We now consider properties like generation, conjugacy, classification, connectedness and dimension of centralizers in connected reductive groups. It turns out that the situation is easiest for semisimple elements.

Semisimple elements

Recall from Corollary 6.11(a) that every semisimple element of a connected group lies in some maximal torus. More precisely we have:

Proposition 14.1Let G be connected, s ∈ G semisimple, T ≤ G a maximal torus. Then s ∈ T if and only if T ≤ CG(s)°. In particular, s ∈ CG(s)°.

Proof As T is abelian, s ∈ T if and only if T ≤ CG(s), which is equivalent to T ≤ CG(s)° as T is connected.

We remark that in contrast, for u ∈ G unipotent, u may not be in CG(u)°.

See Exercise 20.10 for an example in Sp4 over a field of characteristic 2.

We now determine the structure of centralizers of semisimple elements:

Theorem 14.2Let G be connected reductive, s ∈ G semisimple, T ≤ G a maximal torus with corresponding root system Φ. Let s ∈ T and Ψ ≔ {α ∈ Φ | α(s) = 1}. Then:

1. (a) CG(s)° = 〈T,Uα; | α ∈ Ψ〉.

2. (b) CG(s) = 〈T,Uα,ẇ | α ∈ Ψ, w ∈ W with sw = s〉.

Moreover, CG(s)° is reductive with root system Ψ and Weyl group W1 = 〈sα | α ∈ Ψ〉.