Let $F$ be a $p$-adic field and let $G$ be a reductive group over $F$. Associated to an irreducible admissible representation of $\pi$ of $G(F)$, we have a distribution character $\Theta_{\pi}$ defined by $\Theta_{\pi}(f)= \mathrm{tr}( f \mid \pi)$. Then a theorem of Harish-Chandra says there is a locally constant function $f_{\pi}$ on $G(F)_{sr}$, the strongly regular semisimple elements of $G(F)$, such that $\Theta_{\pi}(f)=\int_{G(F)_{sr}} f(g)f_{\pi}(g)dg$.
My question is as follows. What is a proof or reference for a proof of the following statement: If $a_1 \pi_1 + ... a_n \pi_n$ is a virtual representation such that the associated trace distribution is stable, then $a_1f_{\pi_1} + ... +a_n f_{\pi_n}$ is constant on stable conjugacy classes.
