Semisimple Element and Semisimple Conjugacy Class

Let $H$ be a complex reductive algebraic group.

An element $h\in H(\mathbb{C})$ is semisimple if for every finite-dimensional algebraic representation $\rho:H\to \mathrm{GL}(V)$, the linear map $\rho(h)$ is diagonalizable over $\mathbb{C}$.

In a reductive group, semisimple elements are exactly those contained in some maximal torus.

A semisimple conjugacy class is the conjugacy class of a semisimple element.

Example: in $\mathrm{GL}_n(\mathbb{C})$, semisimple means “diagonalizable.”