Strongly regular semisimple element

Let $G$ be a connected reductive group over a field $F$ . An element $\gamma\in G(F)$ is strongly regular semisimple if it is semisimple and its centralizer

G_\gamma=\{g\in G:g\gamma=\gamma g\}

is a maximal torus of $G$ .

Regular versus strongly regular

For a connected reductive group in characteristic zero, regular semisimple elements have torus centralizer, so the two phrases are often used interchangeably. In greater generality, “regular” can be defined by minimal centralizer dimension while “strongly regular” also requires the centralizer to be a torus. The stronger phrase avoids ambiguity in endoscopy.

For the Lie algebra, $X\in\mathfrak g(F)$ is strongly regular semisimple when its centralizer $G_X$ is a maximal torus.

Examples

A matrix in $\operatorname{GL}_n(F)$ is strongly regular semisimple exactly when its characteristic polynomial is separable of degree $n$ , equivalently when it has $n$ distinct eigenvalues over an algebraic closure.

A regular diagonal element in a split reductive group is strongly regular when no root takes the value $1$ on it. In the Lie algebra, the analogous condition is $\alpha(X)\neq 0$ for every root.

Why this locus is used

On the strongly regular semisimple locus:

conjugacy classes are controlled by maximal tori;
centralizer quotients carry natural invariant measures;
orbital integrals are well behaved;
stable conjugacy splits into finitely many rational conjugacy classes over a local field;
endoscopic transfer compares matching classes on different groups.

Singular semisimple and unipotent terms still occur in the trace formula, but their distributions require additional limiting and weighted constructions.

References

Robert E. Kottwitz, “Stable trace formula: cuspidal tempered terms,” Duke Mathematical Journal 51 (1984), 611–650. DOI.
Ngô Bảo Châu, “Survey on the fundamental lemma,” §1.3. PDF.