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Transactions of the American Mathematical Society

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The degenerate residual spectrum of quasi-split forms of $ Spin_8$ associated to the Heisenberg parabolic subgroup


Author: Avner Segal
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11F70; Secondary 11M36, 32N10
DOI: https://doi.org/10.1090/tran/7901
Published electronically: August 1, 2019
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Abstract: In [J. Inst. Math. Jussieu 14 (2015), pp. 149-184] and [Int. Math. Res. Not. imrn 7 (2017), pp. 2014-2099], the twisted standard $ \mathcal {L}$-function $ \mathcal {L}(s,\pi ,\chi ,\mathfrak{st})$ of a cuspidal representation $ \pi $ of the exceptional group of type $ G_2$ was shown to be represented by a family of new-way Rankin-Selberg integrals. These integrals connect the analytic behaviour of $ \mathcal {L}(s,\pi ,\chi ,\mathfrak{st})$ with that of a family of degenerate Eisenstein series $ \mathcal {E}_E(\chi , f_s, s, g)$ on quasi-split forms $ H_E$ of $ Spin_8$, induced from Heisenberg parabolic subgroups. The analytic behaviour of the series $ \mathcal {E}_E(\chi , f_s, s, g)$ in the right half-plane $ \mathfrak{Re}(s)>0$ was studied in [Tran. Amer. Math. Soc. 370 (2018), pp. 5983-6039]. In this paper we study the residual representations associated with $ \mathcal {E}_E(\chi , f_s, s, g)$.


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Additional Information

Avner Segal
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada
Address at time of publication: Department of Mathematics, Bar Ilan University, Ramat Gan 5290002, Israel
Email: segalav4@biu.ac.il

DOI: https://doi.org/10.1090/tran/7901
Received by editor(s): August 20, 2018
Received by editor(s) in revised form: February 17, 2019, and February 22, 2019
Published electronically: August 1, 2019
Article copyright: © Copyright 2019 American Mathematical Society