Morphisms of Character Varieties

Abstract Let k be a field, let H G be (possibly disconnected) reductive groups over k, and let be a finitely generated group. Vinberg and Martin have shown that the induced morphism Hom₊-₆₏ (, H) //H Hom₊-₆₏ (, 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.