Motivic (Representation) Stability of Representation Varieties and Character Stacks

Abstract: In this paper, we introduce the notions of motivic representation stability that is an algebraic counterpart of the notion of representation stability. In the process, we also introduce the notion of motivic decomposition for varieties equipped with an action of a finite group $G$. This motivic decomposition decomposes the virtual class of the variety with respect to irreducible rational representations of $G$. We also formulate conjectures on motivic representation stability in the context of representation varieties and character stacks, and we verify the conjectures for groups whose virtual classes have been extensively studied.