Morphisms of Character Varieties

Let $k$ be a field, let $H \subset G$ be (possibly disconnected) reductive groups over $k$, and let $\Gamma $ be a finitely generated group. Vinberg and Martin have shown that the induced morphism $\underline{\operatorname{Hom}}_{k\textrm{-gp}}(\Gamma , H)//H \to \underline{\operatorname{Hom}}_{k\textrm{-gp}}(\Gamma , G)//G$ is finite. In this note, we generalize this result (with a significantly different proof) by replacing $k$ with an arbitrary locally Noetherian scheme, answering a question of Dat. Along the way, we use Bruhat–Tits theory to establish a few apparently new results about integral models of reductive groups over discrete valuation rings.