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)_{\leq d,\phi} \}$ of finite dimensional subcoalgebras of $\mathcal O(\mathbb G)$ depending upon a closed embedding $\phi: \mathbb G \ \hookrightarrow \ GL_N$. 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 $\mathcal O(\mathbb G)_{\leq d,\phi}$ provide new invariants for $\mathbb G$-modules. For cofinite $\mathbb G$-modules $M$, we explore the the growth of $d \mapsto M_{\cal O(\mathbb G)_{\leq d,\phi}}$.