Loewy structure of G T-Verma modules of singular 1 highest weights ∗ 5 1 be aneda 0 A Noriyuki K Masaharu 2 Hokkaido University Osaka City University r Creative Research Institution (CRIS) Department of Mathematics p A [email protected] [email protected] 1 April 2, 2015 ] T R . h t a Abstract m Let G be a reductive algebraic group over an algebraically closed field of positive characteristic, [ G theFrobeniuskernelof G, and T a maximal torusof G. Weshow that theparabolically induced 1 2 G T-Verma modules of singular highest weights are all rigid, determine their Loewy length, and 1 v describe their Loewy structure using the periodic Kazhdan-Lusztig Q-polynomials. Weassume that 9 thecharacteristicofthefieldislargeenoughthat,inparticular,Lusztig’sconjecturefortheirreducible 2 G T-characters holds. 1 0 7 0 Let G be a reductive algebraic group over an algebraically closed field k of positive . 1 characteristic p. The Frobenius kernel G of G is an analogue of the Lie algebra of G in 1 0 characteristic 0. To keep track of weights, we consider representations of G T with T a 5 1 maximal torus of G. In this paper we study G T-Verma modules, standard objects of the 1 1 : theory. v i X Many years ago Henning Andersen and the second author of the present paper showed r that the G T-Verma modules of p-regular highest weights are all rigid of Loewy length a 1 1 plus the dimension of the flag variety of G, and described their Loewy structure using the periodic Kazhdan-Lusztig Q-polynomials [AK]. For that we assumed the validity of Lusztig’s conjecture on the irreducible characters for G T-modules, or rather Vogan’s 1 equivalent version on the semisimplicity of certain G T-modules, modeling after Irving’s 1 method [I85], [I88]. Lusztig’s conjecture is now a theorem for large p as established by Andersen, Jantzen and Soergel [AJS]. Pushing their graded representation theory, with a machinery of Beilinson, Ginzburg and Soergel [BGS], we showed in [AbK] that the parabolic induction is graded on p-regular blocks, and determined the Loewy structure of parabolically induced G T-Verma modules of p-regular highest weights. In this paper 1 we use Riche’s Koszulity of the G -block algebras [Ri] to uncover the structure of the 1 parabolically induced G T-Verma modules of p-singular highest weights, to complete the 1 entire picture. 2010 Mathematics Subject Classification. 20G05. ∗supportedinpartbyJSPSGrantsinAidforScientificResearch 1 Todescribe our results precisely, let us introduce somenotations. For simplicity we will assumethroughoutthepaperthatGissimplyconnectedandsimple. FixaBorelsubgroup B of G containing T, and choose a positive system R+ of R such that the roots of B are −R+. Let Rs denote the set of simple roots of R+. Let Λ denote the weight lattice of T equipped with a partial order such that λ ≥ µ iff λ−µ ∈ Nα. Put ρ = 1 α. α∈R+ 2 α∈R+ Let W denote the Weyl group of G relative to T, and let W = W⋉ZR be the affine Weyl a group with elements of ZR in W acting on Λ by translatiPons. We let W act onPΛ also via a a x•λ = px1(λ+ρ)−ρ, x ∈ W , λ ∈ Λ. Let R∨ = {α∨ | α ∈ R} denote the set of coroots of p a R, and put Hα,n = {v ∈ Λ⊗ZR | hv+ρ,α∨i = pn}, α ∈ R and n ∈ Z. We call a connected component of (Λ⊗ZR)\∪α∈R,n∈ZHα,n analcove. We say λ ∈ Λ is p-regular iff it belongs to an alcove, otherwise λ is p-singular. Let also Λ+ = {λ ∈ Λ | hλ,α∨i ≥ 0 ∀α ∈ R+} the set of dominant weights. We let A+ = {v | hv +ρ,α∨i ∈ ]0,p[ ∀α ∈ R+} denote the bottom dominant alcove. For a closed subgroup H of G we let H denote its Frobenius kernel. 1 Let ∇ˆ = indG1T denote the induction functor [J, I.3] from the category of B T-modules to B1T 1 the category of G T-modules. The G T-simple modules are parametrized by their highest 1 1 ˆ weights in Λ. We let L(ν) denote the simple G T-module of highest weight ν ∈ Λ. If M 1 is a finite dimensional G T-module, we will write [M : Lˆ(ν)] for the composition factor 1 multiplicity of Lˆ(ν) in M. ALoewyfiltrationofafinitedimensional G T-moduleM isafiltrationofM ofminimal 1 length such that each of its subquotients is semisimple. The length of a Loewy filtration is uniform, called the Loewy length of M, denoted ℓℓ(M). Among the Loewy filtrations, the socle series of M is defined by 0 < soc1M < soc2M < ··· < socℓℓ(M)M = M with soc1M = socM, called the socle of M which is the sum of simple submodules of M, and sociM/soci−1M = soc(M/soci−1M) for i > 1. Also the radical series of M is defined by 0 = radℓℓ(M)M < ··· < rad2M < rad1M < M with rad1M = radM, called the radical of M which is the intersection of the maximal submodules of M, and radiM = rad(radi−1M) for i > 1. If 0 < M1 < ··· < Mℓℓ(M) = M is any Loewy filtration ofM, radℓℓ(M)−iM ≤ Mi ≤ sociM foreachi. WesayM isrigidiffthesocleandtheradical series of M coincide. We put soc M = sociM/soci−1M and rad M = radjM/radj+1M. i j In this paper we show Theorem: Assume p ≫ 0. Let ν ∈ Λ and let N(ν) denote the number of hyperplanes ˆ H on which ν lies. The G T-Verma module ∇(ν) of highest weight ν is rigid of Loewy α,n 1 length 1+dimG/B−N(ν). If x ∈ W is such that ν belongs to the upper closure of x•A+ a and if ν = x−1 •ν, the Loewy structure of ∇ˆ(ν) is given by 0 qd(y,2x)−i[soci+1∇ˆ(ν) : Lˆ(y •ν0)] i∈N X Qy•A+,x•A+ if y ∈ W with y •ν belonging to the upper closure of y •A+, a 0 = 0 else, ( where d(y,x) is the distance from the alcove y • A+ to the alcove x• A+ [L80, 1.4] and Qy•A+,x•A+ is a polynomial from [L80, 1.8]. For this theorem to hold, we assume p ≫ 0 so that Lusztig’s conjecture for the irre- 2 ducible characters of G T-modules and also the conditions [Ri, (10.1.1) and (10.2.1)] from 1 [BMR06] hold. While Fiebig [F] gives an explicit lower, as crude as it may be, bound of p for Lusztig’s conjecture to hold, a recent work of Williamson [W] reveals that p has, in general, to be much bigger than h the Coxeter number of G, which was the original bound for the conjecture to hold. Compared to the restriction required for Lusztig’s conjecture to hold, the other conditions in [Ri] are innocent. We actually obtain, more generally, analogous results for parabolically induced module ∇ˆ (LˆP(ν)) = indG1T(LˆP(ν)) with LˆP(ν) denoting a simple P T-module of highest weight P P1T 1 ν for a parabolic subgroup P of G. For a category C we will denote the set of morphisms from object X to Y in C by C(X,Y) . We are grateful to Wolfgang Soergel for suggesting us the problem and referring to [Ri]. Thanks are also due to Simon Riche for helpful comments to clarify the presentation of the paper. The first author of the paper acknowledges the hospitality of Institute of Mathematics at Jussieu and also Mittag-Leffler Institute, during the visit of which the work has been done. 1◦ Koszulity of the G -block algebras 1 Throughout the paper we will assume p > h the Coxeter number of G unless otherwise specified. In particular, pΛ∩ZR = pZR. All modules we consider are finite dimensional over k. Our basic strategy is to transport the known structure of a G T-block C(λ) of 1 p-regular λ ∈ Λ to an arbitrary block C(µ) by the translation functor. For p ≫ 0, thanks to [Ri], the corresponding translationfunctor for the G -blocks is graded and the G -block 1 1 algebras are all Koszul. ˆ ˆ (1.1) For ν ∈ Λ let L(ν) denote the simple G T-module of highest weight ν, and P(ν) the 1 G T-projective cover of Lˆ(ν). Let Ω be a p-regular orbit of W in Λ and let C(Ω) denote 1 a the corresponding G T-block. Thus C(Ω) = C(ν), ν ∈ Ω, consists of G T-modules whose 1 1 composition factors all have highest weights in Ω. Let Ω′ be a system of representatives of Ω under the translations by pZR, and let Pˆ(Ω) = Pˆ(ν). Then C(Ω)(Pˆ(Ω)⊗ ν∈Ω′ γ∈pZR γ,Pˆ(Ω))formsapZR-gradedk-algebraunderthecompositionusingtheauto-functor?⊗γ, ` ` γ ∈ pZR, on C(Ω). If we let Eˆ(Ω) denote its opposite algebra, C(Ω)(Pˆ(Ω)⊗γ,?) γ∈pZR gives an equivalence of categories from C(Ω) to the category of finite dimensional pZR- ` graded Eˆ(Ω)-modules. Moreover, Eˆ(Ω) admits a Z-grading compatibe with its pZR- gradation [AJS, 18.17.1]. For p large enough that Lusztig’s conjecture holds, [AJS, 18.17] has proved that Eˆ(Ω) is Koszul with respect to its Z-gradation. Let us state Lusztig’s conjecture in an equivalent form as follows: ∀x,y ∈ W , a (L) [∇ˆ(x•0) : Lˆ(y •0)] = Qy•A+,x•A+(1), where Qy•A+,x•A+ is a polynomial from [L80, 1.8]. Assuming (L), let C˜(Ω) denote the category of finite dimensional (pZR × Z)-graded Eˆ(Ω)-modules. For each ν ∈ Ω′ let L˜(ν) be the lift of Lˆ(ν) in C˜(Ω) as a direct summand of 3 the degree 0 part of Eˆ(Ω). If we denote the degree shift of objects in C˜(Ω) by [γ], γ ∈ pZR, and by hii, i ∈ Z, any simple of C˜(Ω) may be written L˜(ν)[γ]hii, ν ∈ Ω′, γ ∈ pZR, and i ∈ Z, in a unique way up to isomorphism. As L˜(ν)[γ] is a lift of Lˆ(ν +γ) = Lˆ(ν) ⊗γ, we will also write L˜(ν + γ) for L˜(ν)[γ]. For each ν ∈ Ω the G T-Verma module ∇ˆ(ν) 1 ˜ ˜ ˜ of highest weight ν admits a lift ∇(ν) in C(Ω) such that its socle is L(ν). Likewise each projective Pˆ(ν) admits a lift P˜(ν) which is the projective cover of L˜(ν). (1.2) Let Λ = {ν ∈ Λ+ | hν,α∨i < p ∀α ∈ Rs}. For ν ∈ Λ we write ν = ν0 +pν1 with p ν0 ∈ Λ and ν1 ∈ Λ. We let L(ν0) denote the simple G-moduleof highest weight ν0, which p remains simple regarded as a G -module. All the simple G -modules are obtained thus. 1 1 One has Lˆ(ν) = L(ν0)⊗pν1, Pˆ(ν) = Pˆ(ν0)⊗pν1, and Pˆ(ν0) provides the G -projective 1 cover of L(ν0), which we will denote by P(ν0). Let now g denote the Lie algebra of G, Ug the universal enveloping algebra of g, and (Ug) the central reduction of Ug with respect to the Frobenius central character 0. As 0 (Ug) coincides with the the algebra of distributions of G , the representation theory of 0 1 G is equivalent to that of (Ug) . For each ν ∈ Λ let (Ug)νˆ be the central reduction of 1 0 0 (Ug) with respect to the Harish-Chandra generalized character νˆ. This is the G -block 0 1 component of (Ug) associated to ν. Let B(ν) denote the category of finite dimensional 0 (Ug)νˆ-modules. The algebra (Ug)νˆ is equipped with a Z-grading [Ri, 6.3 and 10.2 line 16, 0 0 p. 126]. We let Bgr(ν) denote the category of finite dimensional graded (Ug)νˆ-modules. 0 Let Λ(ν) = {(w•ν)0 | w ∈ W}. Each P(η), η ∈ Λ(ν), admits a lift Pgr(η) in Bgr(ν). Let Pν = Pgr(η) and set E(ν) = B(ν)(Pν,Pν)op. As Pν is a projective generator of η∈Λ(ν) B(ν) and as E(ν) = Bgr(ν)(Pνhii,Pν) is equipped with a Z-gradation, hii denoting ` i∈Z the degree shift, B(ν)(Pν, ?) induces an equivalence from Bgr(ν) to the category of finite dimensional Z-graded`E(ν)-modules, which we will denote by B˜(ν). For p ≫ 0, thanks to [Ri, 10.3], all E(ν) are Koszul by a careful choice of graded lift Pgr(η), η ∈ Λ(ν). To be precise, let I ⊆ Rs and let P denote the corresponding standard parabolic subgroupofGwiththeWeylgroupW = hs | α ∈ Ii, wheres isthereflectionassociated I α α to α. Let µ ∈ Λ lying in the closure A+ of the alcove A+ such that C (y •µ) := {x ∈ Wa• W | xy •µ = y •µ} = W for some y ∈ W . Let also λ ∈ A+ such that hy •λ,α∨i = 0 a I a ∀α ∈ I. If p ≫ 0 so that the condition (L) holds, one can take each Pgr((w•λ)0) to satisfy a certain condition [Ri, 8.1(‡)]. With this choice [Ri, Th. 9.5.1] shows that the graded algebra E(λ) is Koszul. For µ assume in addition to (L) two more conditions, which go as follows: the first one [Ri, 10.1.1] coming from [BMR06, Lem. 1.10.9(ii)] reads, with Dλ denoting the sheaf of PD-differential operators on G/P twisted by the invertible G/P sheaf L (λ), G/P (R1) RiΓ(G/P,Dλ ) = 0 ∀i > 0. G/P With(Ug)λ denoting thecentral reductionofUgbytheHarish-Chandracentral character λ, the second condition [Ri, 10.2.1] coming also from [BMR06, Lem. 1.10.9] reads that (R2) the natural morphism (Ug)λ → Γ(G/P,Dλ ) be surjective. G/P If p ≫ 0 so that (L), (R1) and (R2) all hold, one can take each Pgr(η), η ∈ Λ(µ), to satisfy [Ri, Th. 10.2.4], which makes E(µ) also Koszul [Ri, Th. 10.3.1]. For any ν ∈ Λ there is by [BMR06, Lem. 1.5.2] some ξ ∈ Λ such that ν+pξ ∈ W •µ with µ as above. Thus under a 4 the conditions (L), (R1) and (R2) we may assume that all G -block algebras E(ν) are 1 Koszul. For each η ∈ Λ(ν) we denote by L˜(η) the graded lift in B˜(ν) of G -simple L(η) 1 as a direct summand of E(ν) . Let also P˜(η) = Bgr(ν)(Pνhii,Pgr(η)) be a graded 0 i∈Z lift in B˜(ν) of P(η) to form the projective cover of L˜(η). ` (1.3) Assume from now on throughout the rest of the paper that p ≫ 0 so that all the conditions (L), (R1) and (R2) from (1.1) and (1.2) hold, unless otherwise specified. Fix also λ and µ as in (1.2). For our purposes, as tensoring with pη, η ∈ Λ, is an equivalence from the G T-block 1 C(Γ) of a W -orbit Γ to the G T-block C(Γ+pη), we have only to determine the structure a 1 of parabolically induced G T-Verma modules of highest weight x•µ with µ as above and 1 x ∈ W such that hxρ,α∨i ∈ ]0,p[ ∀α ∈ Rs. a If Ω = W • λ, as p > h by the standing hypothesis, E(λ) coincides by the linkage a principle with Eˆ(Ω) from (1.1) as k-algebras. As two Z-gradations on the algebra must agree by their Koszulity [AJS, F.2], there is no ambiguity about the functor from C˜(Ω) to B˜(λ) forgetting the pZR-gradation, which is compatible with the forgetful functor from the category of G T-modules to that of G -modules. Thus one has a commutative 1 1 diagram of forgetful functors C(Ω) oo C˜(Ω) (cid:15)(cid:15) (cid:15)(cid:15) B(λ) oo B˜(λ). (1.4) For each ν ∈ Λ let pr denote the projection from the category of finite dimensional ν G -modulestoitsν-blockB(ν). Forν,η ∈ A+ recallfrom[BMR08]thetranslationfunctor 1 Tη = pr (L((η −ν)+)⊗ ?) : B(ν) → B(η) with (η −ν)+ ∈ W(η−ν)∩Λ+. ν η By [Ri, Prop. 5.4.3 and Th. 6.3.4] the adjoint translation functors Tµ and Tλ are λ µ graded to form a pair of functors Bgr(λ) ⇄ Bgr(µ) such that graded Tµ is right adjoint to λ graded Tλ. In turn, they induce a pair of graded functors, which we will denote by T˜µ µ λ and T˜λ: µ T˜µλ = Bgr(λ)(Pλhii,Tµλ?)◦(Pµ⊗E(µ)?) : B˜(µ) → B˜(λ), i∈N a T˜λµ = Bgr(µ)(Pµhii,Tλµ?)◦(Pλ⊗E(λ)?) : B˜(λ) → B˜(µ)gr. i∈N a Thus T˜µ is right adjoint to T˜λ. λ µ Let N(ν) denote the number of hyperplanes H on which ν ∈ Λ lies, and put, in α,n particular, N = N(λ) = dimG/B, N = N(µ) = dimG/P. A crucial fact to our P results is Riche’s [Ri, 10.2.8] that asserts for each w ∈ W with (w • µ)0 ∈ Λ(µ), i.e., (w •µ)0 belonging to the upper closure of an alcove containing some (w′ •λ)0, w′ ∈ W, TλPgr((w •µ)0) = Pgr((w′ •λ)0)hN −N i, and hence µ P (1) T˜λP˜((w •µ)0) = P˜((w′ •λ)0)hN −N i. µ P 5 (1.5) For each ν ∈ Λ let pr denote the projection from the category of finite dimensional ν G T-modules to the block C(ν). For ν,η ∈ A+ one has as in (1.4) the translation functor 1 Tˆη = pr (L((η − ν)+)⊗ ?) : C(ν) → C(η) [J, II.9.22]. Under the assumption p > h, the ν η b functors Tˆµ and Tµ commute with the forgetful functors as in [Ri, Lem. 4.3.2]: λ λ b Tˆµ C(λ) λ // C(µ) (cid:8) (cid:15)(cid:15) (cid:15)(cid:15) B(λ) //B(µ). Tµ λ (1.6) Under the forgetful functors, ∇ˆ = indG1T yields an induction functor ∇¯ = indG1 B1T B1 from the category of B -modules to the category of G -modules. If M is a G T-module, 1 1 1 it is semisimple iff it is semisimple as G -module [J, I.6.15]. Thus, in order to show that 1 ∇ˆ(x•µ), x ∈ W , is rigid, we have only to show that ∇¯(x•µ) is rigid. a For a facet F in Λ ⊗Z R with respect to Wa let Fˆ denote its upper closure. As ∇ˆ(x • µ) = Tˆµ∇ˆ(x • λ), T˜µ∇˜(x • λ) ∈ B˜(µ) is a graded lift of ∇¯(x • µ), which we will λ λ denote by ∇˜(x • µ)hi + N − Ni if x • µ ∈ x\′ •A+, x′ ∈ W , and if [soc ∇ˆ(x • λ) : P a i+1 Lˆ(x′ • λ)] = [∇˜(x • λ) : L˜(x′ • λ)hii] 6= 0. As ∇¯(x • µ) has a simple socle and a simple head, so does its lift, and hence the lift is rigid by [BGS, Prop. 2.4.1]. There now follows the rigidity of ∇ˆ(x•µ). Proposition: All G T-Verma modules ∇ˆ(ν), ν ∈ Λ, are rigid. 1 (1.7) Let w ∈ W and put wB = wBw−1, ∇ˆ = indG1T . If M is a G T-module, let wM w wB1T 1 denote the G T-module M with the G T-action twisted by w, i.e., we let g ∈ G T act on 1 1 1 m ∈ M by w−1gw. For each ν ∈ Λ one has an isomorphism w∇ˆ(ν) ≃ ∇ˆ (wν) [J, II.9.3]. w Thus Corollary: All ∇ˆ (ν), w ∈ W, ν ∈ Λ, are rigid. w (1.8) Let J ⊆ Rs, Q the standard parabolic subgroup of G associated to J with the Weyl group denoted W , and let ∇ˆ = indG1T denote the induction functor from the J J Q1T category of Q T-modules to the category of G T-modules. Let ν ∈ Λ and let LˆJ(ν) 1 1 denote the simple Q T-module of highest weight ν. Choose a p-regular η ∈ Λ such 1 that ν belongs to the upper closure of the W -alcove containing η. Under the Lusztig J,a conjecture (L) we have shown in [AbK, 3.9] that ∇ˆ (LˆJ(η)) is graded, and in [AbK, 2.3] J that Tˆν(∇ˆ (LˆJ(η))) ≃ ∇ˆ (LˆJ(ν)). As ∇ˆ (LˆJ(ν)) has a simple head and socle [AbK, 1.4], η J J J it follows again from [BGS, Prop. 2.4.1] that Proposition: All parabolically induced G T-Verma modules ∇ˆ (LˆJ(ν)), ν ∈ Λ, are 1 J rigid. 6 2◦ The Loewy structure Keep the notations from §1. ˆ ¯ (2.1) For each ν ∈ Λ we will denote L(ν) by L(ν) when regarded as a G -module. Thus 1 L¯(ν) = L(ν0). Lemma: Let x ∈ W . a (i) One has L˜((x•µ)0)hN −Ni if x•µ ∈ x\•A+, T˜µL˜((x•λ)0) = P λ (0 else. (ii) If x•µ ∈ x\•A+, one has ∀i ∈ N, Tˆµsoci∇ˆ(x•λ) = soci∇ˆ(x•µ). λ \ Proof: (i) We may by (1.5) assume that x•µ ∈ x•A+ [J, II.7.15, 9.22.4], which occurs iff (x•µ)0 lies in the upper closure of the alcove (x•λ)0 belongs to. Thus we are to show in that case that T˜µL˜((x•λ)0) = L˜((x•µ)0)hN −Ni. λ P As P˜((x•µ)0) (resp. P˜((x•λ)0)) is a projective cover of L˜((x•µ)0) (resp. L˜((x•λ)0)), we have for each n ∈ Z B˜(µ)(P˜((x•µ)0)hni,T˜µL˜((x•λ)0)) ≃ B˜(λ)(T˜λP˜((x•µ)0)hni,L˜((x•λ)0)) λ µ ≃ B˜(λ)(P˜((x•λ)0)hn+N −N i,L˜((x•λ)0)) by (1.4.1), P which is nonzero iff n+N −N = 0, and hence the assertion follows. P (ii) Let soci ∇¯((x • λ)0), x ∈ W , denote the i-th term of the G -socle series of G1 a 1 ∇¯((x•λ)0), which is just soci∇ˆ(x•λ) regarded as G -module. As the socle series and the 1 gradation over E(λ) (resp. E(µ)) coincide on ∇˜((x • λ)0) (resp. ∇˜((x • µ)0)) by [BGS, Prop. 2.4.1], we see from (i) that Tµsoci ∇¯((x•λ)0) = soci ∇¯((x•µ)0), and hence the λ G1 G1 assertion. (2.2) ∀x,y ∈ W , let Qy•A+,x•A+(q) = Qy,xqj ∈ Z[q] be the periodic Kazhdan-Lusztig a j j 2 Q-polynomial from [L80]. Put Qy,x = Qy•A+,x•A+(q) for simplicity. Recall from [AK], P [AJS, 18.19]/[AbK, 5.1, 2] qd(y,2x)−i[soci+1∇ˆ(x•λ) : Lˆ(y •λ)] = qd(y,2x)−i[∇˜(x•λ) : L˜(y •λ)h−ii] = Qy,x, i∈N i∈N X X where d(y,x) = d(y • A+,x • A+) is the distance from the alcove y • A+ to the alcove \ x•A+ [L80]. Let W (µ) = {x ∈ W | x•µ ∈ x•A+}. For each x ∈ W (µ), (2.1.ii) shows a a a that qd(y,2x)−i[soci+1∇ˆ(x•µ) : Lˆ(y •µ)] = Qy,x if y ∈ Wa(µ), 0 else. ( i∈N X 7 (2.3) One can do the same with parabolically induced G T-Verma modules ∇ˆ (LˆJ(ν)), 1 J J ⊆ Rs, ν ∈ Λ, from (1.8), via [AbK, 2.3]. Let W = W ⋉ZR denote the affine Weyl J,a J J group for P . J \ Theorem: Let ν ∈ Λ, x ∈ W such that ν ∈ x•A+, and put ν = x−1 •ν. Then a 0 qd(y,2x)−i[soci+1∇ˆJ(LˆJ(ν)) : Lˆ(y •ν0)] i∈N X Qy•A+,z•A+(−1)dJ(z,x)PˆJ if y ∈ W (µ), = z∈WJ,a z•A+,x•A+ a (P0 else, where PˆJ is a Pˆ-polynomial from [Kat] for W and d (z,x) is the distance from z•A+,x•A+ J,a J z •A+ to x•A+ with respect to W . J,a (2.4) Finally we determine the Loewy length of all parabolically induced G T-Verma 1 modules. We first need analogues of [I85, Props. 3.2 and 3.3]. Let ∆ˆ(ν) = Dist(G ) ⊗ ν = ∇ˆ(ν)τ the k-linear dual of ∇ˆ(ν) twisted by the 1 Dist(B1) Chevalley anti-involution τ of G [J, II.2.12]. We say a G T-module M admits a ∇ˆ- 1 filtration iff there is a filtration 0 = M0 < M1 < ··· < Mr = M of G T-modules with 1 each Mi/Mi−1 ≃ ∇ˆ(ν ) for some ν ∈ Λ, in which case one can arrange the filtration such i i that ν 6< ν if i > j [J, II.9.8]. Whenever M admits a ∇ˆ-filtration, we will assume that i j such a rearrangement has been done. Let W = C (ν) and take an alcove A in the closure of which ν lies. Choose η ∈ Λ ν Wa• in A. Let η+ (resp. η−) denote the highest (resp. lowest) weight in W •η. Let us also ν denote by Tˆη : C(W • ν) → C(W • η) and Tˆν : C(W • η) → C(W • ν) the associated ν a a η a a translation functors. Lemma: Assume p ≫ 0 so that (L) holds. (i) ∆ˆ(η+) ≤ radN(ν)Tˆη∆ˆ(ν). ν (ii) Lˆ(η−) ≤ soc ∇ˆ(η+). N(ν)+1 (iii) ℓℓ(TˆηLˆ(ν)) ≥ 2N(ν)+1. ν (iv) ∀M ∈ C(ν), ℓℓ(TˆηM) ≥ 2N(ν)+ℓℓ(M). ν Proof: (i) Recall from [AJS, 18.13] that the translation functors Tˆν and Tˆη admit graded η ν versions, denoted T and T∗, resp. If we let ∆˜(η) denote the graded version of ∆ˆ(η), ! T∗T∆˜(η−) admits by [AJS, 18.15] a filtration with the subquotients ∆˜(w•η−)ho(w•η−)i, ! w ∈ W , where o(w • η−) denotes the number of hyperplanes H ,α ∈ R+,n ∈ Z, on ν α,n which ν lies and such that w •η− belongs to their positive sides [AJS, 15.13]. Thus the graded version of Lˆ(η+) = hd∆ˆ(η+) appears in T∗T∆˜(η−) as L˜(η+)hN(ν)i while that of ! Lˆ(η−) = hd∆ˆ(η−) = hdTˆη∆ˆ(ν) appears as L˜(η−). Under the assumtion (L), ∆˜(η−) is ν 8 graded over the Koszul algebra Eˆ(W •η) from (1.1), and so therefore is T∗T∆˜(η−). As a ! Tˆη∆ˆ(ν) has a simple socle and a simple head, its Loewy series coincides with the grading ν filtration up to degree shift by [BGS]. It follows that Lˆ(η+) appears in rad Tˆη∆ˆ(ν), N(ν) ν and hence ∆ˆ(η+) ≤ radN(ν)Tˆη∆ˆ(ν). ν (ii) Note first that the number of times ∇ˆ(η+) appears in the ∇ˆ-filtration of Pˆ(η−) is by the translation principle equal to (1) [∇ˆ(η+) : Lˆ(η−)] = 1. Thus, if r = max{i ∈ N | η = η+} in the ∇ˆ-filtration M• of Pˆ(η−) with the suquotients i Mi/Mi−1 ≃ ∇ˆ(η ), one has Tˆη∇ˆ(ν) ≤ Mr. By (1) and by [AK, 3.5] there is unique j ∈ N i ν such that [soc Mr : Lˆ(η+)] = [soc ∇ˆ(η+) : Lˆ(η−)] = 1. As [Tˆη∇ˆ(ν) : Lˆ(η+)] 6= 0, we j+1 j+1 ν must have [soc Tˆη∇ˆ(ν) : Lˆ(η+)] = 1. Then taking the τ-dual yields that [rad Tˆη∆ˆ(ν) : j+1 ν j ν Lˆ(η+)] = 1, and hence j = N(ν) by (i). (iii) Consider a filtration of Tˆη∆ˆ(ν) with the subquotients ∆ˆ(w • η), w ∈ W . By ν ν the weight consideration TˆηLˆ(ν) must contain all the composition factors of Tˆη∆ˆ(ν) ν ν isomorphic to Lˆ(η−). On the other hand, [∆ˆ(w•η+) : Lˆ(η−)] = 1 ∀w ∈ W as in (1). Thus TˆηLˆ(ν) contains ν ν a composition factor Lˆ(η−) corresponding to one in each of ∆ˆ(w•η+), w ∈ W . Consider ν the factor corresponding to the one in ∆ˆ(η+). Let θ ∈ G TMod(∆ˆ(η+),TˆηLˆ(ν)) be the 1 ν restriction to ∆ˆ(η+) of the quotient Tˆη∆ˆ(ν) → TˆηLˆ(ν). Then imθ ≤ radN(ν)TˆηLˆ(ν) by ν ν ν (i). As the composition factor Lˆ(η−) comes from the one in rad ∆ˆ(η+) by (ii), it lies N(ν) in rad (imθ). It follows that 2N(ν)+1 ≤ ℓℓ(imθ)+N(ν) ≤ ℓℓ(TˆηLˆ(ν)). N(ν) ν (iv) Consider a nonsplit exact sequence 0 → Lˆ(y•ν) → K → Lˆ(x•ν) → 0, x,y ∈ W , a with x•ν > y •ν. There is an epi ∆ˆ(x•ν) ։ K. As Tˆη∆ˆ(x•ν) has a simple head, so ν does TˆηK. In particular, TˆηK is indecomposable, and so therefore is (TˆηK)τ ≃ Tˆη(Kτ). ν ν ν ν We now argue by induction on ℓℓ(M). We may assume M has a simple head. Let Lˆ(x•ν) = hdM, x ∈ W . Take a quotient M/M′ with radM > M′ > rad2M which fits a in a short exact sequence 0 → Lˆ(y •ν) → M/M′ → Lˆ(x•ν) → 0 for some y ∈ W . As a Tˆη(M/M′) is indecomposable, the exact sequence 0 → Tˆη(radM) → TˆηM → TˆηLˆ(x • ν ν ν ν ν) → 0 cannot split. Thus ℓℓ(TˆηM) ≥ ℓℓ(Tˆη(radM))+1, as desired. ν ν (2.5) Keep the notation of (2.4). Let w denote the longest element of W. ∀x ∈ W , recall 0 a from [AK, 3.4.2] that (1) ℓℓPˆ(ν) ≥ 2ℓℓ(∇ˆ(w •ν))−1 ≥ 2N −2N(ν)+1, 0 and from [AK, 2.3] that ℓℓ∇ˆ(w •ν) ≥ N −N(ν)+1. Thus 0 2N +1 = ℓℓPˆ(η−) by [AK, 5.4] = ℓℓ(TˆηPˆ(ν)) ≥ ℓℓ(Pˆ(ν))+2N(ν) by (2.4) ν ≥ 2N −2N(ν)+1+2N(ν) = 2N +1. 9 It follows that ℓℓPˆ(ν) = 2N −2N(ν)+1, and then ℓℓ∇ˆ(w •ν) = N −N(ν)+1 by (1). 0 As ∇ˆ ((w•ν)hwi) ≃ w∇ˆ(ν)⊗p(w•0) ∀w ∈ W by (1.5), we have w ℓℓ∇ˆ ((w•ν)hwi) = 1+N −N(ν) = 1+dimG/B −N(ν). w Let us also record Theorem: Assume p ≫ 0 so that (L) holds. ∀ν ∈ Λ, ℓℓ(Pˆ(ν)) = 2N −2N(ν)+1. (2.6) Recall the notation of (1.8). To find the Loewy length of ∇ˆ (LˆJ(ν)), we first recall J some identities from [AbK]. These hold without restrictions on p. Let w denote the J longest element of W and put wJ = w w . Let ν ∈ Λ. We will write νhwi, w ∈ W, for J 0 J ν +(p−1)(w•0). One can reformulate [AbK, 1.4] as an isomorphism hd∇ˆJ(LˆJ(ν)) ≃ L((wJ • ν)0) ⊗k p{(wJ)−1•(wJ•ν)1}. Also, [AbK, 4.5] carries over to arbitrary ν ∈ Λ: hdwJ∇ˆ (LˆJ(ν)) ≃ J Lˆ(wJ •ν)⊗k {−p(wJ •0)}. We then have a commutative diagram from [AbK, 4.6.1] ∇ˆ ((wJ •ν)hwJi)⊗{−p(wJ •0)} φwJ⊗{−p(wJ•0)} //∇ˆ(wJ •ν)⊗{−p(wJ •0)} wJ OO OO (cid:31)? (cid:31)? wJ∇ˆ (LˆJ(ν)) ////Lˆ(wJ •ν)⊗{−p(wJ •0)} J and another from [AbK, 4.6.3] ∇ˆ ((wJ •ν)hw i)⊗{−p(wJ •0)} w0 0 ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ **** φ′ ⊗{−p(wJ•0)} wJ∇ˆ (LˆJ(ν)) wI ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤Ff J (cid:15)(cid:15) ss ∇ˆ ((wJ •ν)hwJi)⊗{−p(wJ •0)}. wJ If we write wJ = s ...s in a reduced expression with s denoting the reflection associ- i1 in i ated to the simple root α , the homomorphism φ : ∇ˆ ((wJ •ν)hwJi) → ∇ˆ(wJ •ν) is i wJ wJ the composite ∇ˆ ((wJ •ν)hs ···s i) → ∇ˆ ((wJ •ν)hs ···s i) → ··· si1···sin i1 in si1···sin−1 i1 in−1 → ∇ˆ ((wJ •ν)hs s i) → ∇ˆ ((wJ •ν)hs i) → ∇ˆ((wJ •ν)) si1si2 i1 i2 si1 i1 with each ∇ˆ ((wJ • ν)hs ···s i) → ∇ˆ ((wJ • ν)hs ···s i) bijective iff si1···sir i1 ir si1···sir−1 i1 ir−1 hwJ • ν + ρ,s ···s α∨i ≡ 0 mod p [AK, 2.2]. Thus, if we put R+(w) = {α ∈ R+ | i1 ir−1 ir wα < 0}, w ∈ W, and R+ = {α ∈ R+ | hν+ρ,α∨i ≡ 0 mod p}, then φ ⊗{−p(wJ•0)} ν wJ annihilates socℓ(wJ)−|R+(wJ)∩R+ν|∇ˆ ((wJ •ν)hwJi)⊗{−p(wJ •0)}, and hence wJ (1) ℓℓwJ∇ˆ (LˆJ(ν)) ≥ ℓ(wJ)−|R+(wJ)∩R+|+1. J ν 10