Semisimple element and conjugacy class

Let $G$ be a linear algebraic group over an algebraically closed field. An element $g\in G$ is semisimple if its image under one, equivalently every, faithful algebraic representation is a semisimple linear operator. Over $\mathbb C$ , this means diagonalizable.

In a connected reductive group, an element is semisimple exactly when it lies in a maximal torus. Its conjugacy class is Zariski closed.

Jordan decomposition

Every element $g$ has a commuting Jordan decomposition

g=g_sg_u=g_ug_s

with $g_s$ semisimple and $g_u$ unipotent. Invariant regular functions on $G$ take the same values on $g$ and $g_s$ , so the affine conjugacy quotient records the semisimple class.

Rational versus geometric conjugacy

If $G$ is defined over a non-algebraically-closed field $F$ , elements of $G(F)$ that are conjugate over $\overline F$ need not be conjugate over $F$ . For strongly regular semisimple elements this distinction is organized by stable conjugacy and Galois cohomology.

Examples

In $\operatorname{GL}_n(\mathbb C)$ , semisimple elements are precisely the diagonalizable matrices.
A unipotent matrix other than the identity is not semisimple, even though all of its eigenvalues are $1$ .

Langlands role

Satake and Langlands parameters are taken up to semisimple conjugacy because characters and invariant functions detect closed orbits. A semisimplified parameter can lose inertia or monodromy information, so “semisimple” is not a harmless adjective.

References

T. A. Springer, Linear Algebraic Groups, second edition, Birkhäuser, 1998.