G-complete reducibility and saturation

Let H ⊆ G be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic p > 0. In our first main theorem we show that if a closed subgroup K of H is H-completely reducible, then it is also G-completely reducible in the sense of Serre, under some restrictions on p, generalising the known case for G = GL(V ). Our proof uses R.W. Richardson’s notion of reductive pairs to reduce to the GL(V ) case. We study Serre’s notion of saturation and prove that saturation behaves well with respect to products and regular subgroups. Our second main theorem shows that if K is H-completely reducible, then the saturation of K in G is completely reducible in the saturation of H in G (which is again a connected reductive subgroup of G), under suitable restrictions on p, again generalising the known instance for G = GL(V ). We also study saturation of finite subgroups of Lie type in G. We show that saturation is compatible with standard Frobenius endomorphisms, and we use this to generalise a result due to Nori from 1987 in case G = GL(V ).