EXTENDING HECKE ENDOMORPHISM ALGEBRAS AT ROOTS OF

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Jan 26, 2015 - algebras can be stratified by means of a filtration defined in terms of the ... H-module defined to be the Z -linear dual of the left cell module ...
arXiv:1501.06481v1 [math.RT] 26 Jan 2015

EXTENDING HECKE ENDOMORPHISM ALGEBRAS AT ROOTS OF UNITY JIE DU, BRIAN J. PARSHALL, AND LEONARD L. SCOTT A BSTRACT. Hecke endomorphism algebras are endomorphism algebras over a Hecke algebra associated to a finite Weyl group W of certain q-permutation modules, the “tensor spaces.” Such a space may be defined for any W in terms of a direct sum of certain cyclic modules associated to parabolic subgroups. The associated algebras have important applications to the representations of finite groups of Lie type. In [6], it is proved that these algebras can be stratified by means of a filtration defined in terms of the subsets of the Coxeter generators. It was conjectured that by enlarging the “tensor space” the new resulting endomorphism algebra has a finer “standard” stratification in terms of left cells of the Coxeter group, with associated “strata” corresponding to two-sided cells of W . Using the work [8] on a rational double affine Hecke algebras (RDAHAs)—also known as rational Cherednik algebras—at a key point, we will prove the conjecture in the characteristic zero case at an eth root of unity, e 6= 2. We further prove that each of the new Hecke endomorphism algebras constructed in the paper is quasi-hereditary and that its representation category is equivalent, after a base change, to the category O associated to a corresponding RDAHA. We do not treat the e = 2 case, but expect the conjecture to be true there also (possibly not giving quasi-hereditary algebras, in general).

1. I NTRODUCTION Let G = {G(q)} be a family of finite groups of Lie type having irreducible (finite) Coxeter system (W, S) [3, (68.22)]. Let B(q) be a Borel subgroup of G(q). There are index parameters cs ∈ Z, s ∈ S, defined by [B(q) : s B(q) ∩ B(q)] = q cs ,

s ∈ S.

The generic Hecke algebra H over the ring Z = Z[t, t−1 ] of Laurent polynomials associated to G has basis Tw , w ∈ W , subject to relations ( Tsw , sw > w; (1.0.1) Ts Tw = 2cs t Tsw + (t2cs − 1)Tw , sw < w. We call H a Hecke algebra of Lie type over Z . It is related to the representation theory of the groups in G as follows: for any prime power q, let Hq = C ⊗Z H be the algebra √ G(q) obtained by the map Z → C, t 7→ q. Then Hq ∼ = EndG(q) (indB(q) C). Thus, the generic Hecke algebra H is the quantumization (in the sense of [4, §0.4]) of an infinite family of important endomorphism algebras. Research supported in part by the Australian Research Council and National Science Foundation. 1

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JIE DU, BRIAN J. PARSHALL, AND LEONARD L. SCOTT

Let Ω be the set of left Kazhdan-Lusztig cells for the Coxeter system (W, S). Thus, Ω has a quasi-poset structure ≤L . (A quasi-poset is a set with a transitive and reflexive relation [= preorder] defined on it.) For each ω ∈ Ω, there is a dual left cell module Sω . It is the right H-module defined to be the Z -linear dual of the left cell module defined by ω. In addition, for λ ⊆ S, the L induced modules xλ H have an increasing filtration with sections Sω , ω ∈ Ω. Let T = λ xλ H, and A := EndH (T ). For ω ∈ Ω, ∆(ω) := HomH (Sω , T ) ∈ A-mod. The following conjecture is worded to be the same as in [6, Conj. 2.5.2], but using Z[t, t−1 ] as Z , whereas the latter conjecture used Z[t2 , t−2 ].1 Conjecture 1.1. There exists a right H-module X such that the following statements hold: (1) X has an finite filtration with sections of the form Sω , ω ∈ Ω. + (2) Let T + = T ⊕X , and let R be any commutative Z -algebra. Put A+ R = EndHR (TR ) + + + and, for ω ∈ Ω, ∆ (ω)R := HomH (Sω , T )R . Then {∆ (ω)R }ω∈Ω is a strict + stratifying system relative to the quasi-poset (Ω, ≤op LR ) for the category AR -mod. The idea of a strict stratifying system was first defined for a finite dimensional algebra over a field in [2]. Such algebras with a strict stratifying system are standardly stratified.2 The systems in the conjecture are Z -integral versions [6]. Upon base change to a field they become strict stratifying systems there. The main result of this paper, given in Theorem 5.6, establishes a “special case" of a modified version of this conjecture, using a different quasi-poset (Ω, ≤f ). Actually, the pre-order ≤f used on the set Ω of left cells is a refinement of the pre-order ≤op LR in the conjecture, so the theorem proved is also a “special case" of the original conjecture, in the sense that it would be implied by the latter. A more detailed description of Theorem 5.6 requires some preliminary notation. Throughout this paper, e is an integer > 2. Let Φe denote the (cyclotomic) minimum polynomial for a primitive eth root of unity ζ. Fix a modular system (K, Q, k) by letting  −1] 2  Q := Q[t, t ]m , where m = (Φe (t )) = discrete valuation ring; (1.0.2) K := frac. field(Q) = Q(t);  k := Q/m = Q(ζ 21 ) = residue field 1

The validity of the original formulation over Z[t2 , t−2 ] would imply the validity over Z[t, t−1 ], by easy base change arguments. Using Z = Z[t, t−1 ] gives us the convenient property that the fraction field of Z , namely, Q(t), is a splitting field for H. Also, many results on Hecke algebras quoted from the literature are phrased using Z[t, t−1 ]. 2Standardly stratified algebras are like quasi-hereditary algebras, except that the irreducible head of an indecomposable standard module ∆ may appear with multiplicity > 1 in ∆; see [2, Defn. 6.4.1]. Empirical evidence (e.g., [6], [5]) suggests these algebras play an important role in non-defining representation theory of finite groups of Lie type, especially for small primes p, where the quasi-hereditary notion can be too strong. As in the quasi-hereditary case, the associated module derived categories of a standardly strafied algebra exhibit a “stratification,” though the number of “strata” may be smaller than in the quasi-hereditary case. In the case of algebras of arising from the conjecture and its modification in the Appendix, the number of strata is the number of two-sided cells.

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The base changed algebra HQ(t) is split semisimple, with irreducible modules corresponding to the irreducible modules of the algebra QW . The Q-algebra e := H ⊗Z Q H

has a presentation by elements Tw ⊗ 1 (which will still be denoted Tw , w ∈ W ) completely ek , replacing t2 by ζ. 3 Then analogous to (1.0.1). Finally, similar remarks apply to Hk = H e e which is filtered by dual left cell Theorem 5.6 establishes that there exists a H-module X e modules Sω such that the analogues of conditions (1) and (2) over Q in Conjecture 1.1 hold. The final stratifying system assertion in the conjecture holds with the pre-order ≤op LR on Ω replaced by an explicit refinement pre-order ≤f . With more work, it can be shown that the Q-algebra Ae+ it actually quasi-hereditary. This is done in §6; see Theorem 6.4. Then Corollary 6.5 identifies the module category for a base-changed version of this algebra with a RDAHA-category O in [8]. Finally, an Appendix discusses several variations on Conjecture 1.1. Generally speaking, this paper focuses on the “single parameter" case (i.e., all cs = 1 in (1.0.1)), which covers the Hecke algebras relevant to all untwisted finite Chevalley groups. This avoids a number of complications involving Kazhdan-Lusztig basis elements and Lusztig’s algebra J . In this context, the critical Proposition 3.2 depends on results of [8] which, in part, were only determined in the equal parameter case. Nevertheless, much of our discussion applies in the unequal parameter cases. In particular, we mention the elementary, but important, Lemma 4.3 is stated and proved using unequal parameter notation. This encourages the authors to believe the main results are also provable in the unequal parameter case, though this has not yet been carried out. 2. S OME

PRELIMINARIES

In this section,we recall some mostly well-known facts and fix notation regarding cell theory. Thus, let W be a finite Weyl group associated to a finite root system Φ with a fixed set of simple roots Π. Let S := {sα | α ∈ Π}. Let H is a Hecke algebra over Z defined by (1.0.1). We assume (unless explicitly noted otherwise) that each cs = 1 for s ∈ S. Let X Cw′ = t−l(w) Py,w (t2 )Ty . y≤w

{Cw′ }w∈W

Then is a Kazhdan–Lusztig (or canonical) basis for H. The element Cx′ is denoted cx in [13], a reference we frequently quote. Let hx,y,z ∈ N[t, t−1 ] denote the structure constants. In other words, X hx,y,z Cz′ . Cx′ Cy′ = z∈W

3

From now on in this paper, we will work with the single parameter case, unless stated explicitly otherwise. Thus, we assume that each cs = 1.The unequal parameter situation will be taken up elsewhere.

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JIE DU, BRIAN J. PARSHALL, AND LEONARD L. SCOTT

Using the pre-orders ≤L and ≤R on W , the positivity (see [4, §7.8]) of the coefficients of the hx,y,z implies hx,y,z 6= 0 =⇒ z ≤L y, z ≤R x

(2.0.3)

The Lusztig function a : W −→ N is defined as follows. For z ∈ W , let a(z) be the smallest nonnegative integer such that ta(z) hx,y,z ∈ N[t] for all x, y ∈ W . It may equally be defined as the smallest nonnegative integer such that t−a(x) hx,y,z ∈ N[t−1 ], as used in [13] (or see [4, §7.8]). In fact, each hx,y,z is invariant under the automorphism Z → Z sending t to t−1 . It is not difficult to see that a(z) = a(z −1 ). For x, y, z ∈ W , let γx,y,z be the coefficient of t−a(z) in hx,y,z −1 . Also, by [13, Conjs. 14.2(P8),15.6], (2.0.4)

γx,y,z 6= 0 =⇒ x ∼L y −1, y ∼L z −1 , z ∼L x−1 .

The function a is constant on two-sided cells in W , and so can be regarded as a function (with values in N on (a) the set of two-sided cells; (b) the set of left (or right) cells; and (c) the set Irr(QW ) of irreducible QW -modules. In addition, a is related to the generic degrees dE , E ∈ Irr(QW ). For E ∈ Irr(QE), let dE = btaE +· · ·+ctAE , with aE ≤ AE and bc 6= 0, so that taE (resp., tAE ) is the lowest (resp., largest) power of t appearing nontrivially in dE . Then aE = a(E); cf. [13, Prop. 20.6]. Also, as noted in [8, §6], AE = N − a(E ⊗ det), where N is the number of positive roots in Φ. Following [8, §6], we will use the “sorting function" f : Irr(QW ) → N defined by (2.0.5)

f (E) = aE + AE = a(E) + N − a(E ⊗ det).

This function f is used in [8] to define a poset structure on the set Irr(QW ) of irreducible QW -modules by putting E a(c′ ) and, by [12, Prop.3.3], aE = a(c). Thus, for two left cells ω, ω ′, f (ω) = f (ω ′) if and only if a(ω) = a(ω ′ ). Let

e ∗ = (S(ω) ⊗ Q)∗ Seω = S(ω) Then we have the following result.

Corollary 5.4. For left cells ω, ω ′, we have Ext1He (Seω , Seω′ ) 6= 0 =⇒ f (ω) > f (ω ′).

e ′), S(ω)) e Proof. By the lemma above and Corollary 3.3, Ext1He (S(ω 6= 0 implies f (ω) > ′ f (ω ). The result follows.  Lemma 5.5. For any subset λ ⊆ S, let ωλ be the left cell containing the longest element of the parabolic subgroup Wλ . Suppose

0 = F0λ ⊆ F1λ ⊆ F2λ ⊆ · · · ⊆ Fλm = xλ H λ with ω1 = ωλ . Then f (ωi ) > f (ωλ ). is a filtration by dual left cell modules Sωi ∼ = Fiλ /Fi−1 Proof. Since ω1 ≤L ωi and ω1 6∼L ωi for all 2 ≤ i ≤ m, by [10, Cor. 1.9(b)], we must have a(ω1 ) < a(ωi ) for all i > 1. Let w0 be the longest element of W . For two sided cells c, c′ , c f (ω) = i. Assume f (τ ) = j is minimal with this property. Since Q is a DVR and Ext1 (Seτ , Seω ) is finitely generated, it follows that Ext1 (Seτ , Seω ) is a direct sum of mτ (≥ 0) nonzero cyclic Q-modules. Let Yeτ be the extension of Seτ⊕mτ by Seω . Then by Lemma 4.1 and Corollary 4.2, Ext1 (Seτ , Yeτ ) = 0.

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JIE DU, BRIAN J. PARSHALL, AND LEONARD L. SCOTT

Let Ωj,ω = {ν ∈ Ωj | Ext1 (Seν , Seω ) 6= 0} If ν ∈ Ωj,ω − {τ }, then Ext1 (Seν , Seω ) ∼ = Ext1 (Seν , Yeτ ) by Proposition 3.2 and Corollary 3.3 together with the long exact sequence for Ext. Thus, if Yeτ,ν denote the extension of Seν⊕mν by Yeν , then we have Ext1 (Seω′ , Yeτ,ν ) = 0 for ω ′ = τ, ν.

Note that Yeτ,ν is isomorphic to the extension of Seτ⊕mτ ⊕ Seν⊕mν by Seω and continue. Eventually, we find that, if Yej is the extension of ⊕τ ∈Ωj,ω Seτ⊕mτ by Seω , then Ext1 (Seω′ , Yej ) = 0 for all ω ′ ∈ ∪i≤j Ωi .

Thus, Ext1 (Seω′ , Yej ) 6= 0 implies f (ω ′) > j. Continuing the above construction with the role Seω replaced by Yej1 (j1 = j), we obtain a module Yej1 ,j2 such that j1 < j2 and Ext1 (Seω′ , Yej1,j2 ) = 0 for all ω ′ ∈ ∪i≤j1 Ωi .

Let m be the maximal f -value. This construction will stop after a finite number, say r = e eω := Yej ,j ,··· ,jr such that r(ω), steps, resulting in an H-module X 1 2 eω ) = 0 for all ω ′ ∈ Ω. f (ω) < j1 < j2 < · · · < jr ≤ m, and Ext1 (Seω′ , X

Let Ω′ be the set of all left cells that do not contain the longest element of a parabolic subgroup. Put M M e and Xe = eω . Te = xλ H X λ⊆S

ω∈Ω′

We are now ready to prove the following main result of the paper.

e Theorem 5.6. Let Te + = Te ⊕ Xe, Ae+ = EndHe (Te + ) and ∆(ω) = HomHe (Seω , Te + ) for e e+ ω ∈ Ω. Then {∆(ω)} ω∈Ω is a strict stratifying system for the category A -mod with respect to the quasi-poset (Ω, ≤f ).

e if ω contains the longest element of Wλ , where Proof. For each left cell ω, put Teω = xλ H eω . In the first case, Teω has a filtration by dual λ ⊆ S. If there is no such λ for ω, put Teω = X left cell modules, and Seω appears at the bottom. Moreover, f (ω) < f (ω ′) for any other eω filtration section Seω′ , by Lemma 5.5. This same property holds also in the case Teω = X by construction. Put Te = ⊕ω Teω and note Te + = Te. We will now apply [6, Thm. 1.2.10] to Te and the various Teω . Our Te notation has been chosen to agree with the wording of the cited theorem. We are required the check three conditions (1), (2), (3) and to check the first condition (1) in a strong form to obtain the “strict” property. The discussion above of dual left cell filtrations of the various Teω is precisely what is required for the verification of (1) in its strong form.

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Condition (2) translates directly to the requirement HomHe (Seµ , Teω ) 6= 0 =⇒ ω ≤f µ

for given µ, ω. However, if there is nonzero HomHe (Seµ , Teω ), there must be a nonzero HomHe (Seµ , Seω′ ) for some filtration section Seω′ of Teω . In particular, f (ω ′) ≥ f (ω). Also, (Seµ )F and (Seω′ )F must have a common irreducible constituent, forcing the two-sided cells containing them to agree. This gives f (µ) = f (ω ′ ) ≥ f (ω); so (2) holds. Finally, three is an Ext1He (−, Te) vanishing result implied by the (generally stronger) condition Ext1He (Seµ , Teω ) = 0 for all µ, ω. eω and by Corollary 4.5 in This follows from the construction below Lemma 5.5 for Teω = X e The conclusion of [6, Thm. 1.2.10] now immediately gives the theorem case Teω = xλ H. we are proving here.  6. I DENTIFICATION

OF

Ae+ = EndHe (Te + )

eω in the previous section works just as well using the The construction of the modules X ∗ e eE . module SeE := S(E) in the role of dual left cell modules Seω . This results in modules X eω , we have the following property, with the same proof. As in the case of X eE ) = 0 for all E, E ′ ∈ Irr(W ). Proposition 6.1. Ext1He (SeE ′ , X

eE have strong indecomposability properties, which the modules However, the modules X e Xω do not have, in general.

eE are indecomposable, as is each X eE ⊗ k. The endomorProposition 6.2. The modules X phism algebras of all these modules are “local” (completely primary) with radical quotient k.

Proof. This can be argued without using RDAHAs, but it is faster to quote Rouquier’s 1faithful covering theory, especially [14, Thm. 5.3], which applies to our e 6= 2 case, over R, where R := (C[t, t−1 ](t−ζ 21 ) )∧ 1

is the completion of the localization C[t, t−1 ](t−ζ 12 ) at the maximal ideal (t − ζ 2 ). Note that R is a Q-module via the natural ring homomorphism Q → R. eE )R as X eE,R , etc. It is clear that X eE,R can be constructed from SeE,R in the We write (X eE is constructed from SeE . Also, the proof of [14, Thm. 6.8] shows that same way that X the R-dual of SeE,R is the ZK-image of a standard module in the R-version of O. (Recall the issues in ftn. 4.)

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eE,R )∗ is the image of a dually constructed Consequently, by the 1-faithful property (X module P under the functor KZ, filtered by standard modules, and with Ext1O (P, −) vanishing on all standard modules. Such a module P is projective in O, by, say, [14, Lem. 4.22]. (We remark that both O and KZ would be given a subscript R in [8] though not in [14].) eE,R is indecomposable. However, If we knew P were indecomposable, we could say X 7 this requires proof. Essentially, we want to show P is the projective cover in O of the standard module ∆(E) = ∆O (E). We can, instead, inductively show the truncation Pi , associated to the poset ideal of all E ′ ∈ Irr(W ) with f (E ′ ) ≤ i, is the projective cover of ∆(E) in the associated truncation Oi of O. This requires ∆(E) to be an object of Oi , or equivalently f (E) ≤ i. If f (E) = i, then Pi = ∆(E) is trivially the projective cover of ∆(E). Inductively, Pi−1 is the projective cover of ∆(E) in Oi−1 for some i > f (E). Let P ′ denote the projective cover of ∆(E) in Oi . The truncation (P ′)i−1 to Oi−1 of P ′ — that is, its largest quotient which is an object of Oi−1 — is clearly isomorphic to Pi−1 . Let ϕ : P ′ → Pi be a homomorphism extending a given isomorphism ψ : (P ′ )i−1 → Pi−1 and let τ : Pi → P ′ be a homomorphism extending ψ −1 . Let M, M ′ denote the kernels of the natural surjections Pi ։ Pi−1 and P ′ ։ (P ′ )i−1 . The map τ ϕ : P ′ → P ′ is surjective and, consequently, it is an isomorphism. It induces the identity on (P ′)i−1 . Therefore, the induced map τ |M ϕ|M ′ : M ′ −→ M ′

is an isomorphism, and M = M ′ ⊕ M ′′ for some object M ′′ in O. By construction, M is a direct sum of objects ∆(E ′ ), with f (E ′ ) = i, each appearing with multiplicity mE ′ = rank(Ext1O (Pi , ∆(E ′ ))). However, Ext1O (Pi−1 , ∆(E ′ )) ∼ = HomO (M ′ , ∆(E ′ )). It follows that M ′′ = 0 and Pi ∼ = P ′ is indecomposable. eE,R is indecomposable, as noted. In particular, P is indecomposable and consequently X eE is indecomposable. The 0-faithfulness (or just the covering propIn turn, this implies X erty itself) of the cover given by O and KZ imply e ∗ )op ∼ eE,R )op ∼ EndHeR (X = EndO (P ). = EndHeR (X E,R

Thus, the base changed module P ⊗ C (the tensor is over R) has endomorphism ring EndOC (P ⊗ C) ∼ = EndO (P ) ⊗ C, where OC is the C-version of O. This is a standard consequence of the projectivity of P . By [14, Thm. 5.3], the C versions of KZ and O give eR ⊗ C. So End e (X eE,R ⊗ C)op ∼ a cover for H = EndOC (P ⊗ C) is local, with radical HC quotient C. However, we have eE,R ⊗ C. eE ⊗Q k) ⊗k C ∼ (X =X 7

A similar point should be made regarding the uniqueness claim in [14, Prop. 4.45], which is false without a minimality assumption on Y (M ) there.

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eE ⊗Q k is indecomposable since (by endomorphism ring consideration) the In particular, X e eE,R ⊗C is indecomposable. So the endomorphism ring of X eE ⊗Q k over HR ⊗C-module X e ⊗Q k is local. The radical quotient is a division algebra the finite dimensional algebra H eE,R ⊗ C), D over k with base change under − ⊗k C to a semisimple quotient of EndHeC (X which could only be C itself. Consequently, D = k. eE , X eE ) = 0 implies Finally, the vanishing Ext1He (X eE ⊗Q k). eE ) ⊗Q k ∼ EndHe (X = EndHek (X

eE ) is local with radical quotient k. This completes the proof. So the ring EndHe (X



We remind the reader that we continue to assume e 6= 2.

eE is a direct summand of Te + . Lemma 6.3. Let E ∈ Irr(W ). Then X

e e Proof. Suppose first S(E) is a direct summand of a left cell module S(ω) =: Seω = (Seω )∗ where ω contains the longest element of a parabolic subgroup Wλ , λ ⊆ S. This implies e Consequently, there Seω is the lowest term in the dual left cell module filtration of xλ H. e with cokernel filtered by (sections) SeE ′ , E ′ ∈ Irr(W ). is an inclusion ψ : SeE → xλ H eE may be extended to a map φ : xλ H e → X eE of H-modules. e Thus, ψ −1 : ψ(SeE ) → X eE → xλ H e extending ψ. The Similarly (using e 6= 2 and Corollary 4.5), there is a map τ : X eE . composite τ φ restricts to the identity on SeE ⊆ X e e On the other hand, restriction from XE to SE defines a homomorphism eE ) −→ End e (SeE ) EndHe (X H

e Q K-module X eE ⊗Q K. since (SeE )K is a unique summand of the (completely reducible) H⊗ eE ∩ (SeE )K , since the Q-torsion module (X eE ∩ (SeE )K )/SeE must be zero (Observe SeE = X e e in the Q-torsion free module XE /SE .) Thus, τ φ is a unit in the local endomorphism ring eE ), so X eE is a direct summand of xλ H, e and hence of Te . EndHe (X Next consider the case in which SeE is a summand of a dual left cell module Seω (this always happens for some ω), but ω does not contain the longest element of any parabolic eω is one of the summands of Xe by construction. The argument subgroup. In this case, X eω playing the role of xλ H. e In the same way, X eE is a direct above may be repeated with X e e summand of Xω , and thus of X . eE is a direct summand of Te ⊕ Xe = Te + . In both cases, we conclude that X  e Theorem 6.4. The Q-algebra Ae+ is quasi-hereditary, with standard modules ∆(E) = e e HomHe (SE , T ), E ∈ Irr(W ), and partial order f (E). ∆(E ), E ∈ Irr(W ) (rather than X ⋄ e e Next, we claim that ∆(E) := HomAe+ (∆(E), Te + ) is naturally isomorphic to SeE . More ev precisely, we claim that the natural map SeE → (SeE )⋄⋄ is an isomorphism. We showed above that the sequence eE /SeE )⋄ −→ (X eE )⋄ −→ (SeE )⋄ −→ 0 0 −→ (X

is exact. Applying (−)⋄ once more, we get an injection

eE )⋄⋄ 0 −→ (SeE )⋄⋄ −→ (X

ev eE → eE )⋄⋄ an isomorphism. This gives inclusions with X (X

eE )⋄⋄ ∼ eE . SeE ∼ = ev(SeE ) ⊆ (SeE )⋄⋄ ⊆ (X =X

If (−) ⊗Q K is applied, the first inclusion becomes an isomorphism. This gives eE )⋄⋄ ∩ (SeE )K = SeE (SeE )⋄⋄ ⊆ (X

eE with (X eE )⋄⋄ and SeE with its image in (X eE )⋄⋄ . Consequently, ev(SeE ) = identifying X (SeE )⋄⋄ , proving the claim. e 6= 0. Using the Finally, we suppose E ∼ 6= E ′ ∈ Irr(W ) and HomAe+ (Pe(E ′ ), ∆(E)) ⋄ ⋄ ′ ⋄ ′ ∼ e e e e e e e identifications P (E ) = (XE ′ ) , ∆(E) = (SE ) , P (E ) = XE ′ , and ∆(E)⋄ ∼ = SeE , we have ∼ e eE ′ ⊗Q K). eE ′ ) ⊆ Hom e (SeE ⊗Q K, X 0 6= Hom e+ (Pe(E ′ ), ∆(E)) = Hom e (SeE , X A

H

HK

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This implies f (E ′ ) < f (E). It follows that now from [1] or [14, Thm. 4.15] that Ae+ is quasihereditary over Q.  Corollary 6.5. The category of left modules over the base-changed algebra Ae+ := Ae+ ⊗Q R R

is equivalent to the R-category O of modules, as defined in [14] for the RDAHA associated to W over R. Proof. Continuing the proof of the theorem above, the projective indecomposable Ae+ eE )⋄ . Consequently, Te + = (Ae+ )⋄ is the direct sum of modules are the various Pe(E) = (X eE , each with nonzero multiplicities. The modules X eE,R remain indecomthe modules, X eE above. By posable, as observed in the proof of the indecomposablility of the modules X eE ) = 0 for all E, E ′ ∈ Irr(W ). Thus, there is a similar vanishconstruction, Ext1He (SeE ′ , X eE,R , and—in the reverse order—for their R-linear duals. Observe that ing for SeE ′ ,R and X ∗ ′ ∼ e )⊗Q R is KZ(∆(E ′ )), taking ∆(E ′ ) = ∆O (E ′ ) to be the standard module (SeE ′ ,R ) = S(E for the category O over R as discussed in [14] together with KZ for this category. Put M eE,R )∗ Y = (X E

eE,R ) . This notation imitates that of [14, Prop. 4.45]. The first = (X and set Y part of this cited proposition is missing a minimality assumption on the rank of Y (M), in the terminology there (see ftn. 3 there). However, this is satisfied for M = (SeE,R )∗ and eE,R )∗ because (X eE,R )∗ is indecomposable. Y (M) = (X Several other corrections, in addition to the minimality requirement, should be made to [14, Prop. 4.45]: • A′ should be redefined as EndB (Y ))op ; • P ′ should be redefined as HomB (Y, B)op . In addition B in [14, 4.2.1] should be redefined as EndA (P )op . The “op”s here and above insure action on the left, and consistency with [8, Thm. 5.15, Thm. 5.15]. The definition of P ′ is given to be consistent with the basis covering property EndA′ (P ′)op ∼ = B, as in [8, Thm. 5.15]. (We do not need this below.) With these changes, [14, Thm. 5.3, Prop. 4.45, Cor. 4.46] guarantees that A′ -mod is equivalent to O, where A′ = EndHeR (Y ). (All we really need for this are the 0- and 1faithfulness of the O version of the KZ functor.) However, EndHeR (Y ) ∼ = EndHeR (Y ∗ )op , eE,R . Hence, and Y ∗ is the direct sum ⊕E X M M eE,R )⋄ ∼ Pe(E) ⊗Q R. (X Y ∗⋄ ∼ = = ∗ ) (SeE,R



E

E

eE,R , so that the analogous property holds for Y ∗ . Thus, eE,R )⋄⋄ ∼ Recall that (X = X ∗ op ∼ ∗⋄ EndHeR (Y ) = EndAe+ (Y ). Since the module Y ∗⋄ as displayed above is clearly a R

18

JIE DU, BRIAN J. PARSHALL, AND LEONARD L. SCOTT

′ e+ projective generator for Ae+ R , there is a Morita equivalence over R of AR with A . Hence,  Ae+ R -mod is equivalent to O, as R-categories.

7. A PPENDIX : A

REVISED CONJECTURE

Given the present statement of Conjecture 1.1, the most natural way to formulate Theorem 5.5 would have been to use ≤op LR instead of ≤f . There are several remarks to make:

(1) The formulation with ≤LR implies formally the present formulation, given that ≤f is a refinement of ≤op LR . (2) However, we do not know if the formulation with ≤op LR is true.

(3) Both formulations formally imply the existence of an underlying standardly stratified algebra with strata indexed by the two-sided cells. The original form of Conjecture 1.1, namely, [6, Conj. 2.5.2], arose in a context focusing on (3) above. The order ≤op LR was used in the statement simply because it was the most obvious way of getting a pre-order on Ω whose associated equivalence classes were naturally indexed by the set of two-sided cells. We had no idea there were more sophisticated possibilities, like ≤f , that could achieve the same thing. In view of the experience of this paper, we present the following revision of [6, Conj. 2.5.2], eliminating the dependence on ≤op LR . The notation is the same as in Conjecture 1.1, except that Z[t2 , t−2 ] is used for Z . In particular, the right H-module X has an finite filtration with sections of the form Sω , ω ∈ Ω (the set of left cells in W ). Define T + := T ⊕ X , and let A+ := EndHR (TR+ ), etc. be as stated for a commutative ring R. Conjecture 7.1. The module X can be chosen so that there is a pre-order ≤ on Ω such that: (a) elements ω, ω ′ ∈ Ω are equivalent with respect to ≤ if and only if ω, ω ′ are contained in the same two-sided cell; (b) the indexed set {∆+ (ω)}ω∈Ω is a strict stratifying system relative to the quasi-poset (Ω, ≤). Using Z = Z[t, t−1 ] in Conjecture 7.1 gives a modification of Conjecture 1.1. Other modifications are possible. For example, Conjecture 7.1, with either version of Z , remains quite interesting if R above is always required to be a field. Nevertheless, the “integral setting," allowing R to be a commutative ring is quite useful. It has been somewhat vetted in [6], which checked the original conjecture in all rank 2 cases, even allowing “unequal parameters" and twisted types, including type 2 F4 . These cases are still too simplistic to distinguish ≤f from ≤op LR as regards to the assertion (b) above. However, in view of Theorem 6.4, such a distinction might well be made in the future, using examples arising from the categories O for RDAHAs. Finally, we mention that the original conjecture [6, Conj. 2.5.2] was checked in type A for all ranks using ≤op LR ; see [7].

HECKE ENDOMORPHISM ALGEBRAS

19

R EFERENCES [1] E. Cline, B. Parshall and L. Scott, Integral and graded quasi-hereditary algebras, I, J. Algebra 131 (1990), 126–160. [2] E. Cline, B. Parshall, and L. Scott, Stratifying endomorphism algebras, Memoirs Amer. Math. Soc. 591, 119+iv pages. [3] C. Curtis and I. Reiner, “Methods of Representation Theory," vol. 1,2 Wiley, New York 1981, 1987. [4] B. Deng, J. Du, B. Parshall, and J.-P Wang, Finite Dimensional Algebras and Quantum Groups, Math. Surveys and Monographs 150, Amer. Math. Soc. (2008). [5] J. Du and L. Scott, Stratifying q-Schur algebras of type D, in Representations and quantizations (Shanghai, 1998), 167–197, China Higher Ed. Press, Beijing, 2000. [6] J. Du, B. Parshall, and L. Scott, Stratifying endomorphism algebras associated to Hecke algebras," J. Algebra 203 (1998), 169–210. [7] J. Du, B. Parshall, and L. Scott, Cells and q-Schur algebras, Trans. Groups 3 (1998), 33–49. [8] V. Ginzburg, N. Guay, E. Opdam, R. Rouquier, On the category O for rational Cherednik algebras, Invent. math. 156 (2003), 617–651. [9] G. Lusztig, Cells in affine Weyl groups, I, Adv. Studies Pure Math. 6 (1985), 255-287. [10] G. Lusztig, Cells in affine Weyl groups, II, J. Algebra 109 (1987), 536–548. [11] G. Lusztig, Cells in affine Weyl groups, III, J. Fac. Sci. Univ. Tokyo 34 (1987), 223–243. [12] G. Lusztig, Leading coefficients of character values of Heck algebras, Proc. Symp. Pure Math. 47 (1987), 235–262. [13] G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series 18, Amer. Math. Soc. (2003). [14] R. Rouquier, q-Schur algebras and complex reflection groups, Moscow Math. J. 8 (2008), 119–158. S CHOOL OF M ATHEMATICS AND S TATISTICS , U NIVERSITY 2052 E-mail address: [email protected]

OF

N EW S OUTH WALES , UNSW S YD -

NEY

D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF V IRGINIA , C HARLOTTESVILLE , VA 22903 E-mail address: [email protected] (Parshall) D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF V IRGINIA , C HARLOTTESVILLE , VA 22903 E-mail address: [email protected] (Scott)