Borel–Mostow Normalizer Representative (Semisimple Class)

An element of a complex reductive group is semisimple if it is diagonalizable in every finite-dimensional complex representation (equivalently, it lies in some complex torus).

In a connected complex reductive group, every semisimple element is conjugate into a fixed maximal torus $T$.

In the letter one works in a (typically disconnected) group like $\Gamma\ltimes \widehat G$; a Borel–Mostow result is invoked to ensure a semisimple conjugacy class with a fixed $\Gamma$-component has a representative in the normalizer $N_{\Gamma\ltimes \widehat G}(\widehat T)$, and can be chosen to preserve a chosen set of positive roots (a “dominant” representative).

Use: this supports the bijection “Hecke eigencharacters $\leftrightarrow$ semisimple conjugacy classes with Frobenius projection.”