{s}^{∗}π_{s∗}F) = h(F)(T_{s} + 1).

Corollary 3.2.4. 1. Any F ∈ K can be written as F = F′ ⊕ F′′[1]

with F′, F′′ ∈ D_{B}(G/B)^{ev}.

2. For F ∈ K, s ∈ S we have h(π_{s}^{∗}π_{s∗}F) = h(F)(T_{s} + 1).

Proof. Left to the reader.

Let us now consider as in [KL79] on the Hecke algebra H the involution i : H → H mit i(t) = t^{−1}, i(T_{x}) = (T_{x^{−1}})^{−1}.

Lemma 3.2.5. F ∈ K ⇒ h(𝔻F) = ih(F).

Proof. It is sufficient to establish the following three claims: (1) The formula holds for j_{e∗}\underline{B/B} and (2) if the formula holds for F, then it also holds (a) for F[1] and (b) for π_{s}^{∗}π_{s∗}F. Here (1) and (2)(a) are trivial, and we only have to check (2)(b). But indeed we have

h(𝔻π_{s}^{∗}π_{s∗}F) = h(π_{s}^{!}π_{s!}𝔻F) = h(π_{s}^{∗}π_{s∗}𝔻F[2]) = (ih(F))(T_{s} + 1)t^{−2}

and

ih(π_{s}^{∗}π_{s∗}F) = i(h(F)(T_{s} + 1)) = h(𝔻F)(T_{s} + 1)t^{−2}

what was to be shown.

For arbitrary X let us consider the hypercohomology functor

c_{∗} = ℍ^{•} = ⨁_{ν} Hom_{D}(\underline{X}[ν], ) : D(X) → k-mod^{fl}.

Proposition 3.2.6. For all F, E ∈ K hypercohomology induces an injection

Hom_{D}(F, E) → Hom_{k}(ℍ^{•}F, ℍ^{•}E).

Proof. Let in full generality X be a topological space. For F ∈ D^{+}ShX, its hypercohomology ℍ^{•}F can be computed as follows: Let us take an injective resolution F → I^{•} and then ℍ^{•}F is the cohomology of the complex Γ(I^{•}) of global sections. Now if X = X_{0} ⊃ X_{1} ⊃ … ⊃ X_{r} = ∅ is a filtration by closed subsets, then Γ(I^{•}) is filtered by sections with support in the X_{i},

0 = Γ_{X_{r}}(I^{•}) ⊂ … ⊂ Γ_{X_{0}}(I^{•}) = Γ(I^{•}).

Therefore ℍ^{•}F is the limit of a spectral sequence with E_{1}-term

E_{1}^{p,q} = ℍ_{X_{p}−X_{p+1}}^{p+q}(F)

the local hypercohomology of F along X_{p} − X_{p+1}. If v : Y → X is a locally closed embedding of algebraic varieties and F ∈ D(X), then we have ℍ_{Y}^{•} F = ℍ^{•}v^{!}F in our formalism.

Let us now consider on X = G/B the filtration by the

X_{p} = ⋃_{l(x)+p≤dimX} BxB/B.

Let j_{p} : (X_{p} − X_{p+1}) ↪ X be the inclusion. For F ∈ K our spectral sequence

ℍ^{p+q}j_{p}^{!}F ⇒ ℍ^{•}F

degenerates already at this E_{1}-term: The reason is that we may assume without restriction of generality 𝔻F ∈ D_{B}(G/B)^{ev}, and then our E_{1}- terms vanish as a chessboard.

Take now F, E ∈ K and f ∈ Hom_{D}(F, E). If ℍ^{•}f vanishes, then necessarily 0 = j_{p}^{!}f ∈ Hom_{D}(j_{p}^{!}F, j_{p}^{!}E) for all p, since our spectral sequence degenerates. Let us denote by i_{p} : X_{p} ↪ X the inclusion. We have a decomposition

X_{p} − X_{p+1} ↪^{u} X_{p} ↩^{i} X_{p+1}

into an open and a closed subset and a distinguished triangle

(i_{!}i^{!}, id, u_{∗}u^{∗})i_{p}^{!}

from which we deduce that i_{p+1}^{!}f = 0 = j_{p+1}^{!}f ⇒ i_{p}^{!}f = 0. Thus by induction j_{p}^{!} = 0 ∀p implies f = i_{0}^{!}f = 0.

Next we calculate the dimension of morphism spaces between objects of K. We can define on the Hecke algebra an involutive antiautomorphism j : H → H by the prescription j(t) = t^{−1}, j(T_{x}) = (T_{x})^{−1}. For any two elements A, B ∈ H let us define ⟨A, B⟩(t) ∈ ℤ[t, t^{−1}] as the coefficient of T_{e} in the product Aj(B).

Theorem 3.2.7. For F, E ∈ K we have

∑_{ν} dim_{k}Hom_{D}^{ν}(F, E)t^{−ν} = ⟨h(F), h(E)⟩(t).

Proof. For E = j_{e∗}B/B this follows from the definitions and the adjunction (j_{e}^{∗}, j_{e∗}). To get from this the general case we only have to show that our formula is valid for a pair F, π_{s}^{∗}π_{s∗}E[2] if and only if it is valid for the pair π_{s}^{∗}π_{s∗}F, E.

But now (π_{s}^{∗}, π_{s∗}) is an adjoint pair and so is (π_{s!}, π_{s}^{!}) = (π_{s∗}, π_{s}^{∗}[2]), therefore also (π_{s}^{∗}π_{s∗}, π_{s}^{∗}π_{s∗}[2]) is an adjoint pair. Thus the left hand side of the formula in the theorem is the same for both pairs. We therefore only have to show that

⟨a, bt^{−2}(T_{s} + 1)⟩ = ⟨a(T_{s} + 1), b⟩

for all a, b ∈ H. But indeed j(t^{−2}(T_{s} + 1)) = T_{s} + 1.

3.3. Perverse sheaves on G/B and G/P_{s}. Let P(X) ⊂ D(X) be the category of perverse sheaves. Let us denote by H^{i} : D(X) → P(X) the perverse cohomology functors. Let P_{B}(G/P_{s}) be the perverse sheaves on G/P_{s} which are smooth along the B-orbits. Let us furthermore denote by P_{P_{s}^{−1}}(G/B) the perverse sheaves on G/B which are smooth along all strata BxP_{s}/B for x ∈ W. We want to show:

Theorem 3.3.1. Suppose chark ≠ 2. Then the functors π_{s}^{!}[−1] and H^{1}π_{s!} define an equivalence of categories P_{P_{s}^{−1}}(G/B) ≅ P_{B}(G/P_{s}).

Proof. First we show that H^{1}π_{s!} : P(G/B) → P(G/P_{s}) is right adjoint to π_{s}^{!}[−1] : P(G/P_{s}) → P(G/B). Indeed for F ∈ P(G/B), E ∈ P(G/P_{s}) we have an equation

Hom_{D}(F, π_{s}^{!}[−1]E) = Hom_{D}(π_{s!}[1]F, E) = Hom_{D}(H^{1}π_{s!}[1]F, E),

the second equality since H^{i}π_{s!}[1]F = 0 for i > 0. Now we need

Lemma 3.3.2. Suppose chark ≠ 2. Then we have a decomposition π_{s∗}\underline{G/B} ≅ \underline{G/P_{s}} ⊕ \underline{G/P_{s}}[−2].

Proof. We have completely canonical morphisms

\underline{G/P_{s}} →^{a} π_{s∗}π_{s}^{∗}\underline{G/P_{s}} = π_{s∗}\underline{G/B} = π_{s!}π_{s}^{!}\underline{G/P_{s}}[−2] →^{b} \underline{G/P_{s}}[−2].

Now we choose noncanonically a compact real form K ⊂ G with maximal torus T = K ∩ B. So we get G/B = K/T and s ∈ S acts on K/T stabilizing the fibres of π_{s}. Therefore s defines an involution on π_{s∗}\underline{G/B} and even an involution in the category of all complexes of sheaves on π_{s∗}I^{•}, where I^{•} denotes the canonical flasque resolution of \underline{G/B} in the sense of Godement. Now π_{s∗}I^{•} and therefore π_{s∗}\underline{G/B} decompose into eigenspaces under this involution, i.e. π_{s∗}\underline{G/B} = E^{+} ⊕ E^{−}. Denote by i_{+}, i_{−}, p_{+}, p_{−} the projections and injections corresponding to this decomposition. It is easy to see that 0 = p_{−} ∘ a = b ∘ i_{+} and that p_{+} ∘ a and b ∘ i_{−} are isomorphisms. The lemma follows.

Lemma 3.3.3. Suppose chark ≠ 2. Then there are natural isomorphisms π_{s!}π_{s}^{!}F ≅ F ⊕ F[2] for F ∈ D(G/P_{s}).

Proof. We find

π_{s!}π_{s}^{!}F = RHom(\underline{G/P_{s}}, π_{s!}π_{s}^{!}F)
= π_{s∗}RHom(\underline{G/B}, π_{s}^{!}F)
= RHom(\underline{π_{s∗}G/B}, F)
= F ⊕ F[2]

where the third step is Verdier duality and the last step the preceeding lemma.

Remark 3.3.4. More precisely we have shown: Any choice of a decomposition

π_{s∗}\underline{G/B} ≅ \underline{G/P_{s}} ⊕ \underline{G/P_{s}}[−2]

gives a natural equivalence between the functors F ↦ π_{s!}π_{s}^{!}F and F ↦ F ⊕ F[2].

We deduce that for any F ∈ P(G/P_{s}) the adjointness map

(H^{1}π_{s!})(π_{s}^{!}[−1])F → F

is an isomorphism.

Furthermore we need:

Lemma 3.3.5. The functor H^{1}π_{s!} : P_{P_{s}^{−1}}(G/B) → P_{B}(G/P_{s}) is exact.

Proof. It is sufficient to show that for any object F ∈ P_{P_{s}^{−1}}(G/B) we have H^{2i}(π_{s!}F) = 0 ∀i. It is even sufficient to show this for all simple objects of P_{P_{s}^{−1}}(G/B), i.e. for all IC(\overline{BxB/B}) = π_{s}^{!}[−1]IC(\overline{BxP_{s}/P_{s}}) with x ∈ W, xs < x. For these objects however this follows from the preceeding lemma.

Finally we show that for all F ∈ P_{P_{s}^{−1}}(G/B) the adjointness map F → (π_{s}^{!}[−1])(H^{1}π_{s!})F is an isomorphism. This is immediate for all F of the form π_{s}^{!}[−1]E with E ∈ P_{B}(G/P_{s}), thus for all simple F. It follows for all F by the five lemma.

Corollary 3.3.6. Take E ∈ P_{B}(G/P_{s}). Then we also have π_{s}^{!}[−1]E ∈ P(G/B), and E is semisimple if and only if π_{s}^{!}[−1]E is semisimple.

4. Topology and representation theory

Let from now on the charakteristic of k be again bigger than the Coxeter number. (In the following subsection 4.1 this condition is not yet needed.)

4.1. Preliminary remarks. If X is a topological space, we can interprete its cohomology ring as the endomorphism algebra of the constant sheaf in the derived category

H^{•}X = End_{D}^{•}\underline{X} = ⨁_{ν} Hom_{D}(\underline{X}, \underline{X}[ν]).

In the same way we can interprete the hypercohomology of a complex of sheaves F ∈ D^{+}ShX as

ℍ^{•}F = Hom_{D}^{•}(\underline{X}, F) = ⨁_{ν} Hom_{D}(\underline{X}, F[ν]).

The composition of morphisms in the derived category thus makes ℍ^{•}F in a natural way into a right module over H^{•}X. If f : X → Y is a continous map and F ∈ D^{+}ShX, then we clearly have

ℍ^{•}f_{∗}F = Hom_{D}^{•}(\underline{Y} , f_{∗}F) = Hom_{D}(\underline{X}, F) = ℍ^{•}F

via \underline{X} = f^{∗}\underline{Y}, and one sees that the H^{•}Y -action on ℍ^{•}f_{∗}F comes by restriction via H^{•}Y → H^{•}X from the H^{•}X-action on ℍ^{•}F.

If on the other hand F ∈ D^{+}ShY is given, we have an obvious morphism

ℍ^{•}F ⊗_{H^{•}Y} H^{•}X → ℍ^{•}f^{∗}F

or written differently

Hom_{D}^{•}(\underline{Y} , F) ⊗_{End}_{D}^{•}_{\underline{Y}} End_{D}^{•}\underline{X} → Hom_{D}^{•}(\underline{X}, f^{∗}F)
γ ⊗ a  ↦ f^{∗}(γ) ∘ a.

We show that this map is an isomorphism in a special case.

Proposition 4.1.1. Suppose chark ≠ 2. Let π_{s} : G/B → G/P_{s} be the projection map. Then for all F ∈ D(G/P_{s}) the map

ℍ^{•}F ⊗_{H^{•}(G/P_{s})} H^{•}(G/B) → ℍ^{•}(π_{s}^{∗}F)

explained above is an isomorphism of ℤ-graded right ℍ^{•}(G/B)-modules.

Proof. For this proof we abreviate π_{s} : G/B → G/P_{s} as π : X → Y. Since π is smooth with complex one dimensional fibres we have π^{∗} ≅ π^{!}[−2]. Thus it is sufficient to show that the map

Hom_{D}^{•}(\underline{Y} , F) ⊗_{End}_{D}^{•}_{\underline{Y}} End_{D}^{•}(π^{!}\underline{Y}) → Hom_{D}^{•}(π^{!}\underline{Y} , π^{!}F),
γ ⊗ a  ↦ π^{!}(γ) ∘ a

is an isomorphism. Using the adjunction (π_{!}, π^{!}) it is even sufficient to show that

Hom_{D}^{•}(\underline{Y} , F) ⊗_{End}_{D}^{•}_{\underline{Y}} Hom_{D}^{•}(π_{!}, π^{!}\underline{Y} , \underline{Y}) → Hom_{D}^{•}(π_{!}π^{!}\underline{Y} , F)
γ ⊗ b  ↦ γ ∘ b

is an isomorphism. But now we can use the result π_{!}π^{!}\underline{Y} ≅ \underline{Y} ⊕ \underline{Y} [2] from 3.3.2.

4.2. Equivalence of K and C. Let again C = H^{•}(G/B; k) be the coinvariant algebra alias the cohomology ring of the flag manifold. Certainly hypercohomology is a functor ℍ^{•} : D(G/B) → C-mod^{fl}. We want to show:

Theorem 4.2.1. The hypercohomology functor induces an equivalence of categories ℍ^{•} : K → C.

Proof. We procede in steps.

1. By 4.1.1 for all s ∈ S, F ∈ D(G/B) we have an isomorphism of graded C-modules ℍ^{•}π_{s}^{∗}π_{s∗}F ≅ C ⊗_{C^{s}} ℍ^{•}F. Now certainly we have ℍ^{•}(j_{e∗}\underline{B/B}) = k and from the definitions we deduce easily that ℍ^{•} induces a functor from K to C.

2. Next we show that ℍ^{•} : K → C is fully faithful. Since we know already by proposition 3.2.6 that our functor is faithful we only have to show for all F, E ∈ K that

dim_{k}Hom_{D}^{•}(F, E) = dim_{k}Hom_{C}(ℍ^{•}F, ℍ^{•}E).

To show this we claim that there is a commutative diagram

⟨K⟩ ⟶^{h} H
ℍ^{•} ↓  ↓ t = 1
⟨vC⟩ ⟶^{B} ℤ[W].

Indeed the class of π_{t}^{∗}π_{t∗} … π_{s}^{∗}π_{s∗}j_{e∗}\underline{B/B} gets maped both ways to the same element (t + 1) … (s + 1), for all t, …, s ∈ S. Equality of dimensions now is a consequence of 2.11.2, 3.2.7.

3. Finally we have to show that every object of C is isomorphic to the hypercohomology of an object of K. It will be sufficient to find for all x ∈ W an L_{x} ∈ K such that ℍ^{•}L_{x} ≅ D_{x}. We procede by induction on the length of x. Certainly we can assume L_{e} = j_{e∗}\underline{B/B}. To construct L_{x} we choose a reduced decomposition x = t … s (s, …, t ∈ S) and consider π_{s}^{∗}π_{s∗} … π_{t}^{∗}π_{t∗}L_{e}[l(x)]. We know that

\begin{aligned}ℍ^{•}(π_{s}^{∗}π_{s∗} … π_{t}^{∗}π_{t∗}L_{e}[l(x)]) & = C ⊗_{C^{s}} C … ⊗_{C^{t}} k⟨−l(x)⟩ \\ & = D_{x} ⊕ Rest, \end{aligned}

where Rest is a sum of D_{y}⟨ν⟩ with y < x, ν ∈ ℤ. By the induction hypothesis we have Rest = ℍ^{•}L_{Rest} for suitable L_{Rest} ∈ K. Since we know already that ℍ^{•} : K → C is fully faithful, we get a decomposition of the identity on L_{Rest} into id = p ∘ i,

L_{Rest} →^{i} π_{s}^{∗}π_{s∗} … π_{t}^{∗}π_{t∗}L_{e}[l(x)] →^{p} L_{Rest}.

If we complete i to a distinguished triangle

L_{Rest} →^{i} π_{s}^{∗}π_{s∗} … π_{t}^{∗}π_{t∗}L_{e}[l(x)] → L_{x} →^{d}

the boundery map d = (p∘i)∘d = p∘(i∘d) disappears. Therefore the middle is isomorphic to L_{Rest} ⊕ L_{x} and ℍ^{•}L_{x} ≅ D_{x}.

Corollary 4.2.2. {L_{x}[ν]}_{x∈W,ν∈ℤ} is a system of representatives for the isomorphism classes of indecomposable objects of K.

We also can characterize the L_{x} as follows:

Proposition 4.2.3. The following conditions on L ∈ K are equvivalent:

1. L is indecomposable, suppL ⊂ \overline{BxB/B} and j_{x}^{∗}L ≅ \underline{BxB/B}[l(x)].

2. L ≅ L_{x}.

Proof. For any reduced decomposition x = t … s the object F = π_{s}^{∗}π_{s∗} … π_{t}^{∗}π_{t∗}L_{e}[l(x)] already satisfies j_{x}^{∗}F ≅ \underline{BxB/B}[l(x)] and it satisfies as well suppF ⊂ \overline{BxB/B}. Now the proposition is clear.

Corollary 4.2.4. 𝔻L_{x} ≅ L_{x} ∀x ∈ W.

Proof. Left to the reader.

We now show part 2 of the theorem 1.2 from the introduction.

Theorem 4.2.5. For all x, y ∈ W we have

[M_{x} : L_{y}] = ∑_{i} dim(H^{i}L_{y})_{x}.

Proof. We recall the reciprocity formula [M_{x} : L_{y}] = (P_{y} : M_{x}). Let us consider the commutative diagram

⟨K⟩ →^{ℍ^{•}} ⟨vC⟩ ⟶^{𝕍} ⟨pO⟩
↓ h ↓ B ↓ A
H ⟶^{t=1} ℤ[W] = ℤ[W]

from the proof of 4.2.1 and section 2.11. We know already ℍ^{•}L_{x} ≅ D_{x} ≅ 𝕍P_{x}. We deduce that h(L_{x}) specializes to A(P_{x}) at t = 1.

Finally we show Corollary 1.0.3 from the introduction.

Proof. 2 ⇒ 1. From 2 we deduce by induction that all L_{x} are simple perverse sheaves. By the characterization of intersection cohomology in [BBD82] we have then h(L_{x}) ∈ H_{x} + ∑_{y} vℤ[v]H_{y} in the notations of [Soe97]. But we know already by 4.2.4, 3.2.5 that h(L_{x}) = h(𝔻L_{x}) = ih(L_{x}) and this implies h(L_{x}) = C′_{x} with C′_{x} as in [KL79]. And the conjectures of Lusztig say in our case (after translation by the reciprocity formula) precisely that A(P_{x}) = C′_{x}(1).

1 ⇒ 2. Left to the reader.

References

[AJS94] Henning Haahr Andersen, Jens Carsten Jantzen, and Wolfgang Soergel. Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: Independence of p. Astérisque, 220:1–320, 1994.

[Bas68] Hyman Bass. Algebraic K-theory. Benjamin, 1968.

[BBD82] Alexander A. Beilinson, Joseph N. Bernstein, and Pierre Deligne. Faisceaux pervers. Astérisque, 100:1–172, 1982.

[BGG73] Joseph N. Bernstein, Israel M. Gelfand, and Sergei I. Gelfand. Schubert cells and cohomology of the spaces G/P. Russian Math. Surveys, 28(3):1– 26, 1973.

[Bor84] Armand Borel, editor. Intersection cohomology, volume 50 of Progress in Mathematics. Birkhäuser, 1984.

[Car94] James B. Carrell. The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties. In Proceedings of Symposia in Pure Mathematics 56, Part 1, pages 53–61. AMS, 1994.

[Dem73] M. Demazure. Invariants symétriques des groupes de Weyl et torsion. Invent. math., 21:287–301, 1973.

[Gab62] Peter Gabriel. Des catégories abéliennes. Bull. Soc. Math. France, 90:323– 448, 1962.

[Jan87] Jens Carsten Jantzen. Representations of Algebraic Groups, volume 131 of Pure and applied mathematics. Academic Press, 1987.

[KL79] David Kazhdan and George Lusztig. Representations of Coxeter groups and Hecke algebras. Inventiones, 53:191–213, 1979.

[KS90] Masaki Kashiwara and Pierre Schapira. Sheaves on Manifolds, volume 292 of Grandlehren. Springer, 1990.

[Lus80] George Lusztig. Some problems in the representation theory of finite Chevalley groups. In Proceedings of Symposia in Pure Mathematics 37, pages 313–317. AMS, 1980.

[Soe90] Wolfgang Soergel. Kategorie O, perverse Garben and Moduln über den Koinvarianten zur Weylgruppe. Journal of the AMS, 3:421–445, 1990.

[Soe97] Wolfgang Soergel. Kazhdan-Lusztig-Polynome and eine Kombinatorik for Kipp-Moduln. Representation Theory (An electronic Journal of the AMS), 1:37–68, english 83–114, 1997.

[Spr82] Tonny A. Springer. Quelques applications de la cohomologie d’intersection, exp. 589 in: Séminaire Bourbaki 1981/82. Astérisque, 92-93:249– 273, 1982.