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Murnaghan-Kirillov theory for depth-zero supercuspidal representations: reduction to Lusztig functions

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Abstract

The goal of Murnaghan-Kirillov theory is to associate to an irreducible smooth representation of a reductive p-adic group a family of regular semisimple orbital integrals in the Lie algebra with the following property: the character of π is given, on a well determined set, by an explicit combination of the Fourier transforms of these orbital integrals. Subject to certain restrictions, we adapt arguments of Waldspurger to show that, for depth-zero irreducible smooth supercuspidal representations, this problem may be reduced to a similar one for distributions associated to Lusztig functions.

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References

  1. J. D. Adler, Refined anisotropic K-types and supercuspidal representations, Pacific J. Math. 185 (1998), 1–32.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Adler, S. DeBacker, Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups, J. reine angew. Math. 575 (2004), 1–35.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Adler, S. DeBacker, Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive p-adic group, Michigan Math. J. 50 (2002), no. 2, 263–286.

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Auslander, B. Kostant, Polarization and unitary representations of solvable Lie groups, Invent. Math. 14 (1971), 255–354.

    Article  MathSciNet  MATH  Google Scholar 

  5. C. Cunningham, Characters of depth-zero, supercuspidal representations of the rank-2 symplectic group, Canad. J. Math. 52 (2000), no. 2, 306–331.

    Article  MathSciNet  MATH  Google Scholar 

  6. S. DeBacker, Parametrizing nilpotent orbits via Bruhat-Tits theory, Ann. of Math. (2) 156 (2002), no. 1, 295–332.

    Article  MathSciNet  MATH  Google Scholar 

  7. S. DeBacker, Parametrizing conjugacy classes of maximal unramified tori, Michigan Math. J. 54 (2006), no. 1, 157–178.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. DeBacker, Homogeneity results for invariant distributions of a reductive p-adic group, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 3, 391–422.

    MathSciNet  MATH  Google Scholar 

  9. S. DeBacker, D. Kazhdan, Stable distributions supported on the nilpotent cone for the group G2, in: The Unity of Mathematics, Progress in Mathematics, Vol. 244, Birkhäuser, Boston, 2006, pp. 205–262.

  10. S. DeBacker, M. Reeder, Depth-zero supercuspidal L-packets and their stability, Ann. of Math. (2) 169 (2009), no. 3, 795–901.

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Deligne, G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161.

    Article  MathSciNet  MATH  Google Scholar 

  12. F. Digne, G. Lehrer, J. Michel, The space of unipotently supported class functions on a finite reductive group, J. Algebra 260 (2003), no. 1, 111–137.

    Article  MathSciNet  MATH  Google Scholar 

  13. Harish-Chandra, Harmonic Analysis on Reductive p-Adic Groups, Notes by G. van Dijk, Lecture Notes in Mathematics, Vol. 162, Springer-Verlag, Berlin, 1970.

  14. Harish-Chandra, Admissible Invariant Distributions on Reductive p-Adic Groups, Preface and notes by Stephen DeBacker and Paul J. Sally, Jr., University Lecture Series, Vol. 16, American Mathematical Society, Providence, RI, 1999.

  15. R. Howe, The Fourier transform and germs of characters (case of Gl n over a p-adic field), Math. Ann. 208 (1974), 302–322.

    Article  MathSciNet  Google Scholar 

  16. R. Howe, Two conjectures about reductive p-adic groups, in: Harmonic Analysis on Homogeneous Spaces, Proceedings of Symposia in Pure Mathematics, Vol. 26, American Mathematical Society, 1973, pp. 377–380.

  17. R. Howe, Kirillov theory for compact p-adic groups, Pacific J. Math. 73 (1977), no. 2, 36–381.

    Google Scholar 

  18. D. Kazhdan, Proof of Springer’s hypothesis, Israel J. Math. 28 (1977), 272–286.

    Article  MathSciNet  MATH  Google Scholar 

  19. J.-L. Kim, Supercuspidal representations: an exhaustion theorem, J. Amer. Math. Soc. 20 (2007), no. 2, 273–320.

    Article  MathSciNet  MATH  Google Scholar 

  20. J.-L. Kim, F. Murnaghan, Character expansions and unrefined minimal K-types, Amer. J. Math. 125 (2003), 1199–1234.

    Article  MathSciNet  MATH  Google Scholar 

  21. J.-L. Kim, F. Murnaghan, K-types and Γ-asymptotic expansions, J. reine angew. Math. 592 (2006), 189–236.

    Article  MathSciNet  MATH  Google Scholar 

  22. А. А. Кириллов, Унцтарные представленця нцльпотентных групп Лц, УМН 17 (1962), no. 4 (106), 57–110. Engl. transl.: A. A. Kirillov, Unitary representations of nilpotent Lie groups, Russ. Math. Surv. 17 (1962), no. 4, 53–104.

    Google Scholar 

  23. A. A. Kirillov, Merits and demerits of the orbit method, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 4, 433–488.

    Article  MathSciNet  MATH  Google Scholar 

  24. E. Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Lecture Notes in Mathematics, Vol. 1859, Springer, Berlin, 2005.

  25. G. Lusztig, Green functions and character sheaves, Ann. of Math. 131 (1990), 355–408.

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Lusztig, Fourier transform on a semi-simple Lie algebra over \( {\mathbb{F}_q} \), in: Algebraic Groups, Utrecht 1986, Lecture Notes in Mathematics, Vol. 1271, Springer-Verlag, Berlin, 1987, pp. 177–188.

  27. G. Lusztig, Character sheaves, I-V, Adv. in Math., 56 (1985), no. 3, 193–237; 57 (1985), no. 3, 226–265; 57 (1985), no. 3, 266–315; 59 (1986), no. 1, 1–63; 61 (1986), no. 2, 103–155.

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Moy, G. Prasad, Jacquet functors and unrefined minimal K-types, Comment. Math. Helvetici 71 (1996), 98–121.

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Moy, G. Prasad, Unrefined minimal K-types for p-adic groups, Invent. Math. 116 (1994), 393–408.

    Article  MathSciNet  MATH  Google Scholar 

  30. F. Murnaghan, Characters of supercuspidal representations of SL(n), Pacific J. Math. 170 (1995), no. 1, 217–235.

    MathSciNet  MATH  Google Scholar 

  31. F. Murnaghan, Local character expansions for supercuspidal representations of U(3), Canad. J. Math. 47 (1995), no. 3, 606–640.

    Article  MathSciNet  MATH  Google Scholar 

  32. F. Murnaghan, Characters of supercuspidal representations of classical groups, Ann. Sci. Éc. Norm. Sup. (4) 29 (1996), no. 1, 49–105.

    MathSciNet  MATH  Google Scholar 

  33. F. Murnaghan, Local character expansions and Shalika germs for GL(n), Math. Ann. 304 (1996), no. 3, 423–455.

    Article  MathSciNet  MATH  Google Scholar 

  34. G. Rousseau, Immeubles des groupes réductifs sure les corps locaux, Ph.D. thesis, Paris XI, 1977.

  35. W. Rossmann, Kirillov’s character formula for reductive Lie groups, Invent. Math. 48 (1978), no. 3, 207–220.

    Article  MathSciNet  MATH  Google Scholar 

  36. P. J. Sally, Jr., Some remarks on discrete series characters for reductive p-adic groups, in: Representations of Lie Groups, Kyoto, Hiroshima 1986, Adv. Stud. Pure Math., Vol. 14, Academic Press, Boston, MA, 1988, pp. 337–348.

  37. J.-L. Waldspurger, Homogénéité de certaines distributions sur les groupes p-adiques, Inst. Hautes Études Sci. Publ. Math. 81 (1992), 22–72.

    Google Scholar 

  38. J.-L. Waldspurger, Intégrales Orbitales Nilpotentes et Endoscopie pour les Groupes Classiques non Ramifiés, Astérisque, Vol. 269, 2001.

  39. J.-L. Waldspurger, Quelques resultats de finitude concernant les distributions invariantes sur les algébres de Lie p-adiques, preprint, 1993.

  40. J.-K. Yu, Tame construction of supercuspidal representations, J. Amer. Math. Soc. 14 (2001), no. 3, 579–622.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 530 Church St, 2074 East Hall, Ann Arbor, MI, 48109-1043, USA

    Stephen Debacker

  2. Einstein Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem, 91904, Israel

    David Kazhdan

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  1. Stephen Debacker
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  2. David Kazhdan
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Correspondence to Stephen Debacker.

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To Tony Springer on his 85th birthday

First author partially supported by National Science Foundation Grant Numbers 0345121 and DMS-0854897. Second author supported by ISF.

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Debacker, S., Kazhdan, D. Murnaghan-Kirillov theory for depth-zero supercuspidal representations: reduction to Lusztig functions. Transformation Groups 16, 737–766 (2011). https://doi.org/10.1007/s00031-011-9155-4

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  • DOI: https://doi.org/10.1007/s00031-011-9155-4

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