The degenerate Eisenstein series attached to the Heisenberg parabolic subgroups of quasi-split forms of 𝑆𝑝𝑖𝑛₈

Segal, Avner

doi:10.1090/tran/7293

The degenerate Eisenstein series attached to the Heisenberg parabolic subgroups of quasi-split forms of $Spin_8$
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Trans. Amer. Math. Soc. 370 (2018), 5983-6039 Request permission

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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