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Transactions of the American Mathematical Society

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The degenerate Eisenstein series attached to the Heisenberg parabolic subgroups of quasi-split forms of $ Spin_8$

Author: Avner Segal
Journal: Trans. Amer. Math. Soc. 370 (2018), 5983-6039
MSC (2010): Primary 11F70; Secondary 11M36, 32N10
Published electronically: April 4, 2018
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Abstract: In [J. Inst. Math. Jussieu 14 (2015), 149-184] and [Int. Math. Res. Not. IMRN 7 (2017), 2014-2099] a family of Rankin-Selberg integrals was shown to represent 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$. These integral representations bind the analytic behavior of this $ \mathcal {L}$-function with that of a family of degenerate Eisenstein series for quasi-split forms of $ Spin_8$ associated to an induction from a character on the Heisenberg parabolic subgroup.

This paper is divided into two parts. In Part 1 we study the poles of these degenerate Eisenstein series in the right half-plane $ \mathfrak{Re}(s)>0$. In Part 2 we use the results of Part 1 to prove the conjecture, made by J. Hundley and D. Ginzburg in [Israel J. Math. 207 (2015), 835-879], for stable poles and also to give a criterion for $ \pi $ to be a CAP representation with respect to the Borel subgroup of $ G_2$ in terms of the analytic behavior of $ \mathcal {L}(s,\pi ,\chi ,\mathfrak{st})$ at $ s=\frac {3}{2}$.

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Avner Segal
Affiliation: School of Mathematics, Ben Gurion University of the Negev, POB 653, Be’er Sheva 84105, Israel –and– School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Received by editor(s): August 15, 2016
Received by editor(s) in revised form: January 9, 2017, February 3, 2017, and May 30, 2017
Published electronically: April 4, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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