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 $GL(n)$?



Shimura correspondence for automorphic forms on other groups
There's not a short answer to this question, but here are a few points:


1. 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 $Sp\_{2n}({\mathbb C})$.
2. 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](https://zbmath.org/?q=an:06908304).


