A generalized Theta lifting, CAP representations, and Arthur parameters

Leslie, Spencer

doi:10.1090/tran/7863

A generalized Theta lifting, CAP representations, and Arthur parameters
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by Spencer Leslie PDF

Trans. Amer. Math. Soc. 372 (2019), 5069-5121 Request permission

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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