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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 $GL(n)$?

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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.

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    $\begingroup$ Does this somehow enable us to see some other theta correspondences as instances of Langlands functoriality? $\endgroup$
    – GTA
    Commented Sep 4, 2019 at 7:06
  • $\begingroup$ @GTA -- Well, sort of. First, one has to have an L-group for covering groups, so that functoriality makes sense. And then one can check that various correspondences are functorial. Examples include the metaplectic correspondence (e.g., recent work of Gan and Savin), and Hecke algebra correspondences. For theta correspondences, there are interesting cases to check functoriality -- e.g., $SL_3$ and a double cover of $SL_3$ (or quasisplit $SU_3$ and its double-cover), which form a dual pair in a double cover of $F_4$. Wee Teck Gan and I may finish that someday..... $\endgroup$
    – Marty
    Commented Sep 4, 2019 at 19:36
  • $\begingroup$ Maybe it's a trivial question, but how does Hecke algebra correspondence fit into functoriality? $\endgroup$
    – GTA
    Commented Sep 4, 2019 at 19:41
  • $\begingroup$ @GTA -- Savin describes an isomorphism between the Iwahori-Hecke algebra of a covering group (e.g. an d-fold cover of SL_n) and the Iwahori-Hecke algebra of a related linear group (e.g., a linear quotient of SL_n, determined by d and n). It turns out that this isomorphism reflects an isomorphism of L-groups (using my L-group for the covering group), so it's an instance of functoriality. Presumably some of these may arise from global correspondences between automorphic forms on a covering and a linear group. $\endgroup$
    – Marty
    Commented Sep 4, 2019 at 21:07

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