TY - JOUR

T1 - The degenerate Eisenstein series attached to the Heisenberg parabolic subgroups of quasi-split forms of spin8

AU - Segal, Avner

N1 - Publisher Copyright:
© 2018 American Mathematical Society.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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 L-function L(s, π, χ, st) of a cuspidal representation π of the exceptional group of type G2. These integral representations bind the analytic behavior of this L-function with that of a family of degenerate Eisenstein series for quasi-split forms of Spin8 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 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 π to be a CAP representation with respect to the Borel subgroup of G2 in terms of the analytic behavior of L(s, π, χ, st) at s = 3/2.

AB - 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 L-function L(s, π, χ, st) of a cuspidal representation π of the exceptional group of type G2. These integral representations bind the analytic behavior of this L-function with that of a family of degenerate Eisenstein series for quasi-split forms of Spin8 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 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 π to be a CAP representation with respect to the Borel subgroup of G2 in terms of the analytic behavior of L(s, π, χ, st) at s = 3/2.

UR - http://www.scopus.com/inward/record.url?scp=85047129744&partnerID=8YFLogxK

U2 - 10.1090/tran/7293

DO - 10.1090/tran/7293

M3 - Article

AN - SCOPUS:85047129744

SN - 0002-9947

VL - 370

SP - 5983

EP - 6039

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 8

ER -