Characters Separate Semisimple Conjugacy Classes

For a finite-dimensional representation $\rho:H\to \mathrm{GL}(V)$, its character is the class function $\chi_\rho(h):=\mathrm{tr}(\rho(h))$.

For a connected complex reductive group $H$, two semisimple elements $h,h'$ are conjugate iff $\chi_\rho(h)=\chi_\rho(h') \quad \text{for all irreducible }\rho.$

Reason (used implicitly): semisimple elements are conjugate into a torus, and characters restrict to Weyl-invariant regular functions on the torus that separate Weyl orbits.

In the letter: this justifies recovering the semisimple class of $\alpha_p$ from its values in all representations.