Filtrations and Growth of $\mathbb{G}$-modules

Abstract

We investigate infinite dimensional modules for an affine group scheme $\mathbb{G}$ of finite type over a field of positive characteristic $p$. For any subspace $X\subset \mathcal{O}(\mathbb{G})$ of the coordinate algebra of $\mathbb{G}$, we consider the abelian subcategory $Mod(\mathbb{G},X)\subset Mod(\mathbb{G})$ of ``$X$-comodules" and the left exact functor $(-{)}_X:Mod(\mathbb{G})\to Mod(\mathbb{G},X)$ which is right adjoint to the inclusion functor. We employ ``ascending converging sequences" $\{X_i\}$ of subspaces of $\mathcal{O}(\mathbb{G})$ to provide functorial filtrations $\{M_{X_i}\}$ of each $\mathbb{G}$-module $M$. A $\mathbb{G}$-module $M$ is injective if and only if each $M_{X_i}$ is an injective $X_i$-comodule for some (or, equivalently, for all) such $\{X_i\}$. We consider the explicit ascending converging sequence $\{\mathcal{O}(\mathbb{G}{)}_{\le d,\varphi }\}$ of finite dimensional subcoalgebras of $\mathcal{O}(\mathbb{G})$ depending upon a closed embedding $\varphi :\mathbb{G}\hookrightarrow G{L}_N$. Of particular interest to us are mock injective $\mathbb{G}$-modules, modules whose support varieties are empty. Restrictions of a $\mathbb{G}$-module to each $\mathcal{O}(\mathbb{G}{)}_{\le d,\varphi }$ provide new invariants for $\mathbb{G}$-modules. For cofinite $\mathbb{G}$-modules $M$, we explore the the growth of $d\mapsto M_{\mathcal{O}(\mathbb{G}{)}_{\le d,\varphi }}$.

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