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There is also a theorem of I think Hakim (my apologies if this is not Jeff's theorem), generalized by Prasad and Schulze-Pillot, that allows you to globalize representations distinguished with respect to a subgroup. This one uses a simple relative trace formula.

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There is also a theorem of Hakim, generalized by Prasad and Schulze-Pillot, that allows you to globalize representations distinguished with respect to a subgroup. This one uses a simple relative trace formula.