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Classical Theta Lifts for Higher Metaplectic Covering Groups

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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.

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Authors and Affiliations

  1. Department of Mathematics, Boston College, Chestnut Hill, MA, 02467-3806, USA

    Solomon Friedberg

  2. School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, 6997801, Israel

    David Ginzburg

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  1. Solomon Friedberg
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  2. David Ginzburg
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Correspondence to Solomon Friedberg.

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This work was supported by the BSF, Grant Number 2016000, and by the NSF, Grant Number DMS-1801497 (Friedberg).

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Friedberg, S., Ginzburg, D. Classical Theta Lifts for Higher Metaplectic Covering Groups. Geom. Funct. Anal. 30, 1531–1582 (2020). https://doi.org/10.1007/s00039-020-00548-y

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  • DOI: https://doi.org/10.1007/s00039-020-00548-y

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