Brian Parshall

Rational and Generic Cohomology

Let G be a semisimple algebraic group defined and split over k,=GF(p). For q=p”, let G(q) be the subgroup of GF(q)-rational points. The main objective of this paper is to relate the cohomology of the finite groups G(q) to the rational cohomology of the algebraic group G. Let I/ be a finite dimensional rational G-module, and, for a non-negative integer e, let V(e) be the G-module obtained by “twisting” the original G-action on V by the Frobenius endomorphism x++xtPel of G. Theorem (6.6) states that, for sufficiently large q and e (depending on I’ and n), there are isomorphisms H”(G, V(e))gH’(G(q), V(e))rH”(G(q), V) where the first map is restriction. In particular, the cohomology groups H”(G(q), V) have a stable or “generic” value H;,,(G, V). This phenomenon had been observed empirically many times (cf. [6, 203). The computation of generic cohomology reduces essentially to the computation of rational cohomology. One (surprising) consequence is that Hi,,(G, V) does not depend on the exact weight lattice for a group G of a given type cf. (6.10), though this considerably affects the structure of G(q). We also obtain that rational cohomology takes a stable value relative to twisting i.e., for sufficiently large E, we have semilinear isomorphisms H”(G, V(E)) % H”(G, V(e)) for all e 2 F. This paper contains many new results on rational cohomology beyond those required for the proof of the main theorem. We mention in particular the vanishing theorems (2.4) and (3.3), and especially the results (3.9) through (3.11) which relate H2(G, V) and Extk( K W) to the structure of Weyl modules. These results explain for example the generic values of H’ determined in [6], cf. (7.6). Also, it is shown in Theorem (3.12) that every finite dimensional rational G-module has a finite resolution by finite dimensional acyclic G-modules. A key ingredient in the proofs is an important theorem of G. Kempf [I93 on the vanishing of cohomology of certain homogeneous line bundles. This result is translated into the language of rational cohomology in (1.2), and is used in

Cohomology of finite groups of lie type, I

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Derived categories and Morita theory

This paper represents a first attempt to construct a Morita theory for derived categories, analogous to the classical theory for module categories of rings [l, lo]. Our motivation comes from the recent importance of derived categories in the representation theory of Lie algebras and algebraic groups [ 3, 111. However, our point of view here is entirely ring-theoretic, aimed at placing into a broader context the recent theory of tilting modules for finite dimensional algebras as developed by Brenner and Butler [S], Bongartz 141, Happel and Ringel [S], and Happel [9]. In Section 1 we briefly recall some standard notation and constructions from the theory of triangulated categories. In Section 2 we introduce the notion of a generalized tilting module for rings and show how it gives rise to equivalences of certain derived categories. The main theorem of this section , Theorem 2.1, represents an extension and generalization of a result a Happel [9], although we obtained it only after thorough study of Happels work as well as that of Bongartz 141. In Section 3 we show that the tilting module conditions arise naturally and necessarily in characterizing certain equivalences of derived categories. Finally in Section 4 we try to place the Morita theory developed so far in the context of a general Morita theory for derived categories, indicating some of the remaining difficulties. We also use part of the tilting module set up to obtain a localization result in the spirit of [2]. We would like to thank Klaus Roggenkamp for directing us to Happels work, and for sharing with us his unpublished manuscript [ 133 on generalized tilting modules for orders (,f= 2 versions in the sense of Theorem (2.1)). All nghtr of reproduchon m any lorm rewvrd

ON THE 1-COHOMOLOGY OF FINITE GROUPS OF LIE TYPE

Publisher Summary This chapter discusses the problem of calculating H1(Gσ,V) where G is a finite group of Lie type over K = GF(q) and V is a minimal G -module. It discusses two methods used for determining upper bounds on dim H1 (Gσ,V(λ)). The first method assumes r > 3, and is applicable to nonminimal modules (and can in fact be used to obtain the vanishing of H1 (Gσ,V) in many instances). The second method is an inductive procedure that does not depend on the size of the field but depends heavily on the minimality of V (λ). The second method has only been tried for the split Chevalley groups.

On the cohomology of algebraic and related finite groups

In the above theorem, H*(GL,,M~ ~ denotes the rational cohomology of the algebraic group GL, with coefficients in M(, ~), the rational GL.-module obtained from the adjoint module M . = M ~ ~ through twisting by the r th power of the Frobenius endomorphism on GL,. The algebra structure on the cohomology is induced from the associative algebra structure on M,. On the other hand, we provide the following systematic determination of certain rational cohomology groups H*(G, V(r)), where V (r) is the rational Gmodule obtained from the rational G-module V by applying the r th Frobenius twist (see (1.2)). This is a somewhat sharpened version of Theorem 4 of [36].

A mackey imprimitivity theory for algebraic groups

Let G be an affine algebraic group over an algebraically closed field k and let H be a closed subgroup of G. If V is a rational H-module (a comodule for the coordinate ring of H) there is a now well-known notion of an induced module VI G for G, defined as the space Morph/~(G, V) of all H-equivariant morphisms from G to a finite dimensional subspace of V, with obvious G-action. The question arises, given a rational G-module M, how can one recognize M as an induced module VIG? For a finite group G the answer is the Mackey imprimitivity theorem: the module M is induced if and only if it is a direct sum of subspaces permuted transitively by G (with H the stabilizer of one of these subspaces, called V). One uses this result, for example, in proving the famous Mackey decomposition theorem which describes the restriction of any induced module to a second subgroup L as a direct sum of suitable induced modules; given the imprimitivity theorem, the proof is just a matter of grouping the summands permuted by G into their orbits under L. In the case of algebraic groups the situation is quite different. For some subgroups H, all G-modules are induced. This occurs, for example, if G is connected and k [G/H] ,= k, e.g., if H is parabolic. Also, if G is a connected unipotent group, then a rational G-module M is induced from some proper subgroup if and only if its endomorphism ring contains a two dimensional submodule E, for the conjugation action of G, with E___ k. 1 and such that k. l is precisely the subspace annihilated by the action of the Lie algebra of G on E [27] (cf. also (5.5) below). The latter is an application of a general criterion in case G/H is affine: a rational G-module M is induced if and only if there is an action of the coordinate ring A of G/H compatible with the action of G on both A and M. Note that this generalizes the imprimitivity theorem in the case of finite groups, since then A has a k-basis of [G:H] orthogonal idempotents permuted transitively by G.

Monomial bases for -Schur algebras

Using the Beilinson-Lusztig-MacPherson construction of the quantized enveloping algebra of gl n and its associated monomial basis, we investigate q-Schur algebras S q (n,r) as little quantum groups. We give a presentation for S q (n,r) and obtain a new basis for the integral q-Schur algebra S q (n,r), which consists of certain monomials in the original generators. Finally, when n ≥ r, we interpret the Hecke algebra part of the monomial basis for S q (n,r) in terms of Kazhdan-Lusztig basis elements.

Reduced standard modules and cohomology

First cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts, including Wiless famous paper (1995). Internal to group theory, 1-cohomology plays a role in the general theory of maximal subgroups of finite groups, as developed by Aschbacher and Scott (1985).One can pass to the case where the group acts faithfully and the underlying module is absolutely irreducible. In this case, R. Guralnick (1986) conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups. This paper provides the first general positive results on this conjecture, proving that the generic 1-cohomology H 1 gen (G,L) := lim H 1 (G(q), L) (see Cline, Parshall, Scott, and van der Kallen) (1977) of a finite group G(q) of Lie type, with absolutely irreducible coefficients L (in the defining characteristic of G), is bounded by a constant depending only on the root system. In all cases, we are able to improve this result to a bound on H 1 (G(q),L) itself, still depending only on the root system. The generic H 1 result, and related results for Ext 1 , emerge here as a consequence of a general study, of interest in its own right, of the homological properties of certain rational modules Δ red (λ),∇ red (λ), indexed by dominant weights λ, for a reductive group G. The modules Δ red (λ) and ∇ red (λ) arise naturally from irreducible representations of the quantum enveloping algebra U ζ (of the same type as G) at a pth root of unity, where p > 0 is the characteristic of the defining field for G. Finally, we apply our Ext 1 -bounds, results of Bendel, Nakano, and Pillen (2006), as well as results of Sin (1993), (1992), (1994) on the Ree and Suzuki groups to obtain the (non-generic) bounds on H 1 (G(q),L).

On the representation theory of Lie triple systems

In this paper, we take a new look at the representation theory of Lie triple systems. We consider both ordinary Lie triple systems and restricted Lie triple systems in the sense of Hodge (2001). In a final section, we begin a study of the cohomology of Lie triple systems.

Cohomology for quantum groups via the geometry of the nullcone

Preliminaries and statement of results Quantum groups, actions, and cohomology Computation of F 0 and N(F 0 ) Combinatorics and the Steinberg module The cohomology algebra H (u ? (g),C) Finite generation Comparison with positive characteristic Support varieties over u ? for the modules ? ? (?) and ? ? (?) Appendix A Bibliography