Characters separate semisimple conjugacy classes

Let $H$ be a connected complex reductive group. If $h,h'\in H$ are semisimple, then

h\sim_H h' \quad\Longleftrightarrow\quad \operatorname{tr}(\rho(h)) = \operatorname{tr}(\rho(h'))

for every finite-dimensional algebraic representation $\rho$ of $H$ . It suffices to range over irreducible representations.

Invariant-theory explanation

Choose a maximal torus $T$ with Weyl group $W$ . Semisimple conjugacy classes are identified with $W$ -orbits in $T$ , and restriction gives

\mathcal O(H)^H\simeq\mathcal O(T)^W.

The characters of irreducible algebraic representations span the representation ring and generate enough Weyl-invariant regular functions to distinguish the closed, hence semisimple, conjugacy classes.

Closed-orbit qualification

Invariant regular functions separate closed orbits in the affine quotient. For a general element they recover the semisimple part of its Jordan decomposition, not its full conjugacy class. This is why excursion-operator reconstruction naturally produces semisimple parameters.

Disconnected and twisted scope

A Frobenius fiber in an $L$ -group is a coset of the dual group $\widehat G$ , not a connected group. Its conjugacy is twisted by the Weil-group action. The analogous separation statement uses invariant functions on the twisted quotient; the connected theorem should not be applied to that coset without this modification.

Relation to the letter

Values of all dual-group characters on the Satake class determine that semisimple class. This is the invariant-theoretic reason that the collection of unramified Euler factors can record a Satake parameter.

References

T. A. Springer, Linear Algebraic Groups, second edition, Birkhäuser, 1998.
Claudio Procesi, Lie Groups: An Approach through Invariants and Representations, Springer, 2007.