Shimura correspondence for automorphic forms on other groups I've been told recently that the Shimura correspondence does not fit into Langlands functoriality, i.e. does not have a natural generalization to other groups. However, it should have some generalizations to automorphic forms on other groups.
I was trying to find out about it, but couldn't find a list anywhere on the groups to which the Shimura correspondence has been generalized. So I wanted to ask, whether anyone knows a reference or maybe knows to which groups the Shimura correspondence has generalizations. 
In particular, is there a Shimura correspondence to $GL(n)$?
 A: There's not a short answer to this question, but here are a few points:


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*Regarding the claim that "the Shimura correspondence does not fit into Langlands functoriality."  In some sense it does now!  Part of my goal (and others in this field), in generalizing L-groups to covering groups, was to make the Shimura correspondence and its generalizations fit into Langlands functoriality.  I consider this a solved problem now, since the L-groups of metaplectic groups are "what they should be" in order to make the Shimura correspondence functorial.  More precisely, a choice of an additive character determines an isomorphism from the L-group of $Mp_{2n}$ to the direct product of the Weil/Galois group with $Sp_{2n}({\mathbb C})$.

*There are generalizations of the Shimura correspondence to other groups.  I'll mention Kubota and Flicker and Kazhdan-Patterson for covers of $GL_n$, the Crelle paper of Gordan Savin for unramified representations of covers of simply-laced groups over $p$-adic fields, and the JAMS paper of Adams-Barbasch-Paul-Trapa-Vogan for a Shimura correspondence for real groups.
For more details and precise references, see the Asterisque volume (especially the historical introduction):
Gan, Wee Teck; Gao, Fan; Weissman, Martin H., L-groups and the Langlands program for covering groups,  ZBL06908304.
