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In this article, we study Prasad's conjecture for regular supercuspidalrepresentations based on the machinery developed by Hakim and Murnaghan tostudy distinguished representations, and the fundamental work of Kaletha onparameterization of regular supercuspidal representations. For regularsupercuspidal representations, we give some new interpretations of thenumerical quantities appearing in Prasad's formula, and reduce the proof to thecase of tori. The proof of Prasad's conjecture then reduces to a comparison ofvarious quadratic characters appearing naturally in the above process. We alsohave some new observations on these characters and study the relation betweenthem in detail. For some particular examples, we show the coincidence of thesecharacters, which gives a new purely local proof of Prasad's conjecture forregular supercuspidal representations of these groups. We also prove Prasad'sconjecture for regular supercuspidal representations of G(E), when E/F isunramified and G is a general quasi-split reductive group.

Distinction and quadratic base change for regular supercuspidal representations