Semisimple Conjugacy Classes

Setup

Let \(G\) be a connected reductive group over an algebraically closed field \(k\), with maximal torus \(T \subset G\) and Weyl group \(W = N_G(T)/T\).

Parametrisation theorem

Theorem. Every semisimple element of \(G(k)\) is conjugate to an element of \(T(k)\), and two elements \(t, t' \in T(k)\) are \(G(k)\)-conjugate if and only if \(t' = w \cdot t\) for some \(w \in W\). Hence: \[ \{\text{semisimple conjugacy classes in }G\} \xleftrightarrow{\;\sim\;} T(k)/W. \]

This is the group-theoretic analogue of the fact that semisimple matrices are diagonalisable and are conjugate iff they have the same eigenvalues.

Structure of the centraliser

For a semisimple element \(s \in T(k)\), the identity component of its centraliser \[ C_G(s)^\circ \] is a connected reductive group (a theorem of Steinberg in characteristic \(p\), and classical otherwise). Its root system is the sub-root-system \(\{\alpha \in \Phi : \alpha(s) = 1\}\).

Density of semisimple elements

The set of semisimple elements is dense in \(G\) (in the Zariski topology). Equivalently, unipotent elements form a closed nowhere-dense subset (the unipotent variety) of positive codimension \(\mathrm{rank}(G)\).

Over non-algebraically-closed fields

Over a non-closed field \(F\), semisimple conjugacy classes in \(G(F)\) are no longer parametrised simply by \(T(F)/W\): one must consider all maximal \(F\)-tori (not just one split torus), and the parametrisation involves Galois cohomology. For the Langlands correspondence, semisimple conjugacy classes in the dual group \({}^L G\) parametrise unramified principal series representations.