TEMPERED SPECTRAL TRANSFER IN THE TWISTED ENDOSCOPY OF REAL GROUPS

Suppose that (Formula presented.) is a connected reductive algebraic group defined over (Formula presented.), (Formula presented.) is its group of real points, (Formula presented.) is an automorphism of (Formula presented.), and (Formula presented.) is a quasicharacter of (Formula presented.). Kottwitz and Shelstad defined endoscopic data associated to (Formula presented.), and conjectured a matching of orbital integrals between functions on (Formula presented.) 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 (Formula presented.) and (Formula presented.)