Classical Theta Lifts for Higher Metaplectic Covering Groups
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GAFA Geometric And Functional Analysis
CLASSICAL THETA LIFTS FOR HIGHER METAPLECTIC COVERING GROUPS Solomon Friedberg and David Ginzburg
Abstract. The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series on the metaplectic double cover of a symplectic group that is constructed from the Weil representation. There is also an analogous local correspondence. In this work we present an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups. The key issue here is that for higher degree covers there is no analogue of the Weil representation, and additional ingredients are needed. Our work reflects a broader paradigm: constructions in automorphic forms that work for algebraic groups or their double covers should often extend to higher degree metaplectic covers.
1 Introduction Theta series provide a way to construct correspondences between spaces of automorphic forms. For example, suppose that G1 = SO(V1 ) and G2 = Sp(V2 ) are orthogonal and symplectic groups, resp. Then these groups embed in a symplectic group G := Sp(V1 ⊗ V2 ). Let F be a number field and A its ring of adeles. There is a family of theta functions θφ on the adelic metaplectic double cover M p(V1 ⊗ V2 )(A) (depending on some additional data φ), and these functions may be used to create a correspondence between automorphic forms f1 on G1 (A) and automorphic forms f2 on G2 (A) or its double cover: f2 (g2 , φ) = θφ (g1 · g2 ) f1 (g1 ) dg1 , where the integral is over the adelic quotient G1 (F )\G1 (A). This correspondence, which in a low-rank case can be used to recreate the Shimura correspondence, has This work was supported by the BSF, Grant Number 2016000, and by the NSF, Grant Number DMS-1801497 (Friedberg). Keywords and phrases: Metaplectic cover, Theta lifting, Weil representation, Unipotent orbit Mathematics Subject Classification Primary 11F27; Secondary 11F70
S. FRIEDBERG, D. GINZBURG
GAFA
been studied by many authors over the past half century (see for example Rallis [Ra2] and the references therein). There is also a local correspondence of irreducible smooth representations, the Howe correspondence, that is obtained by restricting the Weil representation to the image of G1 · G2 in G (see Howe [Ho]), that has likewise received a great deal of attention. The goal of this paper is to extend these constructions to higher metaplectic covers of the groups G1 , G2 (beyond the double cover M p) and to initiate an analysis of the resulting map of representations. The double cover of the symplectic group, M p, was introduced by Weil [We] in his treatment of theta series. Over local or global fields F with enough roots of unity, there are higher degree covers of classical groups as well, related to the work of Bass-Milnor-Serre on the congruence subgroup problem. Th
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