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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. The authors first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring. They 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., Comment: The paper has been published

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 $G$ 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 $\Ext^\bullet_{G_1}(V,W)$ 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 $\Ext^n_{G_1}(L,L')^{[-1]}$ 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., Comment: Numerous typos and other corrections have been made

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple simply-connected algebraic group over $k$ that is defined and split over the prime field $\mathbb{F}_p$. In this paper we investigate situations where the dimension of a rational cohomology group for $G$ can be bounded by a constant times the dimension of the coefficient module. We then demonstrate how our results can be applied to obtain effective bounds on the first cohomology of the symmetric group. We also show how, for finite Chevalley groups, our methods permit significant improvements over previous estimates for the dimensions of second cohomology groups., Comment: 13 pages

Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let \sigma:G->G be a strict endomorphism (i. e., the subgroup G(\sigma) of \sigma-fixed points is finite). Also, let G_\sigma\ be the scheme-theoretic kernel of \sigma, an infinitesimal subgroup of G. This paper shows that the degree m cohomology H^m(G(\sigma),L) of any irreducible kG(\sigma)-module L is bounded by a constant depending on the root system \Phi\ of G and the integer m. A similar result holds for the degree m cohomology of G_\sigma. These bounds are actually established for the degree m extension groups Ext^m_{G(\sigma)}(L,L') between irreducible kG(\sigma)-modules L and L', with again a similar result holding for G_\sigma. In these Ext^m results, of interest in their own right, the bounds depend also on L, or, more precisely, on length of the p-adic expansion of the highest weight associated to L. All bounds are, nevertheless, independent of the characteristic p. These results extend earlier work of Parshall and Scott for rational representations of algebraic groups G. We also show that one can find bounds independent of the prime for the Cartan invariants of G(\sigma) and G_{\sigma}, and even for the lengths of the underlying PIMs. These bounds, which depend only on the root system of G and the "height" of \sigma, provide in a strong way an affirmative answer to a question of Hiss, for the special case of finite groups G(\sigma) of Lie type in the defining characteristic., Comment: 22 pages

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., Comment: 21 pages

Let $\zeta$ be a complex $\ell$th root of unity for an odd integer $\ell>1$. For any complex simple Lie algebra $\mathfrak g$, let $u_\zeta=u_\zeta({\mathfrak g})$ be the associated "small" quantum enveloping algebra. In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when $l$ (resp., $p$) is smaller than the Coxeter number $h$ of the underlying root system. For example, Lusztig's conjecture concerning the characters of the rational irreducible $G$-modules stipulates that $p \geq h$. The main result in this paper provides a surprisingly uniform answer for the cohomology algebra $\opH^\bullet(u_\zeta,{\mathbb C})$ of the small quantum group. When $\ell>h$, this cohomology algebra has been calculated by Ginzburg and Kumar \cite{GK}. Our result requires powerful tools from complex geometry and a detailed knowledge of the geometry of the nullcone of $\mathfrak g$. In this way, the methods point out difficulties present in obtaining similar results for the restricted enveloping algebra $u$ in small characteristics, though they do provide some clarification of known results there also. Finally, we establish that if $M$ is a finite dimensional $u_\zeta$-module, then $\opH^\bullet(u_\zeta,M)$ is a finitely generated $\opH^\bullet(u_\zeta,\mathbb C)$-module, and we obtain new results on the theory of support varieties for $u_\zeta$.

The authors compute the support varieties of all irreducible modules for the small quantum group $u_\zeta(\mathfrak{g})$, where $\mathfrak{g}$ is a simple complex Lie algebra, and $\zeta$ is a primitive $\ell$-th root of unity with $\ell$ larger than the Coxeter number of $\mathfrak{g}$. The calculation employs the prior calculations and techniques of Ostrik and of Nakano--Parshall--Vella, as well as deep results involving the validity of the Lusztig character formula for quantum groups and the positivity of parabolic Kazhdan-Lusztig polynomials for the affine Weyl group. Analogous support variety calculations are provided for the first Frobenius kernel $G_1$ of a reductive algebraic group scheme $G$ defined over the prime field $\mathbb{F}_p$., Comment: 10 pages, various typos corrected, references updated