Conjugacy Classes
For an algebraic group \(G\) acting on itself by conjugation \((g, h) \mapsto ghg^{-1}\), the orbits are the conjugacy classes. These carry rich structure used throughout the theory.
The conjugation action
The conjugation morphism
\[ c : G \times G \to G, \qquad (g, h) \mapsto g h g^{-1} \]
is a morphism of schemes. For each fixed \(g \in G\), conjugation by \(g\) is an automorphism \(c_g : G \to G\) (the inner automorphism). The map \(g \mapsto c_g\) is a homomorphism \(G \to \operatorname{Aut}(G)\) whose image is the inner automorphism group \(\operatorname{Inn}(G)\) and whose kernel is the centre \(Z(G)\).
Orbits and stabilisers
For \(h \in G(K)\) (a point with values in a field \(K\)):
- the conjugacy class of \(h\) is the orbit \(\{c_g(h) : g \in G(K')\}\) for varying \(K'/K\) extensions;
- the centraliser \(Z_G(h)\) is the closed subscheme of \(G\) fixing \(h\) under conjugation;
- \(\dim(\text{orbit}) + \dim(\text{centraliser}) = \dim G\).
Semisimple vs. unipotent elements
Over a field, an element \(g \in G\) of an affine algebraic group admits a Jordan decomposition \(g = g_s \cdot g_u = g_u \cdot g_s\) with \(g_s\) semisimple and \(g_u\) unipotent. The decomposition is functorial under group morphisms.
- Semisimple conjugacy classes. Classified (over an algebraically closed field) by characters of a maximal torus modulo Weyl-group action — see root data.
- Unipotent conjugacy classes. Far richer; for classical groups there is a combinatorial classification by partitions. See nilpotent orbits.
- General conjugacy classes. Determined by a pair (semisimple class, unipotent class in the centraliser), via Jordan decomposition.
Steinberg's theorem
For a connected reductive group \(G\) over an algebraically closed field, two semisimple elements are conjugate iff they lie in a common maximal torus and are conjugate by the Weyl group. The Weyl-orbit structure on a maximal torus is therefore the key combinatorial data.
Conjugacy of subgroups
A parallel notion is conjugacy of subgroups: two closed subgroups \(H, H' \subseteq G\) are conjugate if \(H' = g H g^{-1}\) for some \(g \in G\). Key examples:
- Maximal tori are all conjugate over \(\bar{k}\) — this is the uniqueness statement for the abstract torus structure.
- Borel subgroups of a connected reductive group are all conjugate over \(\bar{k}\) — this is the cornerstone of the Bruhat decomposition.
Kottwitz / twisted conjugacy
In the local Langlands program, an important variant is \(\sigma\)-twisted conjugacy for an automorphism (or Frobenius) \(\sigma\): the classes of the action \(g \cdot h = g h \sigma(g)^{-1}\). See Kottwitz structures.