Split metaplectic groups and their L-groups

During the preparation of an earlier paper on metaplectic tori [Pacific J. Math. 241 (2009), 169–200], we benefited greatly from a correspondence with P. Deligne. His advice on that paper was very helpful, and suggested many reasons why the results there are not entirely satisfactory. Deligne's recommendation to consider the geometric work of Finkelberg–Lysenko [J. Inst. Math. Jussieu 9 (2010), 719–739], seconded by a later recommendation of B. Gross to consider the geometric setting in the recent thesis of R. Reich [Represent. Theory 16 (2012), 345–449], was crucial to making a reasonable guess at an L-group. We also thank B. Gross for pointing us towards the recent work of Gan, Gross, and Prasad [Astérisque 346 (2012), 1–109], §11, in which there are precise conjectures on a local Langlands parameterization for the metaplectic group 𝑀𝑝2n. We are also thankful for the advice and personal communication with Wee Teck Gan and Gordan Savin. Both have thought deeply about metaplectic groups; Savin's earlier paper [J. reine angew. Math. 566 (2004), 111–134] and their recent work together [Compos. Math. 148 (2012), 1655–1694] were very helpful, as was their encouragement.