A bound for the orders of centralizers of irreducible subgroups of algebraic groups

Abstract We prove that if 𝐺 is a connected semisimple algebraic group of rank 𝑟, and 𝐻 is a subgroup of 𝐺 that is contained in no proper parabolic subgroup, then we have | C G ⁢ ( H ) | < c r ⁢ | Z ⁢ ( G ) | \lvert C_{G}(H)\rvert<c^{r}\lvert Z(G)\rvert , where 𝑐 is an absolute constant ( c = 16 c=16 if all simple factors of 𝐺 are classical, and c ≤ 197 c\leq 197 in general).