(Edited after discussion in comments):
If a group $G$ acts on a smooth algebraic variety $V$, then the cohomology ring $H^*(V;\mathbb C)$ is a representation of $G$. If, furthermore, $V$ admits a stratification by affines, then the associated graded $\mathrm{gr}K(V)$ of the Grothendieck ring of $V$ is isomorphic to $H^*(V)$. Thus, one has a representation of $G$ on a filtered vector space $K(V)\otimes\mathbb C$ whose associated graded is the representation $H^*(V;\mathbb C)$.
What happens for sheaf cohomology $H^*(V;\mathscr F)$? If $\mathscr F$ has a $G$-equivariant structure then $H^*(V;\mathscr F)$ is a representation of $G$. Is there a "$K$-theoretic upgrade", i.e. a representation on a filtered vector space $K$ whose associated graded is $H^*(V;\mathscr F)$?