The degenerate residual spectrum of quasi-split forms of 𝑆𝑝𝑖𝑛₈ associated to the Heisenberg parabolic subgroup

Segal, Avner

doi:10.1090/tran/7901

The degenerate residual spectrum of quasi-split forms of $Spin_8$ associated to the Heisenberg parabolic subgroup
HTML articles powered by AMS MathViewer

by Avner Segal PDF

Trans. Amer. Math. Soc. 372 (2019), 6703-6754 Request permission

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 $\mathcal {E}_E(\chi , f_s, s, g)$.

References

Jeffrey Adams, Guide to the Atlas software: computational representation theory of real reductive groups, Representation theory of real reductive Lie groups, Contemp. Math., vol. 472, Amer. Math. Soc., Providence, RI, 2008, pp. 1–37. MR 2454331, DOI 10.1090/conm/472/09235
Jeffrey Adams and Fokko du Cloux, Algorithms for representation theory of real reductive groups, J. Inst. Math. Jussieu 8 (2009), no. 2, 209–259. MR 2485793, DOI 10.1017/S1474748008000352
Jeffrey Adams, Wee Teck Gan, Annegret Paul, and Gordan Savin, untitled, unpublished.
James Arthur, Eisenstein series and the trace formula, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 253–274. MR 546601
Jeffrey D. Adams, Marc van Leeuwen, Peter E. Trapa, and David Vogan, Jr. Unitary representations of real reductive groups, http://www.liegroups.org/papers/hermitianForms.pdf.
Dubravka Ban and Chris Jantzen, Jacquet modules and the Langlands classification, Michigan Math. J. 56 (2008), no. 3, 637–653. MR 2490651, DOI 10.1307/mmj/1231770365
Armand Borel, Introduction to automorphic forms, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 199–210. MR 0207650
F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376 (French). MR 756316
A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, 2nd ed., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000. MR 1721403, DOI 10.1090/surv/067
F. du Cloux and M. van Leeuwen, Software for structure and representations of real reductive groups, Atlas of Lie groups and representations, http://www.liegroups.org.
Wee Teck Gan, Nadya Gurevich, and Dihua Jiang, Cubic unipotent Arthur parameters and multiplicities of square integrable automorphic forms, Invent. Math. 149 (2002), no. 2, 225–265. MR 1918673, DOI 10.1007/s002220200210
Wee Teck Gan and Joseph Hundley, The spin $L$-function of quasi-split $D_4$, IMRP Int. Math. Res. Pap. (2006), Art. ID 68213, 74. MR 2268487
S. S. Gelbart and A. W. Knapp, Irreducible constituents of principal series of $\textrm {SL}_{n}(k)$, Duke Math. J. 48 (1981), no. 2, 313–326. MR 620252
David Ginzburg, Stephen Rallis, and David Soudry, On the automorphic theta representation for simply laced groups, Israel J. Math. 100 (1997), 61–116. MR 1469105, DOI 10.1007/BF02773635
Wee Teck Gan and Gordan Savin, A family of Arthur packets of triality $Spin(8)$, unpublished.
Nadya Gurevich and Avner Segal, Poles of the standard $\mathcal {L}$-function of $G_2$ and the Rallis-Schiffmann lift, Canadian J. Math., DOI 10.4153/CJM-2018-019-7.
Nadya Gurevich and Avner Segal, The Rankin-Selberg integral with a non-unique model for the standard $\scr L$-function of $G_2$, J. Inst. Math. Jussieu 14 (2015), no. 1, 149–184. MR 3284482, DOI 10.1017/S147474801300039X
Marcela Hanzer, Degenerate Eisenstein series for symplectic groups, Glas. Mat. Ser. III 50(70) (2015), no. 2, 289–332. MR 3437492, DOI 10.3336/gm.50.2.04
Marcela Hanzer, Non-Siegel Eisenstein series for symplectic groups, Manuscripta Math. 155 (2018), no. 1-2, 229–302. MR 3742780, DOI 10.1007/s00229-017-0927-6
Charles David Keys, On the decomposition of reducible principal series representations of $p$-adic Chevalley groups, Pacific J. Math. 101 (1982), no. 2, 351–388. MR 675406
Henry H. Kim, Residual spectrum of split classical groups; contribution from Borel subgroups, Pacific J. Math. 199 (2001), no. 2, 417–445. MR 1847140, DOI 10.2140/pjm.2001.199.417
A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups. II, Invent. Math. 60 (1980), no. 1, 9–84. MR 582703, DOI 10.1007/BF01389898
C. David Keys and Freydoon Shahidi, Artin $L$-functions and normalization of intertwining operators, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 1, 67–89. MR 944102
Robert P. Langlands, Euler products, Yale Mathematical Monographs, vol. 1, Yale University Press, New Haven, Conn.-London, 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1967. MR 0419366
Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. MR 0579181
Erez Lapid, On Arthur’s asymptotic inner product formula of truncated Eisenstein series, On certain $L$-functions, Clay Math. Proc., vol. 13, Amer. Math. Soc., Providence, RI, 2011, pp. 309–331. MR 2767521
Jing Feng Lau, Residual spectrum of quasi-split $Spin(8)$ defined by a cubic extension, unpublished.
Colette Mœglin, Représentations unipotentes et formes automorphes de carré intégrable, Forum Math. 6 (1994), no. 6, 651–744 (French, with English summary). MR 1300285, DOI 10.1515/form.1994.6.651
C. Mœglin and J.-L. Waldspurger, Spectral decomposition and Eisenstein series, Cambridge Tracts in Mathematics, vol. 113, Cambridge University Press, Cambridge, 1995. Une paraphrase de l’Écriture [A paraphrase of Scripture]. MR 1361168, DOI 10.1017/CBO9780511470905
Taeuk Nam, Avner Segal, and Lior Silberman, Singularities of intertwining operators and certain decompositions of principal series representations, preprint, https://arxiv.org/abs/1811.00803, 2018.
David E. Rohrlich, Root numbers, Arithmetic of $L$-functions, IAS/Park City Math. Ser., vol. 18, Amer. Math. Soc., Providence, RI, 2011, pp. 353–448. MR 2882696, DOI 10.1090/pcms/018/13
Siddhartha Sahi, Jordan algebras and degenerate principal series, J. Reine Angew. Math. 462 (1995), 1–18. MR 1329899, DOI 10.1515/crll.1995.462.1
Avner Segal, A family of new-way integrals for the standard $\mathcal L$-function of cuspidal representations of the exceptional group of type $G_2$, Int. Math. Res. Not. IMRN 7 (2017), 2014–2099. MR 3658191, DOI 10.1093/imrn/rnw090
Avner Segal, The degenerate Eisenstein series attached to the Heisenberg parabolic subgroups of quasi-split forms of $Spin_8$, Trans. Amer. Math. Soc. 370 (2018), no. 8, 5983–6039. MR 3803152, DOI 10.1090/tran/7293
Marko Tadić, Notes on representations of non-Archimedean $\textrm {SL}(n)$, Pacific J. Math. 152 (1992), no. 2, 375–396. MR 1141803
Norman Winarsky, Reducibility of principal series representations of $p$-adic Chevalley groups, Amer. J. Math. 100 (1978), no. 5, 941–956. MR 517138, DOI 10.2307/2373955