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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A generalized Theta lifting, CAP representations, and Arthur parameters


Author: Spencer Leslie
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11F70; Secondary 11F30, 22E50, 22E55
DOI: https://doi.org/10.1090/tran/7863
Published electronically: June 21, 2019
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Abstract: We study a new lifting of automorphic representations using the theta representation $ \Theta $ on the $ 4$-fold cover of the symplectic group $ \overline {\operatorname {Sp}}_{2r}(\mathbb{A})$. This lifting produces the first examples of CAP representations on higher-degree metaplectic covering groups. Central to our analysis is the identification of the maximal nilpotent orbit associated to $ \Theta $.

We conjecture a natural extension of Arthur's parameterization of the discrete spectrum to $ \overline {\operatorname {Sp}}_{2r}(\mathbb{A})$. Assuming this, we compute the effect of our lift on Arthur parameters and show that the parameter of a representation in the image of the lift is non-tempered. We conclude by relating the lifting to the dimension equation of Ginzburg to predict the first non-trivial lift of a generic cuspidal representation of $ \overline {\operatorname {Sp}}_{2r}(\mathbb{A})$.


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Additional Information

Spencer Leslie
Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467-3806
Address at time of publication: Department of Mathematics, Duke University, Durham, North Carolina 27710
Email: lesliew@math.duke.edu

DOI: https://doi.org/10.1090/tran/7863
Keywords: Automorphic representation, Brylinski-Deligne covering group, theta correspondence, unipotent orbit, Arthur parameter
Received by editor(s): April 26, 2018
Received by editor(s) in revised form: March 4, 2019
Published electronically: June 21, 2019
Article copyright: © Copyright 2019 American Mathematical Society