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Suppose that 
$G$ is a connected reductive algebraic group defined over 
$\mathbf{R}$, 
$G(\mathbf{R})$ is its group of real points, 
${\it\theta}$ is an automorphism of 
$G$, and 
${\it\omega}$ is a quasicharacter of 
$G(\mathbf{R})$. Kottwitz and Shelstad defined endoscopic data associated to 
$(G,{\it\theta},{\it\omega})$, and conjectured a matching of orbital integrals between functions on 
$G(\mathbf{R})$ and its endoscopic groups. This matching has been proved by Shelstad, and it yields a dual map on stable distributions. We express the values of this dual map on stable tempered characters as a linear combination of twisted characters, under some additional hypotheses on 
$G$ and 
${\it\theta}$.
