Morphisms of Character Varieties

Let k be a field, let H ⊂ G be (possibly disconnected) reductive groups over k, and let Γ be a finitely generated group. Vinberg and Martin have shown that the induced morphism Homk-gp(Γ , H)//H → Homk-gp(Γ , 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.