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Semisimple Element and Semisimple Conjugacy Class

Elements diagonalizable in representations; conjugacy classes used for Satake parameters

Let HH be a complex reductive algebraic group.

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

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

Remarks

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

Examples
  • in GLn(C)\mathrm{GL}_n(\mathbb{C}), semisimple means "diagonalizable."