[Paper Review] Contraction par Frobenius et modules de Steinberg

M. Gros, Masaharu Kaneda|arXiv (Cornell University)|Jul 4, 2017

Advanced Algebra and Geometry3 citations

TL;DR

This paper establishes that the Frobenius contraction functor on reductive group modules in positive characteristic is right adjoint to the double tensor product with the Steinberg module of the Frobenius twist. It proves that this functor preserves injectivity and good filtrations, but not semisimplicity, extending results to $G_r$-modules and providing a categorical framework for Frobenius-related module constructions via a split Frobenius on distribution algebras.

ABSTRACT

For a reductive group G defined over an algebraically closed field of positive characteristic, we show that the Frobenius contraction functor of G-modules is right adjoint to the Frobenius twist of the modules tensored with the Steinberg module twice. It follows that the Frobenius contraction functor preserves injectivity, good filtrations, but not semisiplicity.

Motivation & Objective

• To characterize the Frobenius contraction functor on $G$-modules via adjunction with the double tensor product with the Steinberg module.
• To establish that the Frobenius contraction functor preserves injective modules and modules with good filtrations.
• To show that the functor does not preserve semisimplicity in general.
• To extend the results to $G_r$-modules, where $G_r$ is the $r$-th Frobenius kernel of $G$, and to $G_rB$-modules.
• To provide a categorical and structural understanding of Frobenius contraction using a split Frobenius on the distribution algebra $\mathrm{Dist}(G)$.

Proposed method

• The authors use a split Frobenius morphism on $\mathrm{Dist}(G)$, constructed via a unique idempotent $\mu_0$ in $\mathrm{Dist}(T_1)$, to define the Frobenius contraction functor $M \mapsto M^\phi$ on $G$-modules.
• The contraction functor is defined by pulling back the $G$-action on $M$ via the algebra homomorphism $\phi: \mathrm{Dist}(G) \to \mu_0\mathrm{Dist}(G)\mu_0$, which satisfies $\mathrm{Dist}(F) \circ \phi = \mathrm{id}$.
• The key adjunction is established between the Frobenius contraction functor and the functor $M \mapsto \mathrm{St}^{[1]} \otimes \mathrm{St}^{[1]} \otimes M^{[1]}$, where $\mathrm{St}$ is the Steinberg module and $[1]$ denotes Frobenius twist.
• The preservation of good filtrations is shown via Ext vanishing and the fact that $\mathrm{St}_s \otimes \mathrm{St}_s \otimes \hat{\Delta}_{r-s}(\lambda)^{[s]}$ admits a $\hat{\Delta}_r$-filtration.
• The results are extended to $G_r$-modules using the $G_rT$-module structure and the fact that $\tilde{\nabla}_r(\lambda)^{\phi^s}$ admits a $\tilde{\nabla}_{r-s}$-filtration.
• The proof relies on standard results from Jantzen's representation theory of algebraic groups, including tensor identities and induction functors.

Experimental results

Research questions

• RQ1 Is the Frobenius contraction functor on $G$-modules right adjoint to the double tensor product with the Frobenius twist of the Steinberg module?
• RQ2 Does the Frobenius contraction functor preserve injective modules and modules with good filtrations?
• RQ3 Does the Frobenius contraction functor preserve semisimplicity in general?
• RQ4 Can the results on $G$-modules be extended to $G_r$-modules and $G_rB$-modules?
• RQ5 What is the role of the split Frobenius on $\mathrm{Dist}(G)$ in defining and characterizing the contraction functor?

Key findings

• The Frobenius contraction functor $M \mapsto M^\phi$ is right adjoint to the functor $M \mapsto \mathrm{St}^{[1]} \otimes \mathrm{St}^{[1]} \otimes M^{[1]}$, establishing a fundamental categorical duality.
• The functor preserves injective modules, as shown via the adjunction and the fact that the right adjoint preserves injectivity.
• The functor preserves the existence of good filtrations, as demonstrated by the vanishing of $\mathrm{Ext}^1$ groups and the $\hat{\Delta}_r$-filtration of relevant tensor products.
• The functor does not preserve semisimplicity in general, as shown by counterexamples in the paper.
• For $G_r$-modules, the contraction functor $M^{\phi^s}$ preserves $\nabla_{r-s}$-filtrations when $M$ has a $\nabla_r$-filtration.
• For $G_rB$-modules, the contraction functor $M^{\phi^s}$ preserves $\tilde{\nabla}_{r-s}$-filtrations when $M$ has a $\tilde{\nabla}_r$-filtration.