Stability of local gamma factors arising from the doubling method for general spin groups

Wagh, Siddhesh

doi:10.1090/proc/16991

Stability of local gamma factors arising from the doubling method for general spin groups
HTML articles powered by AMS MathViewer

by Siddhesh Wagh;

Proc. Amer. Math. Soc. 153 (2025), 2649-2665

DOI: https://doi.org/10.1090/proc/16991

Published electronically: March 24, 2025

HTML | PDF | Request permission

Abstract:

In this work we prove that the local $\gamma$-factor arising from the doubling integrals for split general spin groups is stable. This deep property of the $\gamma$-factor constitutes an important ingredient in the application of the (generalized) doubling method to the construction of a global functorial lift. We obtain our result by adapting the arguments of Rallis and Soudry who proved the stability property for symplectic and orthogonal groups.

References

Mahdi Asgari, Local $L$-functions for split spinor groups, Canad. J. Math. 54 (2002), no. 4, 673–693. MR 1913914, DOI 10.4153/CJM-2002-025-8
Mahdi Asgari and Freydoon Shahidi, Generic transfer for general spin groups, Duke Math. J. 132 (2006), no. 1, 137–190. MR 2219256, DOI 10.1215/S0012-7094-06-13214-3
Yuanqing Cai, Solomon Friedberg, David Ginzburg, and Eyal Kaplan, Doubling constructions and tensor product $L$-functions: the linear case, Invent. Math. 217 (2019), no. 3, 985–1068. MR 3989257, DOI 10.1007/s00222-019-00883-4
Yuanqing Cai, Solomon Friedberg, and Eyal Kaplan, The generalized doubling method: local theory, Geom. Funct. Anal. 32 (2022), no. 6, 1233–1333. MR 4536463, DOI 10.1007/s00039-022-00609-4
J. W. Cogdell and I. I. Piatetski-Shapiro, Stability of gamma factors for $\textrm {SO}(2n+1)$, Manuscripta Math. 95 (1998), no. 4, 437–461. MR 1618194, DOI 10.1007/s002290050040
J. W. Cogdell, I. I. Piatetski-Shapiro, and F. Shahidi, Stability of $\gamma$-factors for quasi-split groups, J. Inst. Math. Jussieu 7 (2008), no. 1, 27–66. MR 2398146, DOI 10.1017/S1474748007000163
Pierre Deligne, La conjecture de Weil pour les surfaces $K3$, Invent. Math. 15 (1972), 206–226 (French). MR 296076, DOI 10.1007/BF01404126
Wee Teck Gan, Doubling zeta integrals and local factors for metaplectic groups, Nagoya Math. J. 208 (2012), 67–95. MR 3006697, DOI 10.1017/S002776300001059X
Dmitry Gourevitch and Eyal Kaplan, Multiplicity one theorems for the generalized doubling method (with an appendix by Avraham Aizenbud and Dmitry Gourevitch), J. Eur. Math. Soc. (JEMS) 25 (2023), no. 8, 3007–3092. MR 4612108, DOI 10.4171/jems/1207
Joseph Hundley and Eitan Sayag, Descent construction for GSpin groups, Mem. Amer. Math. Soc. 243 (2016), no. 1148, v+124. MR 3517154, DOI 10.1090/memo/1148
Hervé Jacquet and Joseph Shalika, A lemma on highly ramified $\epsilon$-factors, Math. Ann. 271 (1985), no. 3, 319–332. MR 787183, DOI 10.1007/BF01456070
Erez M. Lapid and Stephen Rallis, On the local factors of representations of classical groups, Automorphic representations, $L$-functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ., vol. 11, de Gruyter, Berlin, 2005, pp. 309–359. MR 2192828, DOI 10.1515/9783110892703.309
I. Piatetski-Shapiro and S. Rallis, $\epsilon$ factor of representations of classical groups, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 13, 4589–4593. MR 848553, DOI 10.1073/pnas.83.13.4589
Stephen Rallis and David Soudry, Stability of the local gamma factor arising from the doubling method, Math. Ann. 333 (2005), no. 2, 291–313. MR 2195117, DOI 10.1007/s00208-005-0674-y
Shunsuke Yamana, L-functions and theta correspondence for classical groups, Invent. Math. 196 (2014), no. 3, 651–732. MR 3211043, DOI 10.1007/s00222-013-0476-x
Qing Zhang, Stability of Rankin-Selberg gamma factors for $\textrm {Sp}(2n)$, $\widetilde \textrm {Sp}(2n)$ and $\textrm {U}(n,n)$, Int. J. Number Theory 13 (2017), no. 9, 2393–2432. MR 3704368, DOI 10.1142/S1793042117501317

AMS Website Down

Proceedings of the American Mathematical Society

Stability of local gamma factors arising from the doubling method for general spin groups
HTML articles powered by AMS MathViewer

Abstract: