Morphisms of character varieties

Abstract: 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 of character varieties [ \underline{\mathrm{Hom}}{k\textrm{-gp}}(\Gamma, H)//H \to \underline{\mathrm{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.