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Siegel series for skew Hermitian forms over quaternion algebras

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Abstract

We prove a functional equation of Siegel series associated to nondegenerate semi-integral skew Hermitian forms over quaternion algebras over nonarchimedean local fields of characteristic not 2.

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References

  1. Böcherer, S., Kohnen, W.: On the functional equation of singular series. Abh. Math. Sem. Univ. Hamburg 70, 281–286 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Gan, W.T., Ichino, A.: Formal degree and theta correspondence. Invent. Math. 195(3), 509–672 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Gan, W.T., Yu, J.-K.: Group schemes and local densities. Duke Math. J. 105(3), 497–524 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Harris, M., Kudla, S., Sweet Jr., W.J.: Theta dichotomy for unitary groups. J. Am. Math. Soc. 9, 941–1004 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hirai, Y.: On Eisenstein series on quaternion unitary groups of degree \(2\). J. Math. Soc. Japan 51(1), 93–128 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hironaka, Y., Sato, F.: The Siegel series and spherical functions on \({\rm O}(2n)/({\rm O}(n) \times {\rm O}(n))\). In: Automorphic Forms and Zeta Functions, pp. 150–169. World Sci. Publ, Hackensack (2006)

  7. Igusa, J.: On functional equations of complex powers. Invent. Math. 85, 1–29 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ikeda, T.: On the lifting of elliptic cusp forms to Siegel cusp forms of degree \(2n\). Ann. Math. 154, 641–681 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ikeda, T.: On the lifting of Hermitian modular forms. Compos. Math. 144(5), 1107–1154 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ikeda, T.: On the functional equation of the Siegel series (preprint)

  11. Kahn, B.: Sommes de Gauss attachées aux caractères quadratiques: une conjecture de Pierre Conner. Comment. Math. Helv. 62, 532–541 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Karel, M.: Functional equations of Whittaker functions on \(p\)-adic groups. Am. J. Math. 101, 1303–1325 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  13. Katsurada, H.: An explicit formula for Siegel series. Am. J. Math. 121, 415–452 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kottwitz, R.: Sign changes in harmonic analysis on reductive groups. Trans. Am. Math. Soc. 278, 289–297 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kudla, S., Sweet Jr., W.J.: Degenerate principal series representations for \(U(n, n)\). Israel J. Math. 98, 253–306 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lapid, E., Rallis, S.: On the local factors of representations of classical groups. In: Automorphic Representations. \(L\)-Functions and Applications: Progress and Prospects, pp. 309–359. de Gruyter, Berlin (2005)

  17. Piatetski-Shapiro, I., Rallis, S.: Rankin triple \(L\)-functions. Compos. Math. 64, 31–115 (1987)

    MathSciNet  MATH  Google Scholar 

  18. Shimura, G.: Euler products and Eisenstein series. In: CBMS Regional Conference Series in Mathematics, vol. 93. Amer. Math. Soc., New York (1997)

  19. Shimura, G.: Some exact formulas on quaternion unitary groups. J. Reine Angew. Math. 509, 67–102 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sugano, T.: On holomorphic cusp forms on quaternion unitary groups of degree \(2\). J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31(3), 521–568 (1985)

  21. Sweet Jr., W.J.: A computation of the gamma matrix of a family of \(p\)-adic zeta integrals. J. Number Theory 55, 222–260 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sweet, W.J. Jr.: Functional equations of \(p\)-adic zeta integrals and representations of the metaplectic group (preprint)

  23. Weil, A.: Sur la formule de Siegel dans la théorie des groupes classiques. Acta Math. 113, 1–87 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yamana, S.: On the lifting of elliptic cusp forms to cusp forms on quaternionic unitary groups. J. Number Theory 130(11), 2480–2527 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yamana, S.: Degenerate principal series representations for quaternionic unitary groups. Israel J. Math. 185, 77–124 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yamana, S.: On the Siegel–Weil formula for quaternionic unitary groups. Am. J. Math. 135(5), 1383–1432 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yamana, S.: \(L\)-functions and theta correspondence for classical groups. Invent. Math. 196(3), 651–732 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The idea of the proof of Lemma 5.5 was suggested by Wee Teck Gan. We thank him for sharing his insight with us. The author is partially supported by JSPS Grant-in-Aid for Young Scientists (B) 26800017. This paper was written during the author’s stay at University of Rijeka. The author would like to thank the staffs of University of Rijeka, especially Neven Grbac, for an excellent working environment. We are grateful to the anonymous referee for a very careful reading and detailed comments, which helped improve the exposition of the earlier version.

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Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan

    Shunsuke Yamana

  2. Hakubi Center, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501, Japan

    Shunsuke Yamana

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  1. Shunsuke Yamana
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Correspondence to Shunsuke Yamana.

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Communicated by Jens Funke.

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Yamana, S. Siegel series for skew Hermitian forms over quaternion algebras. Abh. Math. Semin. Univ. Hambg. 87, 43–59 (2017). https://doi.org/10.1007/s12188-016-0127-4

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