Let \(G_n={\mathrm{GL}}_n({\mathbb F}_q)\) and \(T_n\simeq{\mathbb F}_{q^n}^{\times}\) be an elliptic torus. A regular torus character determines an irreducible cuspidal representation \(\kappa_\theta\) by the classical finite-field construction. If its character is \(\chi_\theta\) and degree is \(d_\theta\), then
is its primitive central idempotent in \({\mathbb C}[G_n]\).
On an irreducible representation with character \(\psi\), this central element acts by a scalar. Taking traces and using character orthogonality makes the scalar 1 on \(\kappa_\theta\) and 0 on every other irreducible block.
For a proper parabolic \(P=LU\), averaging over its unipotent radical gives
The average selects \(U\)-fixed vectors; the cuspidal block has none.
Scope
The existence and classification of \(\kappa_\theta\) are classical inputs. The idempotent calculation explains its exact finite selector.