@WillSawin Thank you. Of course, the question is not for a subgroup of the group itself (but for the Galois model one can do this). In general, functoriality starts with a map ${}^LH \rightarrow {}^LG$, and one hopes to characterize the image by using some period integral on another group (e.g the example of Shalika period integral as above). There are some convergence issues, in the local setting if we assume the representation is supercuspidal then this is OK.

I find Théorème 3.2.3. in “Théorème de l’Indice et Formule des Traces" by Axel Ferrari contains some good results.

@Kimball Sorry I shall be more specific, thank you!

Thank you! The usual proof of existence of smooth transfer is done locally and using germ expansion, so I am interested in explicit construction.