Restrictions to G(𝔽_p) and G_(r) of rational G-modules

Abstract

We fix a prime p and consider a connected reductive algebraic group G over a perfect field k which is defined over F p . Let M be a finite-dimensional rational G-module M , a comodule for k [G]. We seek to somewhat unravel the relationship between the restriction of M to the finite Chevalley subgroup G(F p ) ⊂ G and the family of restrictions of M to Frobenius kernels G (r) ⊂ G. In particular, we confront the conundrum that if M is the Frobenius twist of a rational G-module N, M = N (1) , then the restrictions of M and N to G(F p ) are equal whereas the restriction of M to G (1) is trivial. Our analysis enables us to compare support varieties (and the finer non-maximal support varieties) for G(F p ) and G (r) of a rational G-module M where the choice of r depends explicitly on M .

On projective modules for Frobenius kernels and finite Chevalley groups

“…and by Friedlander [Fri1,Fri2] has sought to better understand the relationship between the varieties |G(F p )| M and |G 1 | M . In this note we provide an affirmative answer to Parshall's question for all r ≥ 1.…”

Filtrations, 1-parameter subgroups, and rational injectivity

“…Namely, this is a formal consequence of the fact that (−) [d] has an exact left adjoint. Thus, condition (1) implies condition (2) which clearly implies condition (3). Assume now that the rational G-module has the property that each The following proposition contrasts the behavior of injectives and mock injectives.…”

“…Since any rational G-module is a union of its finite dimensional submodules, property (4) follows from property (3).…”

“…uses restrictions of M to 1-parameter subgroups G a → G, and thus is based upon actions of G on M at p-unipotent elements of G. The role of 1-parameter subgroups to study rational G-modules was introduced in [3]; the property of p-unipotent degree introduced in [3, 2.5] is the precursor to our filtration by exponential degree. The origins of this approach to filtrations lie in considerations of support varieties…”

“…The construction of the filtration by exponential degree {M [0] ⊂ M [1] ⊂ · · · ⊂ M } uses restrictions of M to 1-parameter subgroups G a → G, and thus is based upon actions of G on M at p-unipotent elements of G. The role of 1-parameter subgroups to study rational G-modules was introduced in [3]; the property of p-unipotent degree introduced in [3, 2.5] is the precursor to our filtration by exponential degree. The origins of this approach to filtrations lie in considerations of support varieties for modules for infinitesimal group schemes, varieties which are defined in terms of p-nilpotent actions on such modules.…”