A generalized theta lifting, CAP representations, and Arthur parameters. (English) Zbl 1479.11082
The paper concerns a lifting of automorphic representations using the theta representation \(\Theta\) on the 4-fold cover \(\overline{\mathrm{Sp}}(2r,\mathbb A)\) of the symplectic group. This lifting produces the first examples of CAP representations on higher degree metaplectic covering groups. Central to the analysis here is the identification of the maximal nilpotent orbit associated to \(\Theta\). The paper conjectures a natural extension to \(\overline{\mathrm{Sp}}(2r,\mathbb A)\) of Arthur’s parameterization of the discrete spectrum. Assuming this, the paper computes the effect of this lift on Arthur parameters and show that the parameter of a representation in the image of the lift is non-tempered. The paper concludes by relating the lifting to a dimension equation of Ginzburg to predict the first non-trivial lift of a generic cuspidal representation of \(\overline{\mathrm{Sp}}(2r,\mathbb A)\).
Reviewer: Yuval Z. Flicker (Ariel)
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11F30 | Fourier coefficients of automorphic forms |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |
Keywords:
automorphic representation; Brylinski-Deligne covering group; theta correspondence; unipotent orbit; Arthur parameterReferences:
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