Let G be a connected reductive group over coplex numbers whose derived subgroup is simply connected. Let u be a unipotent element of G. The centrelizer of u in G is denoted by Z_G( Let F_(u) be a maximal connected reductive subgroup of Z_(u). My question is that whether the derived subgroup of F_(u) is simply connected.

Let G be a connected reductive group over complex numbers whose derived subgroup is simply connected. Let u be a unipotent element of G. The centralizer of u in G is denoted by Z_(u). Let F_(u) be a maximal connected reductive subgroup of Z_(u). My question is whether the derived subgroup of F_(u) is simply connected.