Kazhdan-Lusztig-polynomials and indecomposable bimodules over polynomial

rings

Wolfgang Soergel January 9, 2006

Abstract

We develop a strategy to prove the positivity of coefficient of Kazhdan-Lusztig polynomials for arbitrary Coxeter groups. Thanks go to Martin H”arterich and Catharina Stroppel for corrections to previous versions. In particular I thank Patrick Polo, whose Remarks led to a sustancial improvement of this article.

Contents

1 Realization of Hecke algebras via bimodules 2 2 A reflection faithful representation 5 3 Further notations and formulas 6 4 The case of a dieder group 7 5 Back from bimodules to the Hecke algebra 11 6 Classification of indecomposable special bimodules 17 7 Discussion of the main conjecture 24

1 Realization of Hecke algebras via bimodules

Notation 1.1. Given a svelte additive categary A denote its split Grothendieckgroup, thus the free abelian group over the objects with relations M = M′ + M′′ whenever we have M ≅ M′ ⊕ M′′. Given a object M let ⟨M⟩ denote its class in ⟨A⟩.

Notation 1.2. Given a ℤ-graded ring A let A -mod_{ℤ}^{f} denote the category of all finitely generated ℤ-graded A-modules. We write M[n] for the object M with its ℤ-grading shifted by n, in formulas (M[n])_{i} = M_{i+n}.

Remark 1.3. Given a field k and a finitely generated commutative nonnegatively ℤ-graded k-algebra A with A_{0} = k the endomorphism ring of a finitely ℤ-graded A-module is always of finite dimension. The endomorphism ring of an indecomposable such module has no indempotents except zero and one and thus is indecomposable as a right module over itself. But it is also the endomorphism ring of this indecomposable right module of finite length and is thus local. Hence we have in A -mod_{ℤ}^{f} the Krull-Schmid theorem with the same proof as in [Pie82] 5.4 and the isomorphy classes of indecomposable objects form a basis of ⟨A -mod_{ℤ}^{f}⟩, so in particular we have ⟨N⟩ = ⟨M⟩ ⇔ N ≅ M.

Notation 1.4. Let (W, S) be a Coxeter system with a finite number of generators, |S| < ∞. Let l : W → ℕ be its length function and ≤ the Bruhat order on W. In particular x < y means x ≤ y, x ≠ y. On the free ℤ[v, v^{−1}]-mdule

H = H(W, S) = ⨁_{x∈W} ℤ[v, v^{−1}]T_{x}

over W there is exactly one structure of associative ℤ[v, v^{−1}]-algebra with T_{x}T_{y} = T_{xy} if l(x) + l(y) = l(xy) and T_{s}^{2} = v^{−2}T_{e} + (v^{−2} − 1)T_{s} for all s ∈ S, see [Bou81], IV, §2, Exercice 23. This associative algebra H is called the Hecke algebra of (W, S). It is unitary with unit T_{e}, we abreviate often T_{e} = 1. In fact it might be more natural to use q = v^{−2}, and the first sections of this article would gain clarity thereby. In the later sections however it is essential for a transparent exposition to have a root of q at our disposal. wichtig, eine Wurzel von q zur Verf”ugung zu haben. The Hecke algebra can also be described as the assocative unitary ℤ[v, v^{−1}] algebra with generators {T_{s}}_{s∈S}, the quadratic relations T_{s}^{2} = v^{−2}T_{e} + (v^{−2} − 1)T_{s} and the so-called braid relations T_{s}T_{t} … T_{s} = T_{t}T_{s} … T_{t} resp. T_{s}T_{t}T_{s} … T_{t} = T_{t}T_{s}T_{t} … T_{s} in case st … s = ts … t resp. sts … t = tst … s for s, t ∈ S.

Definition 1.5. Let (W, S) be a Coxeter system. Let T ⊂ W denote the set of all “reflections”, i.e. all elements of W, which are conjugate to elements of S. By a reflection faithful representation of our Coxeter system we mean a representation W ↪ GL(V) in a finite dimensional vector space over a field k of characteristic char k ≠ 2 with the following properties:

1. Our representation is faithful.

2. For x ∈ W we have dim(V/V^{x}) = 1 ⇔ x ∈ T . Thus exactly the reflections from W have a fixed point set of codimension one in V.

Remark 1.6. Given a reflection faithful representation the elements of T are precisely those elements of W, which act as reflections on V, thus which decompose in V in a onedimensional eigenspace of eigenvalue −1 and a hyperplane of invariand vectors. Indeed x ∈ T cannot act unipotently, since an automorphism of order two of a vector space over a field of charcteristic ≠ 2 is diagonalizable and our condition excludes that x acts as the identity. Furthermore reflections from W can be distinguished as well by their eigenspaces of eigenvalue (−1) as well as by their reflecting hyperplanes, so that for t, r ∈ T we have

V^{t} = V^{r} ⇔ V^{−t} = V^{−r} ⇔ t = r.

For example in case G = V^{−t} = V^{−r}, we consider the short exact sequence G ↪ V ↠ V/G. Since both t and r act as the identity on V/G and since taking the adjoint map doesn’t change the dimension of the eigenspaces we see that tr fixes a hyperplane. But since tr has determinant 1 it must by condition 2 act as the identity on V and thus by condition 2 is the identity of W. We show in the next section, that the representation of a Coxeter group generalizing the representation of an affine Weyl group on the Cartan of a Kac-Moody algebra is always reflection faithful.

Definition 1.7. A representation of W, in which reflections act as reflections and where different reflections have different (−1)-eigenspaces will be called reflection vector faithful.

Example 1.8. The geometric representation of an infinite dieder group is reflection vector faithful. However it is not faithful, since all elements expect the neutral element have a fixed point set of codimension one, not only the reflections.

Notation 1.9. Let now for the sake of simplicity of notation k be an infinite field and let R = R(V) donote the k-algebra of all regular functions on the space V underlying a reflection vector faithful representation. We equip R with a ℤ-grading R = ⨁_{i∈ℤ} R_{i} such, that we have R_{2} = V^{∗} and R_{i} = 0 for uneven i. Again it would be more natural here to work with the usual grading, but in the long run we need these conventions. Let

R = R_{V} ⊂ R -mod_{ℤ}- R

denote the categary of all ℤ-graded R-bimodules, which are finitely generated from the right as well as from the left, and where the action of k from the right and the left is the same. Tensoring ⊗_{R} makes

⟨R⟩

into a ring. For s ∈ S let R^{s} ⊂ R denote the subring of all s-invariants.

Theorem 1.10. Let V be a reflection vector faithful representation of the Coxeter system (W, S) over an infinite field. Let H be the Hecke algebra and R the ring of polynomial functions on V. There is exactely one ring homomorphism

E : H → ⟨R⟩

such that we have E(v) = ⟨R[1]⟩ and E(T_{s} + 1) = ⟨R ⊗_{R^{s}} R⟩ ∀s ∈ S.

Remark 1.11. This theorem is a variant of theorem 1 from [Soe92]. I prefer the proof given here, since it does not assume any knowledge of Demazure operators.

Proof. Unicity is evident, since H is generated by T_{s} + 1 as an algebra over ℤ[v, v^{−1}]. Existence needs only to be proven in case of a dieder group, since H can be described by relations involving each time only two generators. The dieder case will be discussed in section 4.

Definition 1.12. We consicer in the Hecke algebra the elements T̃_{x} = v^{l(x)}T_{x}. By [KL79] there is exactly one involution d : H → H with d(v) = v^{−1} and d(T_{x}) = (T_{x^{−1}})^{−1} and for x ∈ W there is a unique C′_{x} ∈ H with d(C′_{x}) = C′_{x} and

C′_{x} ∈ T̃_{x} + ∑_{y} vℤ[v]T̃_{y}

These elements form the so-called Kazhdan-Lusztig-basis of the Hecke algebra from [KL79].

Conjecture 1.13. Let V be a reflection faithful representation of W over an infinite field ans let R be the ring of regular functions on V. Then there is, at least in case k = ℂ, an indecomposable ℤ-graded R-bimodule B_{x} ∈ R such that

E(C′_{x}) = ⟨B_{x}⟩.

Remark 1.14. In case k = ℂ and W a finite Weyl group this is shown in [Soe92]. For W a finite Weyl group and char k at least the Coxeter number it is proven in [Soe00], that this conjecture is equivalent to a part of Lusztig’s conjecture concerning characters of irreducible representations of algebaric groups over k. For the case of the action of a Weyl group on the Cartan of an affine Kac-Moody algebra the conjecture is established in [Här99]. In 5.3 we construct a left inverse to E, whose explicite description shows, how the preceding conjecture implies the positivity of all coefficients of all Kazhdan-Lusztig polynomials.

Remark 1.15. In 6.16 we show, that the “indecomposable bimodules in the image of E” can at least be parametrized by W. Furthermore we determine in 5.15 the dimension of the Hom-spaces between any two “bimodules im the image of E” and show in which way our formulas are compatible with conjecture 1.13.

Remark 1.16. Still open is in particular the case of universal Coxeter groups (here the KL-polynomials were described by [Dye88]) and the case op general finite Coxeter groups, which should be accessible by computer.

Remark 1.17. Related investigations are in [BM01] and in partially unpublished work of Dyer [Dye94] and in unpublished work of Peter Fiebig.

2 A reflection faithful representation

Proposition 2.1. Given a finite set S and a Coxete rmatrix of type S of the formbe a finite dimensional real vector space and let be given linearly independent vectors (e_{s})_{s∈S} in V and linearly independent linear forms (e_{s}^{∨})_{s∈S} on V such that we have

⟨e_{t}, e_{s}^{∨}⟩ = −2 cos(π/m_{s,t}) ∀s, t ∈ S.

If we assume as in [Kac90] that the dimension of V is smallest possible, then the formula ρ(s)(v) = v − ⟨v, e_{s}^{∨}⟩e_{s} defines a reflection faithful representation ρ : W → GL(V) of our Coxeter group.

Proof. That we get in this way a repesentation, i.e. that the braid relations are satisfied can be deduced from the fact that for any pair s, t of simple reflections with st of finite order our space decomposes as the intersection of the kernels of e_{s}^{∨} and e_{t}^{∨}, which is not moved by s and t at all, and the space generated by e_{s} and e_{t}, where s and t act as generators of the usual dieder group. In general the subspace E ⊂ V spanned by the e_{s} is a subrepresentation isomorphic to the “natural representation” of [Bou81], V.4.3 and is in particular faithful. Furthermore W acts trivially on the simultaneous kernel K of all e_{s}^{∨} and the natural action on (V/K)^{∗} is isomorphic again to the natural representation, which means that the action on (V/K) is isomorphic to the contragredient representation E^{∗} of [Bou81], V.4.4. In view of our minimality assumption on the dimension of V we certainly have K ⊂ E and thus a filtration by subrepresentations

0 ⊂ K ⊂ E ⊂ V.

The pairing (e_{s}, e_{t}) = \frac{1}{2}⟨e_{s}, e_{t}^{∨}⟩ now defines an invariant symetric form on E and a nondegenerate invariant symmetric form on E/K. Now let x ∈ W be an element, whose fix point set V^{x} ⊂ V is a hyperplane. We have to show x ∈ T . Let X ∈ End V denote the image of x. We have either det X = 1 or det X = −1 and start treating the case det X = 1. If we choose a complement ℝc of the hyperplane V^{x}, then our endomorphism X necessarily has the form X(v +αc) = v +αv_{0} +αc for some fixed v_{0} ∈ V^{x}. Certainly we have V^{x} ⊃ K. Since our group acts faithfully on E we can choose c ∈ E and deduce v_{0} ∈ E. Since E^{∗} ≅ V/K is a faithful representation as well, we get v_{0} ∉ K. Now we use our invariant bilinear form on E and deduce (v + αc, v + αc) = (v + αv_{0} + αc, v + αv_{0} + αc) alias 2α(v, v_{0}) + α^{2}(2(c, v_{0}) + (v_{0}, v_{0})) = 0 for all v ∈ E and α ∈ ℝ. This gives us (v, v_{0}) = 0 for all v ∈ E contrdicting v_{0} ∉ K. Thus the case det X = 1 is not possible and we can assume det X = −1. Then necessarily x acts as a reflection on V and also on V/K ≅ E^{∗}. Since W acts faithfully and transitively on the alcoves of the Tits cone, the fix point set of x cannot meet any alcove in the Tits cone. If we consider the fundamental dominant alcove of the Tits cone C ⊂ E^{∗} and its image under x, we find that C and xC are separated only by finite number of reflecting hyperplanes V^{z} for some z ∈ T . On the other hand a certain nonempty open part of the refection plane of x in E^{∗} consits of points on segments joining points of C with points of xC, consider for example the image of C under the projection (id +x)/2. This open part then has to be covered by our finite number of reflecting hyperplans, and this is impossible unless the reflecting hyperplane of x coincides with the reflecting hyperplane of some z ∈ T . But then we form zx and come back to the impossible case det = 1 least we have x = z.

3 Further notations and formulas

Notation 3.1. Given any finite dimensional representation V of a group W we consider for any x ∈ W the (reversed) graph

Gr(x) = {(xλ, λ) | λ ∈ V } ⊂ V × V

and form for any finite subset A ⊂ W in V × V the Zariski closed subset

Gr(A) = ⋃_{x∈A} Gr(x).

Remark 3.2. For any y, z ∈ W we get obviously an isomorphic

Gr(y) ∩ Gr(z) →^{∼} V^{yz^{−1}}

by the projection on the first coordinate and we have

(Gr(y) + Gr(z)) ∩ (V × 0) = im(yz^{−1} − id) × 0.

Notation 3.3. Let now k be an infinite field and let as before denote R the k-algebra of all regular functions on V. We always abreviate ⊗_{k} = ⊗. If we identify R⊗R with the regular functions on V ×V via the rule (f ⊗g)(λ, μ) = f(λ)g(μ), the regular functions on Gr(A) as a quotient of R⊗R from naturally a ℤ-graded R-bimodul. This ℤ-graded R-bimodul we denote

R(A) = R(Gr(A)) ∈ R -mod_{ℤ}- R

and it is easy to check that it is finitely generated as a left module and as a right module over R. For A = {x_{1}, …, x_{n}} we also put Gr(A) = Gr(x_{1}, …, x_{n}) and R(A) = R(x_{1}, …, x_{n}) or even shorter R(x) = R_{x}, R(x, y) = R_{x,y}. In the case A = {y | y ≤ x} we put R(A) = R(≤ x). For the right action of W on R we use exponential notation Sr^{y}(λ) = r(yλ) ∀y ∈ W, λ ∈ V, r ∈ R. If 1_{y} ∈ R_{y} denotes the constant function 1 on Gr(y) alias the image of 1 ⊗ 1 in R_{y}, we have r1_{y} = 1_{y}r^{y}. The notations are chosen to get R(x) ⊗_{R} R(y) ≅ R(xy) for all x, y ∈ W.

4 The case of a dieder group

Remark 4.1. If W is a dieder group, by [Hum90, 7.12(a)] the elementes of the selfdual basis are of the form C′_{x} = v^{l(x)} ∑_{y≤x} T_{y}. We prove this as Remark 4.4.

Theorem 4.2. Let (W, S) be a Coxeter system with twoe generators |S| = 2 and let V be a reflection vector faithful representation over an infinite field. Then the homomorphism of additive groups

\begin{aligned}E : & H & → & ⟨R⟩ \\ & v^{n}C′_{x} & ↦ & ⟨R(≤ x)⟩[n + l(x)] \end{aligned}

is a ring homomorphism.

Remark 4.3. For any reflection s : V → V the obvious map R ⊗ R ↠ R(e, s) induces an isomorphism R ⊗_{R^{s}} R →^{∼} R(e, s). To see this, Remark, that this map certainly is a surjection and compare the dimensions of homogeneous parts, where one may use the eigenspace decomposition R = R^{s} ⊕ R^{s}α for α ∈ V^{∗} an equation of the reflection plane of s and a short exact sequence R_{s}[−2] ↪ R(e, s) ↠ R_{e}. Thus the theorem indeed gives the existence in Theorem 1.10. the proof biven here extends arguments of I. Herrmann [Her99].

Proof. Certainly it suffices, to show for all simple reflections s and all x ∈ W the equation

E ((T_{s} + 1) ∑_{y≤x} T_{y}) = E(T_{s} + 1) E (∑_{y≤x} T_{y}) .

If we put A = {y ∈ W | y ≤ x}, then A ∪ sA and A ∩ sA are both of the form {y ∈ W | y ≤ z} for suitable z in our dieder group W, unless we are in the second case and the intersection is empty. An explicite computation in the Hecke algebra now shows

(T_{s} + 1) ∑_{y∈A} T_{y} = ∑_{y∈A∪sA}T_{y} + v^{−2} ∑_{y∈A∩sA}T_{y}.

If we insert this results on the left hand side, our equation transforms into claiming an isomorphism of graded bimodules

R(A ∪ sA) ⊕ R(A ∩ sA)[−2] ≅ R ⊗_{R^{s}} R ⊗_{R} R(A),

whic will be shown after a preliminary Lemma as Proposition 4.6.

Remark 4.4. To prove the formulas claimed in 4.1 for the C′_{x} in the dieder case we only have to show the γ_{x} = v^{l(x)} ∑_{y≤x} T_{y} are selfdual. For any simple reflection s with sx > x however we can rewrite one of the equations of our proof to read

v(T_{s} + 1)γ_{x} = \{ \begin{matrix}γ_{sx} + γ_{z} with z < x & in case l(x) > 1; \\ γ_{sx} & in case l(x) ≤ 1, \end{matrix}

and thus the selfduality of v(T_{s} + 1) implies inductively the selfduality of all γ_{x}.

Lemma 4.5. Let V be a finite dimensional representation of a group W over an infinite field k of charakteristc ≠ 2. Let A ⊂ W be a finite subset and s ∈ W an element with sA = A, acting on V as a reflection. Then we have:

1. There is an isomorphism of graded bimodules

R ⊗_{R^{s}} R(A) ≅ R(A) ⊕ R(A)[−2].

2. For R(A)^{+} ⊂ R(A) the invariants under the action of s × id multiplication induces an isomorphism

R ⊗_{R^{s}} R(A)^{+} →^{∼} R(A).

Proof. Let first W be an arbitray finite dimensional vector space over k. Any reflection t : W → W defines an involution t : R(W) → R(W), and if we choose an equation β ∈ W^{∗} of the reflecting hyperplane W^{t}, we can consider

∂_{t} = ∂_{t}^{β} : R(W) → R(W), f ↦ \frac{f − tf}{2β} .

If now X ⊂ W is a Zariski closed t-stable subset, then t induces an involution on R(X) and we get an eigenspace decomposition of the form R(X) = R(X)^{+} ⊕ R(X)^{−}. If no irreducible component of X is contained in W^{t}, our operator ∂_{t} stabilizes the kernel of the sujection R(W) ↠ R(X) and thus induces a map ∂_{t} : R(X) → R(X). It is easy to see that ∂_{t} and multiplication by β are mutually inverse isomorphisms R(X)^{+} ≅ R(X)^{−}[2] of ℤ-graded R^{t}-modules.

Now let us take W = V × V and t = s × id for our refection s ∈ W. We apply our results to X = Gr(A) and get a decomposition R(A) = R(A)^{+} ⊕ R(A)^{−} as well as an isomorphism R(A)^{+} ≅ R(A)^{−}[2] by multiplication with α⊗1 = β for α ∈ V^{∗} an equation of the reflecting hyperplane of s. With R = R^{s} ⊕ αR^{s} we get part 2, and with our decompositon R(A) = R(A)^{+} ⊕ R(A)^{−} and part 2 we also get part 1.

Proposition 4.6. Let (W, S) be a Coxeter system with two generators |S| = 2 and let V be a reflection faithful representation of W over an infinite field. Let s ∈ S and x ∈ W be given and let A = {y ∈ W | y ≤ x} denote the set of all elements below x. Then in R-Mod_{ℤ}-R we have an isomorphism

R ⊗_{R^{s}} R(A) ≅ R(A ∪ sA) ⊕ R(A ∩ sA)[−2].

Proof. The case A = sA is already done by the preceding Lemma. In case A = {e} our proposition just claims the isomorphism R(≤ s) ≅ R ⊗_{R^{s}} R of 4.3. Thus only the case A ≠ {e}, A ≠ sA remains to be treated, thus A = {y ∈ W | y ≤ x} with x ≠ e and sx > x. By explicite calculation we find A − sA = {x, rx} for some reflection r ∈ T different from s. By our assumptions the (−1)-eigenspaces of reflections of W are pairwise different and span a twodimensional subrepresentation U ⊂ V. By 3.2 there exists a form β ∈ V^{∗} × V^{∗}, which vanishes on Gr(x) + Gr(rx) but not on U × 0. We consider the submodules generated by the cosets \overline{β} and \overline{1} of β und 1 in R(A) over R^{s} ⊗ R, call them M, N and claim

1. M ≅ R(A ∩ sA)^{+}[−2] in R^{s} -mod_{ℤ}- R;

2. N ≅ R(A ∪ sA)^{+} in R^{s} -mod_{ℤ}- R;

3. R(A) = M ⊕ N,

where the upper index + means (s × id) invariants. Once this is shown, the proposition follows from the preceding lemma 4.5. Thus we only need to show our three claims.

1. To show the first claim we Remark, that for three pairwise different elements x, y, z ∈ W, whose lengths do not all have the same parity, we always have

Gr(x) + Gr(y) + Gr(z) ⊃ U × 0.

To show this we may without restriction assume z = e and x, y ∈ T . By section 3 we have (Gr(x) + Gr(e)) ∩ (U × 0) = V^{−x} × 0 and the some for y. Since we assumed our representation reflection vector faithful, we have V^{−x} ≠ V^{−y} and therefore the inclusion. For y ∉ {x, rx} we thus have Gr(y)+ Gr(x) + Gr(rx) ⊃ U × 0, in particular our function β does not vanish on Gr(y) for y ∈ A ∩ sA. Thus an element of R(A) annihilates \overline{β} iff it vanish on Gr(A ∩ sA). We deduce that multiplication by \overline{β} defines an isomorphism R(A ∩ sA)[−2] →^{∼} R(A)\overline{β}. But the image of R^{s} ⊗ R in R(A ∩ sA) consists precisely on the (s × id) invariants, thus our isomorphism restricts to an isomorphism R(A ∩ sA)^{+}[−2] →^{∼} M.

2. The R^{s} ⊗ R-submodule generated by \overline{1} in R(A ∪ sA) is certainly just R(A ∪ sA)^{+} and the restriction onto Gr(A) gives an injection

R(A ∪ sA)^{+} ↪ R(A).

This proves the second claim.

3. We first show R(A) = M + N. If α ∈ V^{∗} denotes an equation of the reflecting plane V^{s}, then we have R = R^{s} ⊕ αR^{s}, thus R ⊗ R is generated as an R^{s} ⊗ R-modul by 1 ⊗ 1 und α ⊗ 1. It follows that it is also generated by 1 ⊗ 1 and an arbitray not (s × id) invariants β ∈ V^{∗} × V^{∗}. Our β however cannot be invariant under (s × id), because otherwise ist would vanish on Gr(sx) and thus by the inclusion of the proof of claim 1 on all of U × 0. This shows R(A) = M +N. We only need in addition M ∩N = 0. This can be seen as follows: For y ∈ A ∩ sA the restriction on Gr(y, sy) = Gr(y) ∪ Gr(sy) of any element of N is invariant under s × id. Thus it suffices to show, that the restriction of an element of M on such a Gr(y, sy) can only be invariant under s × id, if it vanishes on Gr(y, sy). It will be sufficient for that to show, that the restriction of β to Gr(y, sy) is not invariant under s × id or also, that the restriction of β to Gr(y)+Gr(sy) is not invariant under s×id . But the s×id invariant forms on Gr(y) + Gr(sy) have to vanish on the subspace V^{−s} × 0 and (Gr(x) + Gr(rx)) meets U × 0 in the different subspace V^{−r} × 0.

5 Back from bimodules to the Hecke algebra

Notation 5.1. We work from now on always with a fixed reflection faithful representation of a Coxeter system over an infinite field and want in this section give an explicite left inverse for our ring homomorphism E : H → ⟨R⟩. We start with some notations. Given B, B′ ∈ R we set

Hom(B, B′) = Hom_{R⊗R}(B, B′) ∈ R.

It is understood, that the left resp. right action on R on our Hom-space comes from the left resp. right action on B or equivalently on B′, in formulas (rf)(b) = f(rb) = r(f(b)), (fr)(b) = f(br) = (f(b))r, ∀r ∈ R, b ∈ B, f ∈ Hom(B, B′).

Notation 5.2. Given a finite dimensional ℤ-graded vector space V = ⨁ V_{i} we define its graded dimension by

\underline{dim}V = ∑(dim V_{i})v^{−i} ∈ ℤ[v, v^{−1}]

and for a finitely generated ℤ-graded right R-module M we define its graded rank by the rule

\underline{rk}M = \underline{dim}(M/MR_{>0}) ∈ ℤ[v, v^{−1}].

We thus have \underline{dim}(V [1]) = v(\underline{dim}V) and \underline{rk}(M[1]) = v(\underline{rk}M). Certainly our graded rank is only a reasonable notation for free modules, an we will only use it for such. By \overline{\underline{rk}}M we denote the image of \underline{rk}M under the substitution v ↦ v^{−1}. The following theorem motivates this whole section.

Theorem 5.3. Our map E : H → ⟨R⟩ has as a left inverse the map ⟨R⟩ → H given by the rule

⟨B⟩ ↦ ∑_{x∈W} \overline{\underline{rk}} Hom(B, R_{x}) T_{x}.

Proof. This theorem is a direct consequence of 5.7 and 5.16, which we will prove in the sequal without using it.

Definition 5.4. For any R-bimodule B and any subset A ⊂ W we define the subbimodule

Γ_{A}B = {b ∈ B | supp b ⊂ Gr(A)}

of all elements with support in Gr(A). We put Γ_{≥i}B = Γ_{{x∈W|l(x)≥i}}B and define the category

F_{Δ} ⊂ R

as the full subcategory of all graded bimodules B ∈ R such that B has its support in a set of the form Gr(A) for some finite A ⊂ W and tht the quotient Γ_{≥i}B/Γ_{≥i+1}B is for all i isomorphic to a finite direct sum of graded bimodules of the form R_{x}[ν] with l(x) = i and ν ∈ ℤ.

Remark 5.5. Certainly B ↦ Γ_{≥i}B/Γ_{≥i+1}B is additive in B, thus F_{Δ} is stable under forming direct sums and because of Krull-Schmid 1.3 also stable under forming direct summands.

Notation 5.6. It turns out to be convenient to introduce the graded bimodules

Δ_{x} = R_{x}[−l(x)]

and to work in the Hecke algebra with T̃_{x} = v^{l(x)}T_{x}. for the multiplicity of a summand Δ_{x}[ν] in one and every decomposition of Γ_{≥i}B/Γ_{≥i+1}B for i = l(x) we introduce the notation (B : Δ_{x}[ν]). Also we introduce the abreviation R[1] ⊗_{R^{s}} M = θ_{s}M.

Proposition 5.7. Let s ∈ S be a simple reflection.

1. With B also R ⊗_{R^{s}} B belongs to F_{Δ}.

2. If we define the maps h_{Δ} : F_{Δ} → H by the rule B ↦ ∑_{x,ν}(B : Δ_{x}[ν])v^{ν}T̃_{x}, for all s ∈ S we get two commutative diagrams

\begin{matrix}F_{Δ} & ⟶ & H \\ ^{θ_{s}}↓ & & ↓^{(T̃_{s}+v)·} \\ F_{Δ} & ⟶ & H \end{matrix}  \begin{matrix}F_{Δ} & ⟶ & H \\ ^{[1]}↓ & & ↓^{v·} \\ F_{Δ} & ⟶ & H \end{matrix}

3. Our map E factors via a map E : H → ⟨F_{Δ}⟩ and our h_{Δ} : ⟨F_{Δ}⟩ → H is left inverse to this E

Proof. The proof of this proposition needs Lemma 5.8. Let W be a finite dimensional vector space and U, V ⊂ W two affine subspaces. Then Ext_{R(W)}^{1}(R(U), R(V)) is nonzero only if we have V ∩ U = V or if V ∩ U is of codimension one in V. In the later case Ext_{R(W)}^{1}(R(U), R(V)) is a free R(U ∩ V)-modul of rank one, generated by the class of an arbitray short exact sequence of the form

R(V)[−2] ↪^{α} R(U ∪ V) ↠ R(U)

for α ∈ W^{∗} with α|_{U} = 0, α|_{V} ≠ 0.

Proof. If F and G are free modules of finite rank over the k-algebras A and B and if M or N are arbitray modules over A and B, then we obviously (see [Bou70], I §4 and II §11) have

Hom_{A}(F, M) ⊗ Hom_{B}(G, N) →^{∼} Hom_{A⊗B}(F ⊗ G, M ⊗ N).

If our algebras are noetherian and M′ resp. N′ are finitely generated over A resp. B, we find resolutions F^{•} ↠ M′ resp. G^{•} ↠ N′ with F^{i} resp. G^{j} free of finite rank over A resp. B. Then F^{•} ⊗ G^{•} will be a free resolution of M′ ⊗ N′ and we deduce

Ext_{A⊗B}^{n}(M′ ⊗ N′, M ⊗ N) = H^{n} Hom_{A⊗B}(F^{•} ⊗ G^{•}, M ⊗ N)
= H^{n}(Hom_{A}(F^{•}, M) ⊗ Hom_{B}(G^{•}, N))
= ⨁_{i+j=n} Ext_{A}^{i}(M′, M) ⊗ Ext_{B}^{j}(N′, N)

Thus we can restrict in our Lemma to the case with dim_{k} W = 1, and those can easily be done in an explicit way. More precisely we may without restriction assume U and V linear and find a decomposition by W = S⊕U′⊕V′⊕W′ mit U = S ⊕ U′ and V = S ⊕ V′. If we denote the dimension by dim W = s + u + v + w, we will get Ext_{R(W)}^{•}(R(U), R(V)) ≅ Ext_{k[X]}^{•}(k[X], k[X])^{s} ⊗ Ext_{k[X]}^{•}(k[X], k)^{u} ⊗ Ext_{k[X]}^{•}(k, k[X])^{v} ⊗ Ext_{k[X]}^{•}(k, k)^{w}. Thus Ext^{1} is possible only for v ≤ 1 and in case v = 1 we get Ext^{1} = k[X]^{s}.

Now we prove the proposition. Let s ∈ S be our fixed simple reflection. Certainly we can refine our filtration Γ_{≥i} of B by certain Γ_{≥j}B with j ∈ ℤ + 0, 5 such that for all i ∈ ℤ the quotients Γ_{≥i}B/Γ_{≥i+0,5}B resp. Γ_{≥i−0,5}B/Γ_{≥i}B are sums of R(x)[ν] with x > sx resp. x < sx. With these choices the parameters x, y of two possible subquotients of Γ_{≥i−0,5}B/Γ_{≥i+0,5}B only differ by a reflection in case y = sx: Indeed for an arbitray reflection t we always have tx > x or tx < x, and sy > y < x > sx implies y ≤ sx after a result of Deodhar, the so-called property Z of Coxeter groups. However all extensions in Ext_{R⊗R}^{1}(R_{x}, R_{sx}) die after restriction to R^{s} ⊗ R. Indeed the restriction R_{x,sx} ↠ R_{x} splits over R^{s} ⊗ R, since in the notiations of 4.5 it gives an isomorphism R_{x,sx}^{+} →^{∼} R_{x}. By 5.8 thus the restriction onto R^{s} ⊗ R of Γ_{≥i−0,5}B/Γ_{≥i+0,5}B is isomorphic to a direct sum of copies of certain R_{x}[ν] with x > sx and l(x) = i, and these occur with multiplicity (B : R_{x}[ν])+(B : R_{sx}[ν]). If we now tensor with R⊗_{R^{s}} and remember the short exact sequences R_{x}[−2] ↪ R ⊗_{R^{s}} R_{x} ↠ R_{sx}, we get the first claim and (always assuming x > sx) the formulas

\begin{aligned}(R ⊗_{R^{s}} B : R_{x}[ν]) & = & (B : R_{x}[ν + 2]) & + (B : R_{sx}[ν + 2]) \\ (R ⊗_{R^{s}} B : R_{sx}[ν]) & = & (B : R_{x}[ν]) & + (B : R_{sx}[ν]) \end{aligned}

This we can rewrite to

\begin{aligned}(θ_{s}B : Δ_{x}[ν]) & = & (B : Δ_{x}[ν + 1]) & + (B : Δ_{sx}[ν]) \\ (R[1] ⊗_{R^{s}} B : Δ_{sx}[ν]) & = & (B : Δ_{x}[ν]) & + (B : Δ_{sx}[ν − 1]) \end{aligned}

If we now only for this proof write an element H ∈ H in the form H = ∑_{x,ν}(H : v^{ν}T̃_{x}) v^{ν}T̃_{x}, then in the Hecke algebra we get simalarily

\begin{aligned}((T̃_{s} + v)H : v^{ν}T̃_{x}) & = & (H : v^{ν+1}T̃_{x}) & + (H : v^{ν}T̃_{sx}) \\ ((T̃_{s} + v)H : v^{ν}T̃_{sx}) & = & (H : v^{ν}T̃_{x}) & + (H : v^{ν−1}T̃_{sx}) \end{aligned}

This shows the second claim. To prove the last claim, we Remark first, that we may consider ⟨R⟩ via the ring homomorphism E of 1.10 also as a left Hmodul. The first part of the proposition then means, that ⟨F_{Δ}⟩ ⊂ ⟨R⟩ is an H-submodul ist and the second part, that the map h_{Δ} : ⟨F_{Δ}⟩ → H is a homomorphism of H-modules. Since E(1) = ⟨R_{e}⟩ belongs to F_{Δ}, the map E factorizes over ⟨F_{Δ}⟩, and since h_{Δ} ∘ E : H → H is a homomorphism of H-modules mapping one to one, this composition has to be the identity.

To show theorem 5.3 we als need a “dual” approach. More precisely we consider the filtration of our bimodules by the Γ_{≤i}B = Γ_{{x∈W|l(x)≤i}}B and define the category

F_{∇} ⊂ R

as the full subcategory of all graded bimodules B ∈ R such, that the support is in Gr(A) for some finite A ⊂ W and that the quotients are for all i isomorphic to finite direct sum of graded bimodules of the form R_{x}[ν] with l(x) = i and ν ∈ ℤ. In this context it is convenient and natural, to introduce

∇_{x} = R_{x}[l(x)].

For the multiplicitz of ∇_{x}[ν] in a direct sum decomposition of Γ_{≤l(x)}B/Γ_{≤l(x)−1}B we use the notation (B : ∇_{x}[ν]).

Remark that the multiplicities (B : ∇_{x}[ν]) and (B : Δ_{x}[ν]) concern subquotients for different filtrations. Even for B ∈ F_{Δ} ∩ F_{∇} the multiplicities (B : Δ_{x}[l(x) + ν]) and (B : ∇_{x}[−l(x) + ν]) are thus different in general. Analogously we have now

Proposition 5.9. Let s ∈ S be a simple reflection.

1. With B also R ⊗_{R^{s}} B belongs to F_{∇}.

2. When we define th map h_{∇} : F_{∇} → H by the rule B ↦ ∑_{x,μ}(B : ∇_{x}[μ]) v^{−μ}T̃_{x}, then for all s ∈ S the following diagrams are commutative

\begin{matrix}F_{∇} & ⟶ & H \\ ^{θ_{s}}↓ & & ↓^{(T̃_{s}+v)·} \\ F_{∇} & ⟶ & H \end{matrix}  \begin{matrix}F_{∇} & ⟶ & H \\ ^{[1]}↓ & & ↓_{v^{−1}·} \\ F_{∇} & ⟶ & H \end{matrix}

3. The composition d ∘ h_{∇} is a left inverse of E : H → ⟨F_{∇}⟩.

Proof. We consider the functor

D = Hom_{−R}( , R) : R → R -mod_{ℤ}- R,

where we provide our space of homomorphisms of right R-modules with the obvious ℤ-grading and let the right resp. left action on DB be defined via the right resp. left action on the bimodule B, in formulas (rf)(b) = f(rb) and (fr)(b) = f(br) for all b ∈ B, r ∈ R, f ∈ DB. I want to avoid a general discussion whether an object of R lands in R again under D. In any case we have DR_{x} ≅ R_{x} and D(M[ν]) = (DM)[−ν], our functor induces an equivalence of categories D : F_{∇} →^{∼} F_{Δ}^{opp} and evidently we have h_{∇} = h_{Δ} ∘D. Thus we only need to establish for all M ∈ F_{∇} an isomorphism θ_{s}DM ≅ Dθ_{s}M. For that we first show:

Proposition 5.10. 1. The functors from R^{s} -mod_{ℤ} to R -mod_{ℤ} with M ↦ R[2] ⊗_{R^{s}} M and M ↦ Hom_{R^{s}}(R, M) are naturally equivalent.

2. The functor R[1]⊗_{R^{s}} : R -mod_{ℤ} → R -mod_{ℤ} is selfadjoint.

Proof. Since R is free of finite rank over R^{s}, our Hom −functor can can also be written in the form

Hom_{R^{s}}(R, −) = Hom_{R^{s}}(R, R^{s}) ⊗_{R^{s}} −.

Now more precisely R is free over R^{s} with basis 1, α for α an equation of the reflecting hyperplane s(α) = −α. The dual basis of Hom_{R^{s}}(R, R^{s}) over R^{s} will be denoted 1^{∗}, α^{∗}. The multiplication by α ∈ R is given in this basis by αα^{∗} = 1^{∗}, α1^{∗} = α^{2}α^{∗}. The choice of α therefore defines an isomorphism of R-moduls

R[2] →^{∼} Hom_{R^{s}}(R, R^{s}), 1 ↦ α^{∗}, α ↦ 1^{∗},

and the first claim is established. The second claim follows, since our two functors from the first claim are up to a change of grading just the two adjoints of the restriction of R^{s}.

Thus we indeed get isomorphisms

θ_{s}(DM) ≅ R[1] ⊗_{R^{s}} DM
≅ Hom_{R^{s}}(R[1], Hom_{−R}(M, R))
≅ Hom_{−R}(θ_{s}M, R)
≅ D(θ_{s}M)

The third part of Proposition 5.9 now follows, since we have d(T̃_{s}+v) = T̃_{s}+v and thus the composition d ∘ h_{∇} ∘ E : H → H is a homomorphism of Hleftmodules with 1 ↦ 1.

Definition 5.11. We now define the category B ⊂ R as the category of all graded bimodules B ∈ R, whose class ⟨B⟩ lies in the image of our morphism E : H → ⟨R⟩, and call the objects of B our special bimodules.

Remark 5.12. Certainly B is stable under direct sums and shifts of grading. Much later in 6.16 we will be able to show that B is stable under direct summands. To obtain a criterium for when a bimodul B belongs to B, we consider for an arbitrary finite sequence \underline{s} = (r, …, t) of simple reflections in the Hecke algebra the element b(\underline{s}) = (T_{r} + 1) … (T_{t} + 1) and form the bimodule

B(\underline{s}) = R ⊗_{R^{r}} … R ⊗_{R^{t}} R.

Lemma 5.13. A graded bimodule B ∈ R is special iff there exist objects C, D ∈ R, which each are finite direct sums of objects of the form B(\underline{s})[n] and such, that we have

B ⊕ C ≅ D

Proof. Since ⟨B(\underline{s})[n]⟩ = E(v^{n}b(\underline{s})) the B(\underline{s})[n] are special. This shows our criterium to be sufficient. Since the v^{n}b(\underline{s}) generate H as an abelian group, it is necessary.

Remark 5.14. In particular E : H → ⟨R⟩ factorizes over the split Grothendieck group ⟨B⟩ of the additive category B and 5.7 with 5.9 imply B ⊂ F_{Δ} ∩ F_{∇}.

Theorem 5.15. For M ∈ F_{Δ}, N ∈ B and also for M ∈ B, N ∈ F_{∇} the space Hom(M, N) is graded free as a right R-module with rank

\underline{rk} Hom(M, N) = ∑_{x,ν,μ}(M : Δ_{x}[ν])(N : ∇_{x}[μ])v^{μ−ν}.

Proof. We only treat the case M ∈ F_{Δ}, N ∈ B, the other case is similar. Let i : H → H denote the antiinvolution i(v) = v, i(T̃_{x}) = T̃_{x^{−1}}. We consider the symmetric ℤ[v, v^{−1}]-bilinear pairing ⟨ , ⟩ : H × H → ℤ[v, v^{−1}], which to each pair (F, G) associates the coefficient of T̃_{e} = T_{e} in the representation of i(F)G as a linear combination of the T or equivalently the T̃. It can be described explicitly by ⟨T̃_{x}, T̃_{y}⟩ = δ_{xy}, see [Lus85],§1.4. The looked-for formula thus gets the form

\overline{\underline{rk}} Hom(M, N) = ⟨h_{Δ}M, h_{∇}N⟩.

By 5.13 we can restrict to the case N = B(\underline{s}). With 5.10 we can see, that our formula is correct for the pair (R ⊗_{R^{s}} M, N) iff it is correct for the pair (M, R⊗_{R^{s}} N). Without any difficulties we also see, that our formula is correct for (M[1], N) if and only if it is correct for (M, N[−1]). Using this we can even restrict to the case N = R_{e} which is obvious.

Korollar 5.16. Let B ∈ B be a special bimodule. Then we have

h_{Δ}(B) = ∑_{x∈W} \overline{\underline{rk}} Hom(B, R_{x}) T_{x},
h_{∇}(B) = ∑_{x∈W} \overline{\underline{rk}} Hom(R_{x}, B) T_{x}.

Proof. By the preceding 5.15 we have

\underline{rk} Hom(B, R_{x}) = \underline{rk} Hom(B, ∇_{x}[−l(x)])
= ∑_{ν}(B : Δ_{x}[ν])v^{−l(x)−ν}
\underline{rk} Hom(R_{x}, B) = \underline{rk} Hom(Δ_{x}[l(x)], B)
= ∑_{μ}(B : ∇_{x}[μ])v^{−l(x)+μ}

and the corollary follows from the definitions.

6 Classification of indecomposable special bimodules

Notation 6.1. We work with a reflection faithful representation over an infinite field. For B ∈ B and y ∈ W we abbreviate Γ_{≤y}B/Γ_{<y}B = Γ_{y}^{≤}B, and also Γ_{≥y}B/Γ_{>y}B = Γ_{y}^{≥}B and B/Γ_{≠y}B = Γ^{y}B.

Remark 6.2. Given a bimodule with a finite filtration by subbimodules with subquotient R_{x} the support of any section s ∈ B is an union of graphs of elements of our Coxete group. To show this we may without restriction assume supp s irreducible. But any nonzero element s ∈ B gives a nonzero element s̄ in a suitable subquotient of the Δ-flag and we deduce first supp s ⊃ supp s̄ = Gr(x) for suitable x ∈ W and then by the irreducibility of supp s even supp s = Gr(x). In particular we have Γ_{y}B ∩Γ_{≠y}B = 0 and the embeddings Γ_{y}B ↪ Γ_{y}^{≤}B and Γ_{y}^{≥}B ↪ Γ^{y}B. Under the same assumptions on B also the multiplication from the right with a nonzero element p ∈ R doesn’t change the support. We thus have (Γ_{A}B)p = Γ_{A}(Bp) and then (Γ_{y}^{≥}B)p ≅ Γ_{y}^{≥}(Bp) and the like. We use these identifications to omit all brackets in this kind of expression.

Lemma 6.3. Let B ∈ F_{∇} be given and let y_{0}, y_{1}, y_{2}, … be an enumeration of the elements of W such that elements bigger in the Bruhat order have always a bigger index. Let C(k) = {y_{0}, …, y_{k}} denote the set of k elements of our list and let y_{k} = y. Then the obvious map induces an isomorphism

Γ_{≤y}B/Γ_{<y}B →^{∼} Γ_{C(k)}B/Γ_{C(k−1)}B,

both sides are direct sums of objects of the form ∇_{y}[ν], and a ∇_{y}[ν] appears in this quotient with multiplicity (B : ∇_{y}[ν]) as a direct summand. Analogous results hold for F_{Δ}.

Proof. If z_{0}, z_{1}, z_{2}, … is an enumeration of the elements of W such, that the sequence l(z_{0}), l(z_{1}), l(z_{2}), … of the lengths is increasing, and if we put A(j) = {z_{0}, z_{1}, …, z_{j}}, then the Γ_{A(j)}B are a refinement of our filtration Γ_{≤i}B and Γ_{A(j)}B/Γ_{A(j−1)}B is a sum of (B : ∇_{z_{j}}[ν]) copies of certain ∇_{z_{j}}[ν]. Between any two enumerations of elements of W such, that Bruhat-bigger elements come later, we always can go back and forth in a finite number of steps each of which just exchanges two neighbouring incomparable elements. Since incomparable elements connot differ by a reflection, in any of these steps there is not Ext^{1} between the corresponding subquotients, thus the filtration before and after each of these steps lead to the same subquotients. The subquotient Γ_{C(k)}B/Γ_{C(k−1)}B with y_{k} = z_{j} thus is isomorphic to Γ_{A(j)}B/Γ_{A(j−1)}B, and thus Γ_{≤y}B/Γ_{<y}B is a sum of translated copies of ∇_{y}, and ∇_{y}[ν] appear (B : ∇_{y}[ν]) times.

Proposition 6.4. Given B ∈ B and y ∈ W our Γ_{y}^{≤}B, Γ_{y}^{≥}B and Γ^{y}B and Γ_{y}B are graded free right R-modules, on which the R ⊗ R-action factors through R_{y}.

Proof. For the first two modules of our list this easily follows from 6.3. For the last we then deduce from the embedding Γ_{y}B ↪ Γ_{y}^{≤}B from 6.2, that the action of R ⊗ R factors over R_{y}, so that the evaluation at 1_{y} gives an isomorphism Hom(R_{y}, B) →^{∼} Γ_{y}B, which by 5.15 shows Γ_{y}B is free. The claim concerning Γ^{y}B finally for y = e follows from B ∈ F_{Δ}. In general we Remark, the by definition of the category B and 5.7 we even get B ⊗_{R} R_{z} ∈ F_{Δ} for all z ∈ W. If we apply this to z = y^{−1}, we get the general case.

Notation 6.5. Let T ⊂ W the set of all reflections, and for t ∈ T let α_{t} ∈ V^{∗} be an equation of the reflecting hyperplane ker α_{t} = V^{t}. The α_{t} are unique up to scalar, we fix them for the rest of this section arbitrarily. For y ∈ W we now consider in R the element

p_{y} = ∏_{t∈T, yt<y} α_{t}.

Satz 6.6. Let y ∈ W and B ∈ B. Then the obvious morphisms Γ_{y}B → Γ_{y}^{≤}B and Γ_{y}^{≥}B → Γ^{y}B induce isomorphisms

\begin{aligned}Γ_{y}B & →^{∼} & Γ_{y}^{≤}Bp_{y}, \\ Γ_{y}^{≥}B & →^{∼} & Γ^{y}Bp_{y}. \end{aligned}

Remark 6.7. The proof will only be given following 6.10. In [Soe92] it is shown in case |W| < ∞ that there exists c_{y} ∈ R ⊗ R of degree 2l(y), vanishing on Gr(x) for x < y, but not vanishing on Gr(y) itself. Then necessarily c_{y} coincides on Gr(y) up to a scalar with 1 ⊗ p_{y} and we get an up to this scalar commutative diagram of isomorphisms

Γ_{y}^{≤}B →^{c_{y}} Γ_{y}B
·py ↓  ∥
Γ_{y}^{≤}Bp_{y} ← Γ_{y}B

descriping the inverse of the first isomorphism claimed in the theorem somewhat more explicitely. Similarly we also get an up to this scalar commutative diagram with isomorphisms in the leftmost square

Γ_{≠y}B ↪ B ↠ Γ^{y}B →^{∼} Γ^{y}Bp_{y}
↓ ↓ ↓ ↓
Γ_{≰y}B ↪ Γ_{≮y}B ↠ coker ≅ Γ_{y}^{≥}B

where all verticals up to the rightmost vertical are multiplications by c_{y}, the right upper horizontal is multiplication with p_{y} and for the last lower horizontal we use an analogon of 6.3 to get a short exact sequence Γ_{≰y}B ↪ Γ_{≮y}B ↠ Γ_{y}^{≥}B. In this way the inverse of the second isomorphism claimed in the theorem can be described somewhat more explicitely. Remark 6.8. I would like to know, whether for every y of an infinite Coxeter group the function c_{y} : Gr(≤ y) → ℂ is regular, which vanishes on Gr(x) for x < y and on Gr(y) is given by 1 ⊗ p_{y}. Notation 6.9. For a reflection t ∈ W let R_{(t)} denote the localization of R at all homogenous function on V, which do not vanish identically on the reflecting hyperplane V^{t}.

Lemma 6.10. For any special bimodule B ∈ B its localization B ⊗_{R} R_{(t)} in R -mod_{ℤ}- R_{(t)} is isomorphic to a direct summand in a finite direct sum of copies of R_{y,yt} ⊗_{R} R_{(t)} and R_{y} ⊗_{R} R_{(t)} for y ∈ W mit y < yt.

Remark 6.11. The screwed-up formulation helps to avoid a discussion of Krull-Schmid in this context.

Proof. We prove this by induction using the inductive definition of the objects of B. It is sufficient to show that for any simple reflection s ∈ S both R-R_{(t)}- bimodules R ⊗_{R^{s}} R_{y} ⊗_{R} R_{(t)} and R ⊗_{R^{s}} R_{y,yt} ⊗_{R} R_{(t)} admit a decomposition into bimodules of the same type. But R_{y} ⊗_{R} R_{(t)} and R_{x} ⊗_{R} R_{(t)} can only extend in R -mod_{ℤ}- R_{(t)}, if we have y = xt. Indeed after 5.8 in R -mod_{ℤ}- R there are only extensions if we have y = xr for a reflection r, and these extensions die under multiplication from the right with any equation of the reflecting hyperplane of r.

To finish the first case we thus only have to show, that sy ≠ yt implies syt > sy. This however follows from Deodhar’s property Z or alternatively from the geometrix insight, that neighbouring alcoves are separated just by one wall. In the second case there is nothing to do in case sy = yt, in this case we even have R ⊗_{R^{s}} R_{y,yt} ≅ R_{y,yt} ⊕ R_{y,yt}[−2] by 4.5.

If finally we have sy ≠ yt, the short exact sequence of bimodules R[−2] ↪ R ⊗_{R^{s}} R ↠ R_{s} leads to a short exact sequence

R_{y,yt}[−2] ↪ R ⊗_{R^{s}} R_{y,yt} ↠ R_{sy,syt}

and this sequences has to split after applying ⊗_{R}R_{(t)}, since all extensions between the bimodules R_{y}, R_{yt} as submodules and R_{sy}, R_{syt} as quotients split after we apply ⊗_{R}R_{(t)}.

Proof of 6.6. The cokernel of Γ_{y}B ↪ Γ_{≤y}B contains no elements with support Gr(y). Thus the restriction gives an embedding

Hom(Γ_{y}^{≤}B, R_{y}) = Hom(Γ_{≤y}B, R_{y}) ↪ Hom(Γ_{y}B, R_{y}).

We first show, that this embedding even lands in Hom(Γ_{y}B, R_{y}p_{y}). Indeed, if for a reflection t ∈ T with yt < y we pass by ⊗_{R}R_{(t)} to the localizations and decompose B as in lemma 6.10, then only the summands of the form R_{y,yt} ⊗_{R} R_{(t)} contribute to the two Hom-spaces we want to compare and we deduce, that our embedding lands in Hom(Γ_{y}B, R_{y}α_{t} ⊗_{R} R_{(t)}), since this is true when we replace B by R_{y,yt}. Since this holds for all t with yt < y, and since the intersection of all these R_{y}α_{t} ⊗_{R} R_{(t)} with R_{y} is just R_{y}p_{y}, we get the looked-for factorization

Hom(Γ_{y}^{≤}B, R_{y}) = Hom(Γ_{≤y}B, R_{y}) ↪ Hom(Γ_{y}B, R_{y}p_{y}).

Since Γ_{y}^{≤}B is by 6.4 a free R_{y}-module, we further see, that the embedding Γ_{y}B ↪ Γ_{y}^{≤}B lands in Γ_{y}^{≤}Bp_{y}. Now we show by comparing dimensions, that this embedding is in fact an isomorphism. By Lemma 6.3 we get already

\underline{rk}Γ_{y}^{≤}B = ∑(B : ∇_{y}[ν])v^{ν+l(y)}.

On the other hand by 6.2 the action of R ⊗ R on Γ_{y}B factorizes over R_{y} and we thus get Γ_{y}B = Hom(R_{y}, B) = Hom(Δ_{y}[l(y)], B). This however is by 5.15 a free right R-module of rank \underline{rk}Γ_{y}B = ∑(B : ∇_{y}[μ])v^{μ−l(y)}. But since p_{y} lives in degree 2l(y), this coincides with the rank of Γ_{y}^{≤}Bp_{y} and we get the first isomorphism of our theorem.

To deduce the second isomorphism we Remark, that dually the cokernel of the embedding Γ_{≥y}B ↪ B contains no elements with support in Gr(y). Thus the restriction induces an embedding

Hom(B, R_{y}) ↪ Hom(Γ_{≥y}B, R_{y}) = Hom(Γ_{y}^{≥}B, R_{y}).

We first show now as before with Lemma 6.10 on the local structure of the bimodules from B, that this embedding even lands in the subspace Hom(Γ_{≥y}B, R_{y}p_{y}). Then comparing the dimensions of the homogeneous parts we see that this embedding is in fact an isomorphism

Hom(B, R_{y}) →^{∼} Hom(Γ_{y}^{≥}B, R_{y}p_{y}).

From there we get either comparing dimensions or dualizing with 6.4 the second claim of the theorem.

comment 6.12. Remark 6.13. F”ur A ⊂ W eine Teilmenge und M einen R-Bimodul setzen wir ganz allgemein Γ^{A}M = M/Γ_{W−A}M. F”ur M torsionsfrei als R-Rechtsmodul sind auch die Γ^{A}M torionsfreie R-Rechtsmoduln, in der Tat gilt dann n”amlich

Γ_{W−A}M = M ∩ Γ_{W−A}(M ⊗_{R} Quot R)

Remark 6.14. We will soon be able to show that B is stable under the formation of direct summands. However as long as we don’t know this yet, we set add B to be the category of all ℤ-graded R-bimodules occuring as direct summands of special bimodules. Now we generalize 5.15 to add B and show:

Lemma 6.15. For M ∈ F_{Δ}, N ∈ add B and also for M ∈ add B, N ∈ F_{∇} the space Hom(M, N) is graded free as a right R-module and has the rank

\underline{rk} Hom(M, N) = ∑_{x,ν,μ}(M : Δ_{x}[ν])(N : ∇_{x}[μ])v^{μ−ν}.

Proof. We start with the first case. To begin with we Remark that the isomorphism Γ_{y}N →^{∼} Γ_{y}^{≤}Np_{y} of Satz 6.6 also holds for all N ∈ add B. From there we get the claim for M = Δ_{y} and N ∈ add B, since we have

\underline{rk} Hom(Δ_{y}, N) = \underline{rk}Γ_{y}N[l(y)]
= ∑_{μ}(Γ_{y}N[l(y)] : ∇_{y}[μ])v^{μ+l(y)}
= ∑_{μ}(Γ_{y}^{≤}N[−l(y)] : ∇_{y}[μ])v^{μ+l(y)}
= ∑_{μ}(N : ∇_{y}[μ])v^{μ}.

In the general case we argue by induction on the length of a Δ-flag of M. Indeed let x ∈ W be an element of maximal length with (M : Δ_{x}[ν]) ≠ 0 for some ν ∈ ℤ. By maximality of x we have a short exact sequence

Γ_{x}M ↪ M ↠ coker

with Γ_{x}M and coker in F_{Δ} and

(M : Δ_{y}[ν]) = (Γ_{x}M : Δ_{y}[ν]) + (coker : Δ_{y}[ν])

for all y ∈ W and ν ∈ ℤ. Certainly on the right side of this equation the first summand is zero in case y ≠ x and the second in case y = x. But in any case our dimension formulas 5.15 given for all N ∈ B a short exact sequence

Hom(coker, N) ↪ Hom(M, N) ↠ Hom(Γ_{x}M, N).

This sequences then has to be exact for all N ∈ add B as well and thus induction on the length of a Δ-flag of M reversly shows the dimension formulas claimed for all N ∈ add B.

The second case is treated analogously. That 6.6 also holds for all B ∈ add B gives the cases N = ∇_{y}. In general we argue on the length of a ∇-flag of N, take an element x of maximal length with (N : ∇_{x}[ν]) ≠ 0 for at least one ν ∈ ℤ, get by maximality of x a short exact sequence

ker ↪ N ↠ Γ^{x}N

with Γ^{x}N and ker in F_{∇} and

(N : ∇_{y}[ν]) = (ker : ∇_{y}[ν]) + (Γ^{x}N : ∇_{y}[ν])

for all y ∈ W and ν ∈ ℤ, and again our dimension formulas 6.15 for all M ∈ B a short exact sequence

Hom(M, ker) ↪ Hom(M, N) ↠ Hom(M, Γ^{x}N),

which has to be exact for all M ∈ add B and permits the looked-for inductive argument.

Satz 6.16. 1. For all x ∈ W there is up to isomorphism a unique indecomposable special bimodule B_{x} ∈ B with suopport in Gr(≤ x) and (B_{x} : Δ_{x}[ν]) = 1 for ν = 0 and zero for ν ≠ 0.

2. The map (x, ν) ↦ B_{x}[ν] defines a bijection

W × ℤ →^{∼} \{ \begin{matrix}indecomposable objects of B, \\ up to isomorphism \end{matrix} \}

3. The bimodules B_{x} are selfdual, in formulas DB_{x} ≅ B_{x}.

4. The category B is stable under the formation of direct summands, in formulas B = add B.

Proof. We first show the first three points for add B instead of B. Let x = st … r be a reduced word. In R ⊗_{R^{s}} R ⊗_{R^{t}} … R ⊗_{R^{r}} R[l(x)], which by 5.7 up to indecomposability possesses all properties we ask of B_{x}, we choose the indecomposable summand B_{x} with (B_{x} : Δ_{x}) = 1 and thus already get a possible B_{x} ∈ add B, which even has to be selfdual as the unique direct summand of a selfdual bimodule, whose support contains Gr(x).

Now it only remains to show that except these B_{x} and their translates B_{x}[ν] there are no indecomposables in add B. But let M ∈ add B and let x ∈ W be an element of maximal length (M : Δ_{x}[ν]) ≠ 0 for some ν ∈ ℤ. We show that B_{x}[ν] is a direct summand of M. By maximality of x first by 6.15 or more precisely its proof the obvious maps give surjections

Hom(M, B_{x}) ↠ Hom(Γ_{x}M, B_{x}) = Hom(Γ_{x}M, Γ_{x}B_{x}),
Hom(B_{x}, M) ↠ Hom(B_{x}, Γ^{x}M) = Hom(Γ^{x}B_{x}, Γ^{x}M).

The we Remark, that by maximality of x with 6.3 we even get Γ_{x}^{≤}M →^{∼} Γ^{x}M, so that by 6.6 the embedding Γ_{x}M ↪ Γ^{x}M gives isomorphisms

i : Γ_{x}M →^{∼} Γ^{x}Mp_{x} and i : Γ_{x}B_{x} →^{∼} Γ^{x}B_{x}p_{x}

. Given m ∈ Γ^{x}M homogeneous we now find f ∈ Hom(Γ^{x}B_{x}, Γ^{x}M) homogeneous with f(b) = m for b ∈ Γ^{x}B_{x} a homogeneous generator. If we choose m even so, that it is part of a homogeneous basis of the graded free R-module Γ^{x}M, then we also get g ∈ Hom(Γ_{x}M, Γ_{x}B_{x}) with g(i^{−1}(mp_{x})) = i^{−1}(bp_{x}). For homogeneous lifts f̃: B_{x} → M and g̃ : M → B_{x} of f and g we deduce, that g̃ ∘ f̃: B_{x} → B_{x} induces the identity on Γ^{x}B_{x}. Thus g̃ ∘ f̃ can not be nilpotent, thus it is an isomorphism sicne B_{x} is indecomposable. Finally by induction on the length of x, that all B_{x} already belong to B: Indeed our long tensor product from above decomposes into a summand B_{x} along with other summands, which by their support necessarily are of the form B_{y}[ν] with l(y) < l(x). This shows the other claim.

Remark 6.17. If we find B ∈ B with ⟨B⟩ = E(C′_{x}), then we can deduce B ≅ B_{x} and thus ⟨B_{x}⟩ = E(C′_{x}). Indeed such a B is necessarily indecomposable, because calculating the dimension of the homogeneous components of its endomorphism ring we get nothing in negative degrees and but the ground field in degree zero. Remark 6.18. For all x ∈ W at least h_{Δ}(B_{x}) = h_{∇}(B_{x}) is a selfdual element of H of the form C′_{x} + ∑_{y<x} h_{y}C′_{y} for suitable selfdual h_{y} ∈ ℤ[v, v^{−1}]. Let indeed x = st … r be a reduced word. Then

R ⊗_{R^{s}} R ⊗_{R^{t}} … R ⊗_{R^{r}} R[l(x)]

is selfdual and thus decomposes into B_{x} along with summands of the form B_{y} and B_{y}[ν] ⊕ B_{y}[−ν] for y < x. We deduce h_{Δ}(B_{x}) = h_{∇}(B_{x}) is selfdual by induction on the Bruhat order, and then we obviously get

h_{Δ}(B_{x}) = h_{∇}(B_{x}) = C′_{x} + ∑_{y<x} h_{y}C′_{y}

for suitable selfdual h_{y} ∈ ℤ[v, v^{−1}]. However I cannot show that these h_{y} have nonnegative coefficients.

7 Discussion of the main conjecture

A natural approach towards the proofs of conjecture 1.13 seems to me induction on the length of x. Here I formulate some variants of this approach as an omnibus Lemma. Let \underline{Hom}(B, B′) denote the (R ⊗ R)-homomorphism of ℤ-degree zero between two ℤ-graded R-bimodules B, B′.

Lemma 7.1. Let x ∈ W and s ∈ S be given with sx > x. We further assume that ⟨B_{y}⟩ = E(C′_{y}) for all y ∈ W with l(y) ≤ l(x). Then the following are equivalent:

1. We have ⟨B_{sx}⟩ = E(C′_{sx});

2. For all y with l(y) ≤ l(x) the composition of morphisms \underline{Hom}(B_{y}, θ_{s}B_{x})× \underline{Hom}(θ_{s}B_{x}, B_{y}) → \underline{End}B_{y} = k defines a nondegenerate pairing of kvectorspaces.

3. For all y with l(y) ≤ l(x) the map Γ_{y}θ_{s}B_{x} ↪ Γ^{y}θ_{s}B_{x} defines an inclusion (Γ_{y}θ_{s}B_{x})_{l(y)} ↪ Γ^{y}θ_{s}B_{x}p_{y} ⊗_{R} k.

4. For all y with l(y) ≤ l(x) definiert die offensichtliche Abbildung eine Inklusion (Γ_{y}θ_{s}B_{x})_{l(y)} ↪ Γ_{y}^{≥}θ_{s}B_{x} ⊗_{R} k.

5. For all y with l(y) ≤ l(x) the obvious map defines an inclusion (Γ_{y}^{≤}θ_{s}B_{x})_{−l(y)} ↪ Γ^{y}θ_{s}B_{x} ⊗_{R} k.

Proof. 1 ⇒ 2 : Left to the reader. 2 ⇒ 1 : In the notation of [Soe97] we can write

C′_{s}C′_{x} = C′_{sx} + ∑_{y<sx} m_{y}C′_{y}

for certain m_{y} ∈ ℤ and we have E(C′_{s}C′_{x}) = ⟨θ_{s}B_{x}⟩. By 5.15 and the induction hypothesis we thus get

dim \underline{Hom}(B_{y}, θ_{s}B_{x}) = m_{y} = dim \underline{Hom}(θ_{s}B_{x}, B_{y}).

If our pairing is not degenerate, we can for every y split off just m_{y} copies of B_{y} from θ_{s}B_{x} and thus get B ∈ B with ⟨B⟩ = E(C′_{sx}). By Remark 6.17 we then have B ≅ B_{sx}.

3 ⇔ 2 : Comparing dimensions we see, that on the homomorphisms of degree zero the obvious maps are isomorphisms

\underline{Hom}(B_{y}, θ_{s}B_{x}) →^{∼} \underline{Hom}(Δ_{y}, θ_{s}B_{x})
\underline{Hom}(θ_{s}B_{x}, B_{y}) →^{∼} \underline{Hom}(θ_{s}B_{x}, ∇_{y}).

By 6.6 we know, that the composition of morphisms of degree zero Δ_{y} → B_{y} → ∇_{y} is up to a scalar right multiplication by p_{y} and that for all B ∈ B any composition Δ_{y} → B → ∇_{y} lands in ∇_{y}p_{y}. Thus we can rewrite our pairing of part 2 as a pairing

\underline{Hom}(Δ_{y}, θ_{s}B_{x}) × \underline{Hom}(θ_{s}B_{x}, ∇_{y}) → p_{y}k ⊂ \underline{Hom}(Δ_{y}, ∇_{y}).

But now such a pairing between two spaces is nothing else but a map from one to the

\underline{Hom}(Δ_{y}, θ_{s}B_{x}) = (Γ_{y}θ_{s}B_{x})_{l(y)}
\underline{Hom}(θ_{s}B_{x}, ∇_{y}) = \underline{Hom}(Γ^{y}θ_{s}B_{x}, ∇_{y})
= \underline{Hom}(Γ^{y}θ_{s}B_{x}, k[l(y)])
= \underline{Hom}((Γ^{y}θ_{s}B_{x}) ⊗_{R} k, k[l(y)])
\underline{Hom}(θ_{s}B_{x}, ∇_{y})^{∗} = ((Γ^{y}θ_{s}B_{x}) ⊗_{R} k)_{l(y)}

and thus get 3 ⇔ 2. the variants 4 and 5 are reformulations of 3 using 6.6.

Remark 7.2. It seems natural, to investigate the embeddings

Γ_{y}B_{x} ↪ Γ^{y}B_{x}

more closely. If we define a surjection R ↠ k[T] as the restriction on the line kρ, then I expect, that the cokernel of (Γ_{y}B_{x}) ⊗_{R} k[T] ↪ (Γ^{y}B_{x}) ⊗_{R} k[T] is a direct sum of copies of (k[T]/(T^{i+1}))[i], where we have again set deg T = 2. This should even hold more generally for all lines, which are not contained in a reflecting hyperplane.

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