Regular Elements
Definition
Let $G$ be a connected reductive group of rank $r = \mathrm{rank}(G)$. An element $g \in G$ is regular if the dimension of its centraliser is minimal: $$\dim C_G(g) = r.$$
This is equivalent to saying $g$ lies in the open dense subset of $G$ where the function $g \mapsto \dim C_G(g)$ attains its minimum.
Regular semisimple elements
A semisimple element $s \in G$ is regular semisimple if $C_G(s)^\circ$ is a maximal torus (equivalently, $\alpha(s) \ne 1$ for all roots $\alpha$).
Properties:
- The set $G^{\mathrm{rs}}$ of regular semisimple elements is open dense in $G$.
- $s$ is regular semisimple $\iff$ $s$ lies in a unique maximal torus.
- Over an algebraically closed field, $G^{\mathrm{rs}}$ is the complement of the root hyperplanes $\ker(\alpha - 1) \subset T$ (and their conjugates).
Regular unipotent elements
A unipotent element $u \in G$ is regular unipotent if $\dim C_G(u) = r$. There exists a unique regular unipotent conjugacy class $\mathcal{O}_{\mathrm{reg}}$, which is the open dense orbit in the unipotent variety $\mathcal{U}$.
Characterisation: $u$ is regular unipotent $\iff$ for every Borel subgroup $B \supset T$, the projection of $u$ onto each simple root factor $U_{\alpha_i} \cong \mathbb{G}_a$ is non-zero.
Regular elements in general
For a general element $g = g_s g_u$, $g$ is regular $\iff$ $g_u$ is a regular unipotent element of $C_G(g_s)^\circ$.
Strongly regular elements
An element $g$ is strongly regular if $C_G(g)$ is a torus (automatically a maximal torus). Strongly regular $\Rightarrow$ regular semisimple, but the converse fails for non-simply-connected groups where $C_G(s)$ may be non-connected.
Density theorem
The set of regular elements is open dense in $G$, as is the set of regular semisimple elements. The complement has codimension $\ge 1$.