A key result in a 2004 paper by S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg compares the bounded derived category of modules for the principal block of a Lusztig quantum enveloping algebra at anroot of unity with an explicit subcategory of the bounded derived category of integrable type 1 modules for a Borel part of that algebra. Specifically, according to this Induction Theorem the right derived functor of induction yields an equivalence of triangulated categories. The authors ABG of that paper suggest a similar result holds for algebraic groups in positive characteristic p, and this paper provides a statement with proof for such a modular induction theorem. Our argument uses the philosophy of ABG as well as new ingredients. A secondary goal of this paper has been to put the original characteristic zero quantum result of ABG on firmer ground, and we provide arguments as needed to give a complete proof of that result also. Finally, using the modular result, we have been able in a separate preprint to introduce truncation functors, associated to finite weight posets, which effectively commute with the modular induction equivalence, assuming p>2h-2, with h the Coxeter number. This enables interpreting the equivalence at the level of derived categories of modules for suitable finite dimensional quasi-hereditary algebras. We expect similar results to hold in the the quantum setting.

Let $G$ be a finite group of Lie type. In studying the cross-characteristic representation theory of $G$, the (specialized) Hecke algebra $H=\End_G(\ind_B^G1_B)$ has played a important role. In particular, when $G=GL_n(\mathbb F_q)$ is a finite general linear group, this approach led to the Dipper-James theory of $q$-Schur algebras $A$. These algebras can be constructed over $\sZ:=\mathbb Z[t,t^{-1}]$ as the $q$-analog (with $q=t^2$) of an endomorphism algebra larger than $H$, involving parabolic subgroups. The algebra $A$ is quasi-hereditary over $\sZ$. An analogous algebra, still denoted $A$, can always be constructed in other types. However, these algebras have so far been less useful than in the $GL_n$ case, in part because they are not generally quasi-hereditary. Several years ago, reformulating a 1998 conjecture, the authors proposed (for all types) the existence of a $\sZ$-algebra $A^+$ having a stratified derived module category, with strata constructed via Kazhdan-Lusztig cell theory. The algebra $A$ is recovered as $A=eA^+e$ for an idempotent $e\in A^+$. A main goal of this monograph is to prove this conjecture completely. The proof involves several new homological techniques using exact categories. Following the proof, we show that $A^+$ does become quasi-hereditary after the inversion of the bad primes. Some first applications of the result -- e.g., to decomposition matrices -- are presented, together with several open problems.

Parshall died suddenly and unexpectedly on

A key result in a 2004 paper by S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg compares the bounded derived category D-b(block(U)) of finite dimensional modules for the principal block of a Lusztig quantum enveloping algebra U at an lth root of unity with a special full subcategory D-triv(B) of the bounded derived category of integrable type 1 modules for a Borel part B subset of U. Specifically, according to this "Induction Theorem" [1, Theorem 3.5.5] the right derived functor of induction Ind(B)(U) yields an equivalence of categories RInd(B)(U): D-triv(B) (->) over tilde D-b (block(U)). Some restrictions on l are required e.g., l > h, the Coxeter number. It is suggested briefly [1, Remark 3.5.6] that an analog of this equivalence carries over to characteristic p > 0 representations of algebraic groups. Indeed, the authors of the present paper have verified, in a separate preprint [6], that there is such an equivalence RInd(B)(G) : D-triv(B) (->) over tilde D-b (block(G)) relating an analog of D-triv(B), defined using a Borel subgroup B of a simply connected semisimple algebraic group G, to the bounded derived category of the principal block of finite dimensional rational G-modules. The proof is not without difficulty and supplies new, previously missing details even in the quantum case. The present paper continues the study of the modular case, taking the derived category equivalence as a starting point. The main result here is that, assuming p > 2h - 2, the equivalence behaves well with respect to certain weight poset "truncations," making use of a variation by Woodcock [12] on van der Kallen's "excellent order" [10]. This means, in particular, that the equivalence can be reformulated in terms of derived categories of finite dimensional quasi-hereditary algebras. We expect that a similar result holds in the quantum case. In [9, Thm. 2.1] J. Rickard proves a theorem stating existence, in an algebraic groups context, of some derived equivalences with specified corresponding character isometries. In an appendix to this paper, we discuss a similar more recent result [6, Lem. 3.2]. It gives additional information on the behavior of the derived equivalences, giving their action on some natural right-derived induced objects, lifting the given character isometries. (C) 2020 Elsevier Inc. All rights reserved.

The tables of this title are a first attempt to understand empirically the sizes of certain distinguished sets, introduced by Hankyung Ko, of elements in affine Weyl groups. The sizes are relevant to the computational efficiency of direct approaches to computing characters of modular representations of algebraic groups from characters of corresponding irreducible representations of quantum groups.

This paper aims at developing a "local-global" approach for various types of finite dimensional algebras,especially those related to Hecke algebras.The eventual intention is to apply the methods and applications developed here to the cross-characteristic representation theory of finite groups of Lie type.We first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring.We prove that many global properties of these algebras hold if and only if they hold locally at every prime ideal.When the commutative ring is sufficiently good,it is often sufficient to check just the prime ideals of height at most one.These methods are applied to construct certain generalized q-Schur algebras,proving they are often quasi-hereditary (the "good" prime case) but always stratified.Finally,these results are used to prove a triangular decomposition matrix theorem for the modular representations of Hecke algebras at good primes.In the bad prime case,the generalized q-Schur algebras are at least stratified,and a block triangular analogue of the good prime case is proved,where the blocks correspond to Kazhdan-Lusztig cells.

This paper constructs enlargements of Hecke algebras over Z[t,t−1] to certain standardly stratified algebras. The latter are obtained as endomorphism algebras of modules with dual left cell module filtrations in the sense of Kazhdan–Lusztig. A novel feature of the proofs is the use of suitably chosen exact categories to avoid difficult Ext1-vanishing conditions.

In well-known work, Kazhdan and Lusztig (1979) defined a new set of Hecke algebra basis elements (actually two such sets) associated to elements in any Coxeter group. Often these basis elements are computed by a standard recursive algorithm which, for Coxeter group elements of long length, generally involves computing most basis elements corresponding to Coxeter group elements of smaller length. Thus, many calculations simply compute all basis elements associated to a given length or less, even if the interest is in a specific Kazhdan-Lusztig basis element. Similar remarks apply to "parabolic" versions of these basis elements defined later by Deodhar (1987,1990), though the lengths involved are the (smaller) lengths of distinguished coset representatives. We give an algorithm which targets any given Kazhdan-Lusztig basis element or parabolic analog and does not precompute any other Kazhdan-Lusztig basis elements. In particular it does not have to store them. This results in a considerable saving in memory usage, enabling new calculations in an important case (for finite and algebraic group 1-cohomology with irreducible coefficients) analyzed by Scott-Xi (2010).

This paper considers Weyl modules for a semisimple, simply connected algebraic group G over an algebraically closed field k of positive characteristic p not equal 2. The main result proves, if p >= 2h - 2 (where h is the Coxeter number) and if the Lusztig character formula holds for all (irreducible modules with) p-regular p-restricted highest weights, then any Weyl module Delta(lambda) has a Delta(p)-filtration, namely a filtration with sections of the form Delta(p)(mu(0) + p mu(1)) := L(mu(0))circle times Delta(mu(1))([1]), where mu(0) is p-restricted and mu(1) is arbitrary dominant. In case the highest weight lambda of the Weyl module Delta(lambda) is p-regular, the p-filtration is compatible with the G(1)-radical series of the module. The problem of showing that Weyl modules have Delta(p)-filtrations was first proposed by Jantzen in 1980. The proof in this paper is based on new methods involving 'forced gradings' arising from orders associated to quantum enveloping algebras. A new Ext(1)-criterion is proved for Delta(p)-filtrations, but only in the context of such forced gradings. Finally, in subsequent work, these results have already had applications to the G-module structure of Ext-groups for the restricted enveloping algebra of G.

The (Iwahori-)Hecke algebra in the title is a q-deformation H of the group algebra of a finite Weyl group W. The algebra has a natural enlargement to an endomorphism algebra A = End(H)(T) where T is a q-permutation module. In type An (i.e., W congruent to Sn+1), the algebra A is a q-Schur algebra which is quasi-hereditary and plays an important role in the modular representation of the finite groups of Lie type. In other types, A is not always quasi-hereditary, but the authors conjectured 20 year ago that T can be enlarged to an H-module T+ that A(+) = End(H)(T+) is at least standardly stratified, a weaker condition than being quasi-hereditary, but with "strata" corresponding to Kazhdan-Lusztig two-sided cells.The main result of this paper is a "local" version of this conjecture in the equal parameter case, viewing as defined over Z [t, t(-1)], with the localization at a prime ideal generated by a cyclotomic polynomial Phi(2e)(t), e not equal 2. The proof uses the theory of rational Cherednik algebras (also known as RDAHAs) over similar localizations of C[t, t(-1)]. In future papers, the authors hope to prove global versions of the conjecture, maintaining these localizations.

This paper has two parts. The main goal, carried out in Part I, is to survey some recent work by the authors in which "forced" grading constructions have played a significant role in the representation theory of semisimple algebraic groups $G$ in positive characteristic. The constructions begin with natural finite dimensional quotients of the distribution algebras Dist$(G)$, but then "force" gradings into the picture by passing to positively graded algebras constructed from ideal filtrations of these quotients. This process first guaranteed a place for itself by proving, for large primes, that all Weyl modules have $p$-Weyl filtrations. Later it led, under similar circumstances, to a new "good filtration" result for restricted Lie algebra Ext groups between restricted irreducible $G$-modules. In the process of proving these results, a new kind of graded algebra was invented, called a Q-Koszul algebra. Recent conjectures suggest these algebras arise in forced grading constructions as above, from quotients of Dist$(G)$, even for small primes and even in settings possibly involving singular weights. Related conjectures suggest a promising future for using Kazhdan-Lusztig theory to relate quantum and algebraic group cohomology and Ext groups in these same small prime and possibly singular weight settings. A part of one of these conjectures is proved in Part II of this paper. The proof is introduced by remarks of general interest on positively graded algebras and Morita equivalence, followed by a discussion of recent Koszulity results of Shan-Varagnalo-Vasserot, observing some extensions. Version 2 corrects some minor typos and inaccurate references. The paper will appear in PSPUM.

The (Iwahori-)Hecke algebra in the title is a $q$-deformation $\sH$ of the group algebra of a finite Weyl group $W$. The algebra $\sH$ has a natural enlargement to an endomorphism algebra $\sA=\End_\sH(\sT)$ where $\sT$ is a $q$-permutation module. In type $A_n$ (i.e., $W\cong {\mathfrak S}_{n+1}$), the algebra $\sA$ is a $q$-Schur algebra which is quasi-hereditary and plays an important role in the modular representation of the finite groups of Lie type. In other types, $\sA$ is not always quasi-hereditary, but the authors conjectured 20 year ago that $\sT$ can be enlarged to an $\sH$-module $\sT^+$ so that $\sA^+=\End_\sH(\sT^+)$ is at least standardly stratified, a weaker condition than being quasi-hereditary, but with "strata" corresponding to Kazhdan-Lusztig two-sided cells. The main result of this paper is a "local" version of this conjecture in the equal parameter case, viewing $\sH$ as defined over ${\mathbb Z}[t,t^{-1}]$, with the localization at a prime ideal generated by a cyclotomic polynomial $\Phi_{2e}(t)$, $e\not=2$. The proof uses the theory of rational Cherednik algebras (also known as RDAHAs) over similar localizations of ${\mathbb C}[t,t^{-1}]$. In future paper, the authors expect to apply these results to prove global versions of the conjecture, at least in the equal parameter case with bad primes excluded.

Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let σ :G → G be a strict endomorphism (i.e., the subgroup G(σ) of σ-fixed points is finite). Also, let G σ be the scheme-theoretic kernel of σ, an infinitesimal subgroup of G. This paper shows that the dimension of the degree m cohomology group Hm(G(σ),L) for any irreducible k G(σ)-module L is bounded by a constant depending on the root system Φ of G and the integer m. These bounds are actually established for the degree m extension groups \( Ext^{m}_{G(\sigma )}(L,L^{\prime })\) between irreducible k G(σ)-modules \(L,L^{\prime }\), with a similar result holding for G σ . In these Extm results, the bounds also depend on the highest weight associated to L, but are, nevertheless, independent of the characteristic p.

In previous work, the authors introduced the notion of Q-Koszul algebras, as a tool to "model" module categories for semisimple algebraic groups over fields of large characteristics. Here we suggest the model extends to small characteristics as well. In particular, we present several conjectures in the modular representation theory of semisimple groups which these algebras inspire. They provide a new world-view of modular representation theory, potentially valid for some root systems in all characteristics. In fact, we give a non-trivial example in which $p=2$. This paper begins a systematic study of Q-Koszul algebras, viewed as interesting objects in their own right.

Let G be a simple, simply connected algebraic group over an algebraically closed field of positive characteristic p. In recent work, the authors have studied a graded analogue of the category of rational G-modules. These gradings are not natural but are "forced" on related algebras though filtrations, often obtained from appropriate quantum structures. This paper presents new results on Koszul modules for the graded algebras obtained through this forced grading process. Most of these results require that the Lusztig character formula holds for all restricted p-regular weights, but the paper begins to investigate how these and previous results might be established when the Lusztig character formula is only assumed to hold on a proper poset ideal in the Jantzen region. This opens up the possibility of inductive arguments.

Let N be a normal subgroup of a group G . An N -module Q is called G -stable provided that Q is equivalent to the twist Q g of Q by g , for every g ∈ G . If the action of N on Q extends to an action of G on Q , then Q is obviously G -stable, but the converse need not hold. A famous conjecture in the modular representation theory of reductive algebraic groups G asserts that the (obviously G -stable) projective indecomposable modules (PIMs) Q for the Frobenius kernels G r ( r ≥ 1) of G have a G -module structure. It is sometimes just as useful (for a general module Q ) to know that a finite direct sum Q ⊕ n of Q has a compatible G -module structure. In this paper, this property is called numerical stability. In recent work (Parshall and Scott, Adv Math 226:2065–2088, 2011 ), the authors established numerical stability in the special case of PIMs. We provide in this paper a more general context for that result, working in the context of k -group schemes and a suitable version of G -stability, called strong G -stability. Among our results here is the determination of necessary and sufficient conditions for the existence of a compatible G -module structure on a strongly G -stable N -module, in the form of a cohomological obstruction which must be trivial precisely when the G -module structure exists. Our main result is achieved by giving an approach to killing the obstruction by tensoring with certain finite dimensional G / N -modules.

Abstract Given a quasi-hereditary algebra $B$, we present conditions which guarantee that the algebra $\mathrm{gr} \hspace{0.167em} B$ obtained by grading $B$ by its radical filtration is Koszul and at the same time inherits the quasi-hereditary property and other good Lie-theoretic properties that $B$ might possess. The method involves working with a pair $(A, \mathfrak{a})$ consisting of a quasi-hereditary algebra $A$ and a (positively) graded subalgebra $\mathfrak{a}$. The algebra $B$ arises as a quotient $B= A/ J$ of $A$ by a defining ideal $J$ of $A$. Along the way, we also show that the standard (Weyl) modules for $B$ have a structure as graded modules for $\mathfrak{a}$. These results are applied to obtain new information about the finite dimensional algebras (e.g., the $q$-Schur algebras) which arise as quotients of quantum enveloping algebras. Further applications, perhaps the most penetrating, yield results for the finite dimensional algebras associated with semisimple algebraic groups in positive characteristic $p$. These results require, at least at present, considerable restrictions on the size of $p$.

The idea that the cohomology of finite groups might be fruitfully approached via the cohomology of ambient semisimple algebraic groups was first shown to be viable in the papers [CPS75] and [CPSvdK77]. The second paper introduced, through a limiting process, the notion of generic cohomology, as an intermediary between finite Chevalley group and algebraic group cohomology. The present paper shows that, for irreducible modules as coefficients, the limits can be eliminated in all but finitely many cases. These exceptional cases depend only on the root system and cohomological degree. In fact, we show that, for sufficiently large r, depending only on the root system and m, and not on the prime p or the irreducible module L, there are isomorphisms H^m(G(p^r),L) -> H^m(G(p^r),L') -> H^m_gen(G,L') -> H^m(G,L'), where the subscript gen refers to generic cohomology and L' is a constructibly determined irreducible shift of the (arbitrary) irreducible module L for the finite Chevalley group G(p^r). By a famous theorem of Steinberg, both L and L' extend to irreducible modules for the ambient algebraic group G with p^r-restricted highest weights. This leads to the notion of a module or weight being shifted m-generic, and thus to the title of this paper. Our approach is based on questions raised by the third author in [Stea], which we answer here in the cohomology cases. We obtain many additional results, often with formulations in the more general context of Ext^m_G(q) with irreducible coefficients.

Let G be a semisimple algebraic group over an algebraically closed field k of positive characteristic p. Under some restrictions on the size of p, the present paper establishes new results on the G-module structure of ^∙_G_1(V,W) when V,W belong to several important classes of rational G-modules, and G_1 denotes the first Frobenius kernel of G. For example, it is proved that, if L,L' are (p-regular) irreducible G_1-modules, then ^n_G_1(L,L')^[-1] has a good filtration with computable multiplicities. This and many other results depend on the entirely new technique of using methods of what we call forced gradings in the representation theory of G, as developed by the authors in recent papers, and extended here. In addition to providing proofs, these methods lead effectively to a new conceptual framework for the study of rational G-modules, and, in this context, to the introduction of a new class of graded finite dimensional algebras, which we call Q-Koszul algebras. These algebras are similar to Koszul algebras, but are quasi-hereditary, rather than semisimple, in grade 0.

Let $G$ be a semisimple, simply connected algebraic group defined and split over a prime field ${\mathbb F}_p$ of positive characteristic. For a positive integer $r$, let $G_r$ be the $r$th Frobenius kernel of $G$. Let $Q$ be a projective indecomposable (rational) $G_r$-module. The well-known Humprheys-Verma conjecture (cf. \cite{Ballard}) asserts that the $G_r$-action on $Q$ lifts to an rational action of $G$ on $Q$. For $p\geq 2h-2$ (where $h$ is the Coxeter number of $G$), this conjecture was proved by Jantzen in 1980, improving on early work of Ballard. However, it remains open for general characteristics. In this paper, the authors establish several graded analogues of the Humphreys-Verma conjecture, valid for all $p$. The most general of our results, proved in full here, was announced (without proof) in an earlier paper. Another result relates the Humphreys-Verma conjecture to earlier work of Alperin, Collins, and Sibley on finite group representation theory. A key idea in all formulations involves the notion of a forced grading. The latter goes back, in particular, to the recent work of the authors, relating graded structures and $p$-filtrations. The authors anticipate that the Humphreys-Verma conjecture results here will lead to extensions to smaller characteristics of these earlier papers.