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On the Gan–Gross–Prasad problem for finite unitary groups

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In this paper we study the Gan–Gross–Prasad problem for unitary groups over finite fields. Our results provide complete answers for unipotent representations, and we obtain the explicit branching of these representations.

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References

  1. Alvis, D.: The duality operation in the character ring of a finite Chevalley group. Bull. Am. Math. Soc. (N.S.) 1, 907–911 (1979)

    Article  MathSciNet  Google Scholar 

  2. Adams, J., Moy, A.: Unipotent representations and reductive dual pairs over finite fields. Trans. Am. Math. Soc. 340, 309–321 (1993)

    Article  MathSciNet  Google Scholar 

  3. Atobe, H.: The local theta correspondence and the local Gan–Gross–Prasad conjecture for the symplectic-metaplectic case. Math. Ann. 371(1–2), 225–295 (2018)

    Article  MathSciNet  Google Scholar 

  4. Aubert, A.-M., Michel, J., Rouquier, R.: Correspondance de Howe pour les groupes reductifs sur les corps finis. Duke Math. J. 83(2), 353–397 (1996)

    Article  MathSciNet  Google Scholar 

  5. Beuzart-Plessis, R.: La conjecture locale de Gross–Prasad pour les représentations tempérées des groupes unitaires. Mém. Soc. Math. Fr. (N.S.) 149, vii+191 (2016)

  6. Beuzart-Plessis, R.: Endoscopie et conjecture locale raffinée de Gan–Gross–Prasad pour les groupes unitaires. Compos. Math. 151(7), 1309–1371 (2015)

    Article  MathSciNet  Google Scholar 

  7. Carter, R.: Finite Groups of Lie Type, Conjugacy Classes and Complex Characters. Wiley, England (1985)

    MATH  Google Scholar 

  8. Curtis, C.W.: Truncation and duality in the character ring of a finite group of Lie type. J. Algebra 62, 320–C332 (1980)

    Article  MathSciNet  Google Scholar 

  9. Deligne, P., Lusztig, G.: Representations of reductive groups over finite fields. Ann. Math. 103, 103–161 (1976)

    Article  MathSciNet  Google Scholar 

  10. Gan, W.T., Gross, B.H., Prasad, D.: Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Astérisque 346, 1–109 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Gan, W.T., Gross, B.H., Prasad, D.: Restrictions of representations of classical groups: examples. Astérisque 346, 111–170 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Gan, W.T., Ichino, A.: The Gross–Prasad conjecture and local theta correspondence. Invent. Math. 206(3), 705–799 (2016)

    Article  MathSciNet  Google Scholar 

  13. Gérardin, P.: Weil representations associated to finite fields. J. Algebra 46, 54–101 (1977)

    Article  MathSciNet  Google Scholar 

  14. Gross, B., Prasad, D.: On the decomposition of a representation of \({\rm SO}_n\) when restricted to \({\rm SO}_{n-1}\). Can. J. Math. 44, 974–1002 (1992)

    Article  Google Scholar 

  15. Gross, B., Prasad, D.: On irreducible representations of SO\(_{2n+1}\times \) SO\(_{2m}\). Can. J. Math. 46, 930–950 (1994)

    Article  MathSciNet  Google Scholar 

  16. Hiss, G., Zalesski, A.: The Weil–Steinberg character of finite classical groups. Represent. Theory 13, 427–459 (2009)

    Article  MathSciNet  Google Scholar 

  17. Kawanaka, N.: Fourier transforms of nilpotently supported invariant functions on a finite simple Lie algebra. Proc. Japan Acad. Ser. A Math. Sci. 57, 461–464 (1981)

    Article  MathSciNet  Google Scholar 

  18. Lusztig, G.: Irreducible representations of finite classical groups. Invent. Math. 43, 125–175 (1977)

    Article  MathSciNet  Google Scholar 

  19. Lusztig, G., Srinivasan, B.: The characters of the finite unitary groups. J. Algebra 49, 167–171 (1977)

    Article  MathSciNet  Google Scholar 

  20. Liu, D., Wang, Z.: Descents of unipotent representations of finite unitary groups. Trans. Am. Math. Soc. 373, 4223–4253 (2020)

    Article  MathSciNet  Google Scholar 

  21. Mœglin, C., Waldspurger, J.-L.: La conjecture locale de Gross–Prasad pour les groupes spéciaux orthogonaux: le cas général. Sur les conjectures de Gross et Prasad. II. Astérisque 347, 167–216 (2012)

  22. Reeder, M.: On the restriction of Deligne–Lusztig characters. J. Am. Math. Soc. 20, 573–602 (2007)

    Article  MathSciNet  Google Scholar 

  23. Srinivasan, B.: Representations of Finite Chevalley Groups, Lecture Notes in Math., vol. 764. Springer (1979)

  24. Srinivasan, B.: Weil representations of finite classical groups. Invent. Math. 51, 143–153 (1979)

    Article  MathSciNet  Google Scholar 

  25. Waldspurger, J.-L.: Une formule intégral à la conjecture locale de Gross–Prasad. Compos. Math. 146(5), 1180–1290 (2010)

    Article  MathSciNet  Google Scholar 

  26. Waldspurger, J.-L.: Une formule intégrale reliée à la conjecture locale de Gross-Prasad, 2e partie: extension aux représentations tempérées. Sur les conjectures de Gross et Prasad. I. Astérisque 346, 171–312 (2012)

    MATH  Google Scholar 

  27. Waldspurger, J.-L.: La conjecture locale de Gross-Prasad pour les représentations tempérées. des groupes spéciaux orthogonaux. Sur les conjectures de Gross et Prasad. IISur les conjectures de Gross et Prasad. II. Astérisque 347, 103–165 (2012)

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, Zhejiang, People’s Republic of China

    Dongwen Liu & Zhicheng Wang

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  1. Dongwen Liu
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  2. Zhicheng Wang
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Correspondence to Dongwen Liu.

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Liu, D., Wang, Z. On the Gan–Gross–Prasad problem for finite unitary groups. Math. Z. 297, 997–1021 (2021). https://doi.org/10.1007/s00209-020-02543-3

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