A TWO-SIDED q-ANALOGUE OF THE COXETER COMPLEX

MARKUS LINCKELMANN AND SIBYLLE SCHROLL

Abstract. We define a complex of bimodules over the Iwahori-Hecke algebra associated to a finite Coxeter group, calculate its cohomology and show that it induces a derived equivalence over its module category extending the Morita equivalence given by a certain algebra automorphism. We show further that when tensored with the index representation this complex becomes isomorphic to the one-sided q-analogue of the Coxeter complex previously defined by V. Deodhar [Deo] and A. Mathas [Ma].

Let (W, S) be a finite Coxeter system, let R be a unitary commutative ring and let q ∈ R^{×}. Let H be the Iwahori-Hecke algebra of (W, S) over R with parameter q; that is, H is the free R-module with basis {T_{w}|w ∈ W } and multiplication given by

T_{s}T_{w} = \{ \begin{matrix}T_{sw} & if ℓ(sw) > ℓ(w) \\ qT_{sw} + (q − 1)T_{w} & if ℓ(sw) < ℓ(w) \end{matrix}

for s ∈ S, w ∈ W and where ℓ is the usual length function on W. We fix a total order S = {s_{i} | 1 ≤ i ≤ |S|} on the set S. For every subset I ⊂ S let W_{I} be the subgroup of W generated by I (with the convention W_{∅} = 1) and let H_{I} be the parabolic subalgebra of H having as R-basis the set {T_{w} | w ∈ W_{I}}; this algebra is isomorphic in the obvious way to the Iwahori-Hecke algebra of (W_{I}, I) over R with parameter q. The H-module R endowed with the action T_{w}.λ = q^{ℓ(w)}λ for w ∈ W and λ ∈ R is called the index representation of H.

Given an R-algebra A and an algebra automorphism α of A, we write _{α}A for the A-A-bimodule with the regular right action given by multiplication in A and the left action given by a.a′ = α(a)a′, for a, a′ ∈ A. We denote by A-mod the category of finitely generated left A-modules and by D^{♭}(A-mod) the bounded derived category of A-mod.

Definition. Let D be the cochain complex of H-H-bimodules concentrated in degrees 0 to |S| defined by setting

D^{n} = ⨁_{|I|=n^{I⊂S}} H ⊗_{H_{I}} H

for 0 ≤ n ≤ |S|, with differential d^{n} : D^{n} → D^{n+1} given, for any integer n such that 0 ≤ n < |S|, by

d^{n} = ∑_{|I|=n^{I⊂S}} ∑_{s_{i}∈S\I}(−1)^{i}π_{I,I∪{s_{i}}}

where π_{I,J} : H ⊗_{H_{I}} H → H ⊗_{H_{J}} H is the canonical surjective homomorphism of H-H-bimodules, for any subsets I, J of S such that I ⊂ J.

Equivalently, D is the cochain complex associated with the coefficient system on the simplicial complex of subsets of S sending a subset I of S to H⊗_{H_{I}} H and the inclusion I ⊂ J to the canonical map π_{I,J}. One checks easily that d^{n+1} ∘ d^{n} = 0. Note that D^{0} = H ⊗_{R} H and D^{|S|} = H.

Date: May 31, 2004.

Theorem 1. Let α be the algebra automorphism of H sending T_{w} to (−q)^{ℓ(w)}(T_{w^{−1}})^{−1} for all w ∈ W. The complex D has cohomology concentrated in degree zero, isomorphic to the H-H-bimodule _{α}H. In particular, the functor D ⊗_{H} − : D^{♭}(H − mod) → D^{♭}(H − mod) is an equivalence of categories which extends the Morita equivalence on H − mod given by the automorphism α.

Corollary. For any H-module M the complex D ⊗_{H} M has cohomology concentrated in degree 0 isomorphic to _{α}M.

An equivalent result to the above Theorem has independently been obtained by Parshall and Scott [PS, 3.4]. Following [Iw, 5.4] the automorphism α of H has first been considered by O. Goldmann. This automorphism is the composition of the two ring automorphisms σ and ι of H which both send q to q^{−1} and which send T_{w} to (−1)^{ℓ(w)}q^{−ℓ(w)}T_{w} and (T_{w^{−1}})^{−1}, respectively, for any w ∈ W. Furthermore, tensoring D by the index representation yields the q-analogue C_{q} of the Coxeter complex defined by Deodhar in [Deo] and by Mathas in [Ma].

Theorem 2. There is an isomorphism of complexes of left H-modules D ⊗_{H} R ≅ C_{q}.

There are several reasons to think that the derived equivalence in Theorem 1 should in fact be a homotopy equivalence. For q = 1 the complex C_{q} is isomorphic to the Coxeter complex C of RW -modules, and the complex D is isomorphic to the “diagonally induced” Coxeter complex of RW -RW -bimodules Ind_{ΔW}^{W ×W} (C) (see e.g. [LS]), of which it is known that it induces a homotopy equivalence (consequence of [Ri, §8]). Furthermore, the complex D and the complex X inducing the Alvis-Curtis duality defined for example by M.Cabanes and J.Rickard in [CaRi] are shown in [Sch, 4.5.2] to be related via the Schur functor. Since one expects X to induce a homotopy equivalence [CaRi, 5.2] one should certainly also expect D to induce a homotopy equivalence.

Conjecture. The functor D ⊗_{H} − : K^{♭}(H − mod) → K^{♭}(H − mod) is an equivalence of categories.

For J a subset of S, we denote by ^{J}W the full set of left coset representatives of W/W_{J} given by all w in W such that for all s in J, ℓ(ws) > ℓ(w). In particular, the elements of ^{J}W are the unique elements of minimal length in their coset. Therefore whenever we have for w ∈ W that w = xv with x ∈ ^{J}W and v ∈ W_{J} then ℓ(w) = ℓ(x) + ℓ(v). Let w_{0} be the longest element in W. In order to prove Theorem 1 we need the following lemma :

Lemma. For any w ∈ W we have T_{w_{0}}T_{w}T_{w_{0}}^{−1} = T_{w_{0}ww_{0}^{−1}}; in particular the R-linear map on H_{W} sending T_{w} to T_{w_{0}ww_{0}^{−1}} is an inner algebra automorphism of H_{W}.

Proof. We will show that T_{w_{0}}T_{w} = T_{w_{0}ww_{0}^{−1}}T_{w_{0}} by induction over ℓ(w). If s ∈ S then T_{w_{0}}T_{s} = qT_{w_{0}s} +(q −1)T_{w_{0}} = qT_{(w_{0}sw_{0}^{−1})w_{0}} +(q −1)T_{w_{0}} = T_{w_{0}sw_{0}^{−1}}T_{w_{0}}. If w = w′s with s ∈ S and ℓ(w′) < ℓ(w) then by the above calculation and by induction we get T_{w_{0}}T_{w} = T_{w_{0}}T_{w′}T_{s} = T_{w_{0}w′w_{0}^{−1}}T_{w_{0}}T_{s} = T_{w_{0}w′w_{0}^{−1}}T_{w_{0}sw_{0}^{−1}}T_{w_{0}} = T_{w_{0}ww_{0}^{−1}}T_{w_{0}}. The last equality holds since ℓ(w_{0}ww_{0}^{−1}) = ℓ(w_{0}w′w_{0}^{−1}) + ℓ(w_{0}sw_{0}^{−1}). The result follows. □

Proof of Theorem 1. By shifting and renumbering the terms we view the augmented Coxeter complex C associated to the finite Coxeter group (W, S) as a cochain complex concentrated in degrees 0 to |S|. Therefore the degree n component C^{n} of C is equal to the direct sum of the permutation modules R[W/W_{I}], where I runs over the set of subsets of S such that |I| = n (see for example [Hu, 1.15]). In particular, the degree 0 component of C is the regular RW -module RW. It is well-known that C has cohomology concentrated in degree 0, isomorphic to the sign representation of W.

We denote by C̃ the subcomplex of C, which in all positive degrees coincides with C, and whose degree 0 term is the R-submodule of RW with basis W − {w_{0}}. Then C̃ is an acyclic complex of R-modules.

The complex C̃ is known to have a filtration by acyclic subcomplexes C̃_{(p)} defined for any positive integer p as follows: the degree n component of C̃_{(p)} is the direct sum, taken over all subsets I ⊂ S such that |I| = n, of the R-submodule of R[W/W_{I}] generated by the cosets wW_{I} such that ℓ(w) ≤ p.

Let I be a subset of S, the set {xW_{I}|x ∈ ^{I}W } is an R-basis of R[W/W_{I}], and recall that the set of tensors {T_{x} ⊗ T_{w}|x ∈ ^{I}W , w ∈ W } is an R-basis of H ⊗_{H_{I}} H. Thus there is a unique isomorphism of right H-modules

R[W/W_{I}] ⊗_{R} H ≅ H ⊗_{H_{I}} H

sending xW_{I} ⊗ T_{w} to T_{x} ⊗ T_{w}, where x ∈ ^{I}W and w ∈ W. Taking the direct sum over the set of subsets of S of these isomorphisms yields an isomorphism of graded right H-modules

η : C ⊗_{R} H ≅ D

This graded map is not in general compatible with the differentials, so this is not an isomorphism of complexes, in general. But if we set D̃ = η(C̃ ⊗_{R} H) and D̃_{(p)} = η(C̃_{(p)} ⊗_{R} H), it is nevertheless obvious that D̃ and D̃_{(p)} are subcomplexes of right H-modules of D. Then η induces graded isomorphisms between subsequent quotients on both sides,

(1)  (C̃_{(p)} ⊗_{R} H)/(C̃_{(p−1)} ⊗_{R} H) ≅ D̃_{(p)}/D̃_{(p−1)}

for any positive integer p. The crucial point here is to observe that this is actually now an isomorphism of complexes. To see the compatibility with the differentials of the quotient complexes, we apply exactly the argument Mathas used in formula (2.3) in [Ma] combined with Remark 2.8 in that paper : consider subsets I ⊂ J ⊂ S such that |I| + 1 = |J|. Let x ∈ ^{I}W such that ℓ(x) = p and let w ∈ W. We distinguish then two cases : if x ∈ ^{J}W then the isomorphism in (1) commutes with the induced differentials because in this case the element T_{x} ⊗ T_{w} is also a basis element of H ⊗_{H_{J}} H. If x is not in ^{J}W then x = yv for some y ∈ ^{J}W and some v ∈ W_{J}, which implies that ℓ(y) < ℓ(x) = p. Thus xW_{I} ⊗ T_{w} belongs to the subcomplex C̃_{(p−1)} ⊗_{R} H, and similarly, the image of xW_{I} ⊗ T_{w} in H ⊗_{H_{I}} H belongs to the subcomplex D̃_{(p−1)}.

This means that D̃ is filtered by subcomplexes such that subsequent quotients of this filtration are acyclic. By a standard long exact cohomology sequence argument it follows that the complex D̃ is acyclic.

The quotient complex D/D̃ is concentrated in degree 0, its degree 0 term is R-free and has as a basis the set {T_{w_{0}} ⊗ T_{w}|w ∈ W } and thus is isomorphic to H as a right H-module. Therefore, the cohomology of D is concentrated in degree 0 and is isomorphic to H as a right H-module.

In order to calculate the H-H-bimodule structure of H^{0}(D), let φ : H → H^{0}(D) be the isomorphism of right H-modules such that

φ(T_{w}) = T_{w_{0}} ⊗ T_{w} + ∑_{x∈W_{1}} T_{x} ⊗ h_{x}(w)

for some uniquely determined elements h_{x}(w) ∈ H, where W_{1} is the subset of W of all elements x such that ℓ(x) ≤ ℓ(w_{0}) − 1. Denote by W_{2} the subset of all x ∈ W such that ℓ(x) ≤ ℓ(w_{0}) − 2. We can rewrite φ(T_{w}) in terms of w_{0}, S and W_{2} :

(2)  φ(T_{w}) = T_{w_{0}} ⊗ T_{w} + ∑_{s∈S} T_{w_{0}s} ⊗ h_{w_{0}s}(w) + ∑_{x∈W_{2}} T_{x} ⊗ h_{x}(w)

Since the complex D is concentrated in degrees 0 to |S| and the cohomology of D is concentrated in degree 0, φ(T_{w}) is in the kernel of the differential d^{0}. This implies by the following calculation that h_{w_{0}s}(w) = −T_{s}T_{w} : let s ∈ S and denote by d_{s} the component of d^{0} whose image is in H ⊗_{H_{{s}}} H. We start by calculating d_{s}(φ(T_{w})) componentwise. The equality T_{w_{0}} = T_{w_{0}s}T_{s} implies for w ∈ W that

(3)  d_{s}(T_{w_{0}} ⊗ T_{w}) = T_{w_{0}s} ⊗ T_{s}T_{w}

And for t ∈ S we have

(4)  d_{s}(T_{w_{0}t} ⊗ h_{w_{0}t}(w)) = \{ \begin{matrix}T_{w_{0}s} ⊗ h_{w_{0}s}(w) & if & t = s \\ (qT_{w_{0}ts} + (q − 1)T_{w_{0}t}) ⊗ T_{s}^{−1}h_{w_{0}t}(w) & if & t ≠ s \end{matrix}

since for all t ∈ S such that t ≠ s we have ℓ(w_{0}ts) < ℓ(w_{0}t). Finally for x ∈ W_{2} and w ∈ W we obtain

d_{s}(T_{x} ⊗ h_{x}(w)) = \{ \begin{matrix}T_{xs} ⊗ T_{s}^{−1}h_{x}(w) & if & ℓ(xs) > ℓ(x) \\ (qT_{xs} + (q − 1)T_{x}) ⊗ T_{s}^{−1}h_{x}(w) & if & ℓ(xs) < ℓ(x) \end{matrix}

Therefore d_{s} (∑_{x∈W_{2}} T_{x} ⊗ h_{x}(w)) is a linear combination of elements of the form T_{y} ⊗ h_{y} such that ℓ(y) ≤ ℓ(w_{0}) − 2 or y = sw_{0}. This implies that d_{s}(φ(T_{w})) is a linear combination of elements of the form T_{y} ⊗ h_{y} where either ℓ(y) ≤ ℓ(w_{0}) − 2, y = sw_{0} or y = w_{0}s. But since sw_{0} ≠ w_{0}s for all s ∈ S and since d_{s}(φ(T_{w})) = 0 we obtain from (3) and the first line of (4) that h_{w_{0}s}(w) = −T_{s}T_{w}.

In order to calculate the left action of H on H^{0}(D) we make further use of the isomorphism φ. Let t ∈ S and w ∈ W. We denote the left action of an element T_{t} on an element φ(T_{w}) of H^{0}(D) by T_{t}.φ(T_{w}). More precisely, since we would like to calculate the left action of H on H^{0}(D) we have to consider φ^{−1}(T_{t}.φ(T_{w})). Thus if T_{t}.φ(T_{w}) = ∑_{w∈W} T_{w} ⊗ h_{w} we only consider the value of h_{w_{0}} corresponding to T_{w_{0}}. Using the decomposition in (2) we must calculate the different terms of

(5)  T_{t}.φ(T_{w}) = T_{t}.T_{w_{0}} ⊗ T_{w} + ∑_{s∈S} T_{t}.T_{w_{0}s} ⊗ h_{w_{0}s}(w) + ∑_{x∈W_{2}} T_{t}.T_{x} ⊗ h_{x}(w)

By definition of the multiplication in H we have

T_{t}.T_{w_{0}} ⊗ T_{w} = qT_{tw_{0}} ⊗ T_{w} + (q − 1)T_{w_{0}} ⊗ T_{w}

and only the term (q − 1)T_{w_{0}} ⊗ T_{w} is going to contribue. For the middle term of equation (5) we obtain for s ∈ S

T_{t}.T_{w_{0}s} ⊗ h_{w_{0}s}(w) = \{ \begin{matrix}T_{tw_{0}s} ⊗ h_{w_{0}s}(w) & if & ℓ(tw_{0}s) > ℓ(w_{0}s) \\ (qT_{tw_{0}s} + (q − 1)T_{w_{0}s}) ⊗ h_{w_{0}s}(w) & if & ℓ(tw_{0}s) < ℓ(w_{0}s) \end{matrix}

but ℓ(tw_{0}s) > ℓ(w_{0}s) implies that tw_{0}s = w_{0} and therefore T_{tw_{0}s} = T_{w_{0}}. Thus the only term we need to consider is the term T_{t}.T_{w_{0}t′} ⊗h_{w_{0}t′}(w) where t′ is the unique element in S such that w_{0}^{−1}tw_{0} = t′. Therefore t′ is the image of t by the automorphism of W given by conjugation with w_{0}. It now follows by the preceding paragraph that

T_{t}.T_{w_{0}s} ⊗ h_{w_{0}s}(w) = T_{w_{0}} ⊗ h_{w_{0}(w_{0}tw_{0}^{−1})}(w) = −T_{w_{0}} ⊗ T_{w_{0}tw_{0}^{−1}}T_{w}.

The elements of the form T_{t}.T_{x} ⊗ h_{x}(w) with x ∈ W_{2} of the third term in equation (5) are not going to contribute to the H-H-bimodule structure, since for all t ∈ S, w_{0} ∉ tW_{2}.

Finally we have obtained

φ^{−1}(T_{t}.φ(T_{w})) = ((q − 1)T_{1} − T_{w_{0}tw_{0}^{−1}})T_{w}

and since α(T_{t}) = −q(T_{t})^{−1} = (q − 1)T_{1} − T_{t} it follows from the Lemma that

φ^{−1}(T_{t}.φ(T_{w})) = T_{w_{0}}α(T_{t})(T_{w_{0}})^{−1}T_{w}.

That is, H^{0}(D) ≅ _{β}H as H-H-bimodule, where β is the algebra automorphism obtained by composing α with the inner automorphism given by conjugation by T_{w_{0}}. Thus _{β}H ≅ _{α}H and finally H^{0}(D) ≅ _{α}H.

We are now able to prove the last part of the theorem : since H is a symmetric algebra and all the terms of D are finitely generated and projective as left and right H-modules, all we need to show is that

D ⊗_{H} D^{∨} ≅ H

and

D^{∨} ⊗_{H} D ≅ H

in the derived category of H-H-bimodules, where D^{∨} denotes the R-linear dual of D. However, since the cohomology of D is concentrated in degree 0, where it is isomorphic to H, D is exact in every postive degree. Since the differential d^{|S|−1} is surjective onto H, its image is projective as a right H-module. Therefore the short exact sequences of right H-modules

0 → Ker(d^{k}) → D^{k} → Im(d^{k}) → 0

are split for all k > 0 . Thus the cycles Z^{∗}(D) and the boundaries B^{∗}(D) are projective as right H-modules. Therefore by the Künneth formula H^{∗}(D⊗_{H}D^{∨}) ≅ H^{∗}(D)⊗_{H}H^{∗}(D^{∨}). Since H^{0}(D) ≅ αH and H^{∗}(D^{∨}) = H^{0}(D^{∨}) ≅ (_{α}H)^{∨} ≅ H_{α} we obtain H^{∗}(D) ⊗_{H} H^{∗}(D^{∨}) ≅ _{α}H ⊗_{H} H_{α} ≅ _{α}H_{α}. This implies the first isomorphism since _{α}H_{α} ≅ H as H-H-bimodules. The second isomorphism is obtained in the same way by considering left instead of right H-modules. □

Proof of Theorem 2. Let

G = − ⊗_{H} R : (H ⊗_{R} H^{∘})-mod ⟶ H-mod

be the functor induced by tensoring with the index representation of H. Since for I ⊂ S the terms of D are direct sums of the H-H-bimodules H ⊗_{H_{I}} H and the terms of C_{q} are direct sums of Hmodules of the form H ⊗_{H_{I}} R, the terms of the complex G(D) = D ⊗_{H} R are isomorphic as left H-modules to the terms of C_{q}.

In order to show that the differentials coincide we we have to show that for I ⊂ J ⊂ S the maps G(π_{I,J}) induce the differential of C_{q}. But if T_{x} ⊗ T_{y} ∈ H ⊗_{H_{I}} H then its canonical image in H ⊗_{H_{I}} R is equal to T_{x} ⊗ q^{ℓ(y)} and similarly the canonical image of π_{I,J}(T_{x} ⊗ T_{y}) = T_{x} ⊗ T_{y} ∈ H ⊗_{H_{J}} H in H ⊗_{H_{J}} R is equal to G(π_{I,J})(T_{x} ⊗ q^{ℓ(y)}) = T_{x} ⊗ q^{ℓ(y)} and thus G(π_{I,J}) clearly induces the differential of C_{q}. □

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Markus Linckelmann Ohio State University, Math Tower 231 West 18th Avenue Columbus, Ohio 43210

E-mail : linckelm@math.ohio-state.edu

Sibylle Schroll Université Paris 7 Denis Diderot UFR de Mathématiques Case Postale 7012; 2, place Jussieu F-75251 Paris Cedex 05

E-mail : schroll@math.jussieu.fr