Let $G$ be an affine group scheme over a field.
Say that, for every finite-dimensional representation of $G$, I have a $\mathbb{Z}$-grading on the underlying vector space, compatible with tensor products, with the property that every $G$-equivariant map is graded. Then by abstract Tannaka duality I have a map of group schemes $\mathbb{G}_m\to G$.
Now instead of a grading, say I have a $\mathbb{Z}$-indexed filtration, compatible with tensor products, such that every $G$-equivariant map $f:V\to W$ strictly preserves the filtration, meaning for all $n\in\mathbb{Z}$, $$ f\big(\mathrm{Fil}^n V\big)=f(V)\cap\mathrm{Fil}^n W. $$ What does this tell me about $G$?
