Christopher P. Bendel

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

On comparing the cohomology of algebraic groups, finite Chevalley groups and Frobenius kernels

Let G be a semisimple simply connected algebraic group de(ned and split over the (eld Fp with p elements, G(Fq) be the (nite Chevalley group consisting of the Fq-rational points of G where q = p r , and Gr be the rth Frobenius kernel of G. This paper investigates relationships between the extension theories of G, G(Fq), and Gr over the algebraic closure of Fp. First, some qualitative results relating extensions over G(Fq) and Gr are presented. Then certain extensions over G(Fq) and Gr are explicitly identi(ed in terms of extensions over G. c 2001 Elsevier Science B.V. All rights reserved. MSC: Primary 20C; 20G; secondary 20J06; 20G10

Extensions for finite Chevalley groups II

Let G be a semisimple simply connected algebraic group defined and split over the field Fp with p elements, let G(Fq) be the finite Chevalley group consisting of the Fq-rational points of G where q = p r , and let G r be the rth Frobenius kernel. The purpose of this paper is to relate extensions between modules in Mod(G(F q )) and Mod(G r ) with extensions between modules in Mod(G). Among the results obtained are the following: for r > 2 and p > 3(h - 1), the G(F q )-extensions between two simple G(F q )-modules are isomorphic to the G-extensions between two simple p r -restricted G-modules with suitably twisted highest weights. For p > 3(h - 1), we provide a complete characterization of H 1 (G(F q ), H0(λ)) where H 0 (λ) = ind G B A and A is p r -restricted. Furthermore, for p > 3(h - 1), necessary and sufficient bounds on the size of the highest weight of a G-module V are given to insure that the restriction map H 1 (G, V) → H 1 (G(F q ), V) is an isomorphism. Finally, it is shown that the extensions between two simple p r -restricted G-modules coincide in all three categories provided the highest weights are close together.

Extensions for finite Chevalley groups I

Let G be a connected semisimple algebraic group defined and split over the field Fp with p elements, and k be the algebraic closure of Fp. Assume further that G is almost simple and simply connected and let G(Fq) be the finite Chevalley group consisting of Fq-rational points of G where q=pr for a non-negative integer r. In this paper, formulas are found relating extensions between simple kG(Fq)-modules and extensions over G (considered as an algebraic group over k). One of these formulas, which only holds for primes p⩾3(h−1) (where h is the Coxeter number of G), is then used to show the vanishing of self-extensions between simple kG(Fq)-modules except for certain simple modules when r=1 and the underlying root system is of type A1 or Cn.

Projectivity of modules for infinitesimal unipotent group schemes

In this paper, it is shown that the projectivity of a rational module for an infinitesimal unipotent group scheme over an algebraically closed field of positive characteristic can be detected on a family of closed subgroups. Let k be an algebraically closed field of characteristic p > 0 and G be an infinitesimal group scheme over k, that is, an affine group scheme G over k whose coordinate (Hopf) algebra k[G] is a finite-dimensional local k-algebra. A rational G-module is equivalent to a k[G]-comodule and further equivalent to a module for the finitedimensional cocommutative Hopf algebra k[G]∗ ≡ Homk(k[G], k). Since k[G]∗ is a Frobenius algebra (cf. [Jan]), a rational G-module (even infinite-dimensional) is in fact projective if and only if it is injective (cf. [FW]). Further, for any rational G-module M and any closed subgroup scheme H ⊂ G, if M is projective over G, then it remains projective upon restriction to H (cf. [Jan]). We consider the question of whether there is a “nice” collection of closed subgroups of G upon which projectivity (over G) can be detected. For an example of what we mean by a “nice” collection, consider the situation of modules over a finite group. Over a field of characteristic p > 0, a module over a finite group is projective if and only if it is projective upon restriction to a p-Sylow subgroup (cf. [Rim]). For a p-group (and hence for any finite group), L. Chouinard [Ch] showed that a module is projective if and only if it is projective upon restriction to every elementary abelian subgroup. If the module is assumed to be finite-dimensional, this result follows from the theory of varieties for finite groups (cf. [Ca] or [Ben]). Indeed, elementary abelian subgroups play an essential role in this theory. In work of A. Suslin, E. Friedlander, and the author [SFB1], [SFB2], a theory of varieties for infinitesimal group schemes was developed. In this setting, subgroups of the form Ga(r) (the rth Frobenius kernel of the additive group scheme Ga) play the role analogous to that of elementary abelian subgroups in the case of finite groups. Not surprisingly then, for finite-dimensional modules, one obtains the following analogue of Chouinard’s Theorem. Proposition 1 ([SFB2, Proposition 7.6]). Let k be an algebraically closed field of characteristic p > 0, r > 0 be an integer, G be an infinitesimal group scheme over k of height ≤ r, and M be a finite-dimensional rational G-module. Then M is Received by the editors January 27, 1998 and, in revised form, March 24, 1998 and May 19, 1999. 2000 Mathematics Subject Classification. Primary 14L15, 20G05; Secondary 17B50. c ©2000 American Mathematical Society

Bounding the Dimensions of Rational Cohomology Groups

Let k be an algebraically closed field of characteristic p > 0, and let G be a simple simply-connected algebraic group over k. 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. As an application, effective bounds on the first cohomology of the symmetric group are obtained. We also show how, for finite Chevalley groups, our methods permit significant improvements over previous estimates for the dimensions of second cohomology groups.