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Loss of cuspidality by Langlands tranfer
Given an $L$-homomorphism of Langlands dual groups
$${}^LG \to {}^LG'$$
Langlands functoriality contectures predicts the existence of a tranfer map of automorphic representations
$$Aut(G) \to Aut(G')$ … $
However, nothing in the functoriality results or conjectures seems to be concerned with cuspidality. …