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An exceptional Siegel–Weil formula and poles of the Spin L-function of $\text{PGSp}_{6}$

Published online by Cambridge University Press:  29 May 2020

Wee Teck Gan
Affiliation:
Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, 119076, Singapore email matgwt@nus.edu.sg
Gordan Savin
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA email savin@math.utah.edu

Abstract

We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of $\text{PGSp}_{6}$ discovered by Pollack, we prove that a cuspidal representation of $\text{PGSp}_{6}$ is a (weak) functorial lift from the exceptional group $G_{2}$ if its (partial) Spin L-function has a pole at $s=1$.

Type
Research Article
Copyright
© The Authors 2020

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References

Chenevier, G., Subgroups of Spin(7) or SO(7) with each element conjugate to some element of G 2 , and applications to automorphic forms, Doc. Math. 24 (2019), 95161.Google Scholar
Dvorsky, A. and Sahi, S., Explicit Hilbert spaces for certain unipotent representations II, Invent. Math. 138 (1999), 203224.10.1007/s002220050347CrossRefGoogle Scholar
Gan, W. T., A regularized Siegel–Weil formula for exceptional groups, in Arithmetic geometry and automorphic forms, Advanced Lectures in Mathematics, vol. 19 (International Press, Somerville, MA, 2011), 155182.Google Scholar
Gan, W. T. and Gurevich, N., Non-tempered Arthur packets of G 2 : liftings from SL̃2, Amer. J. Math. 128 (2006), 11051185.10.1353/ajm.2006.0040CrossRefGoogle Scholar
Gan, W. T. and Gurevich, N., CAP representations of G 2 and the Spin L-function of PGSp6, Israel J. Math. 170 (2009), 152.CrossRefGoogle Scholar
Gan, W. T. and Savin, G., On minimal representations definitions and properties, Represent. Theory 9 (2005), 4693.CrossRefGoogle Scholar
Ginzburg, D. and Jiang, D., Periods and lifting from G 2 to C 3, Israel J. Math. 123 (2001), 2959.10.1007/BF02784119CrossRefGoogle Scholar
Gross, B. H. and Savin, G., Motives with Galois group of type G2: an exceptional theta-correspondence, Compos. Math. 114 (1998), 153217.10.1023/A:1000456731715CrossRefGoogle Scholar
Hanzer, M. and Savin, G., Eisenstein series arising from Jordan algebras, Canad. J. Math. 72 (2020), 183201.10.4153/CJM-2018-033-2CrossRefGoogle Scholar
Hilgert, J., Kobayashi, T. and Möllers, J., Minimal representations via Bessel operators, J. Math. Soc. Japan 66 (2014), 349414.CrossRefGoogle Scholar
Huang, J.-S., Pandžić, P. and Savin, G., New dual pair correspondences, Duke Math. J. 82 (1996), 447471.CrossRefGoogle Scholar
Kobayashi, T. and Savin, G., Global uniqueness of small representations, Math. Z. 281 (2015), 215239.CrossRefGoogle Scholar
Kudla, S. and Rallis, S., A regularized Siegel–Weil formula: the first term identity, Ann. of Math. (2) 140 (1994), 180.CrossRefGoogle Scholar
Lapid, E. M., A remark on Eisenstein series, in Eisenstein series and applications, Progress in Mathematics, vol. 258 (Birkhäuser, Boston, MA, 2008), 239249.CrossRefGoogle Scholar
Li, J.-S., The correspondences of infinitesimal characters for reductive dual pairs in simple Lie groups, Duke Math. J. 97 (1999), 347377.CrossRefGoogle Scholar
Loke, H. Y. and Savin, G., Duality for spherical representations in exceptional theta correspondences, Trans. Amer. Math. Soc. 371 (2019), 63596375.CrossRefGoogle Scholar
Magaard, K. and Savin, G., Exceptional 𝛩-correspondences. I, Compos. Math. 107 (1997), 89123.CrossRefGoogle Scholar
Moeglin, C. and Waldspurger, J.-L., Spectral decomposition and Eisenstein series (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
Möllers, J. and Schwarz, B., Bessel operators on Jordan pairs and small representations of semisimple Lie groups, J. Funct. Anal. 272 (2017), 18921955.CrossRefGoogle Scholar
Pollack, A., The spin L-function on GSp6 for Siegel modular forms, Compos. Math. 153 (2017), 13911432.CrossRefGoogle Scholar
Pollack, A. and Shah, S., The spin L-function on GSp6 via a non-unique model, Amer. J. Math. 153 (2018), 753788.CrossRefGoogle Scholar
Sahi, S., Explicit Hilbert spaces for certain unipotent representations, Invent. Math. 110 (1992), 409418.CrossRefGoogle Scholar
Sahi, S., Unitary representations on the Shilov boundary of a symmetric tube domain, in Representations of groups and algebras, Contemporary Mathematics, vol. 145 (American Mathematical Society, Providence, RI, 1993), 275286.CrossRefGoogle Scholar
Sahi, S., Jordan algebras and degenerate principal series, J. reine angew. Math. 462 (1995), 118.Google Scholar
Savin, G. and Woodbury, M., Structure of internal modules and a formula for the spherical vector of minimal representations, J. Algebra 312 (2007), 755772.CrossRefGoogle Scholar
Savin, G. and Woodbury, M., Matching of Hecke operators for exceptional dual pair correspondences, J. Number Theory 146 (2015), 534556.CrossRefGoogle Scholar
Vogan, D., Singular unitary representations, in Non-commutative harmonic analysis and Lie groups, Lecture Notes in Mathematics, vol. 880, eds Carmona, J. and Vergne, M. (1981), 506535.CrossRefGoogle Scholar
Weissman, M., The Fourier–Jacobi map and small representations, Represent. Theory 7 (2003), 275299.CrossRefGoogle Scholar
Yamana, S., Degenerate principal series representations for quaternionic unitary groups, Israel J. Math. 185 (2011), 77124.CrossRefGoogle Scholar

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An exceptional Siegel–Weil formula and poles of the Spin L-function of $\text{PGSp}_{6}$
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