Advertisement

Mathematische Zeitschrift

, Volume 283, Issue 1–2, pp 169–196 | Cite as

Local theta correspondences between epipelagic supercuspidal representations

  • Hung Yean Loke
  • Jia-Jun Ma
  • Gordan Savin
Article
  • 216 Downloads

Abstract

In this paper we study the local theta correspondences between epipelagic supercupsidal representations of a type I classical dual pair \((G,G')\) over p-adic fields. We show that, besides an exceptional case, an epipelagic supercupsidal representation \(\pi \) of \({\widetilde{G}}\) lifts to an epipelagic supercupsidal representation \(\pi '\) of \({\widetilde{G}}'\) if and only if the epipelagic data of \(\pi \) and \(\pi '\) are related by the moment maps.

Keywords

Local theta correspondence Stable vector Moment map Epipelagic supercuspidal representation 

Mathematics Subject Classification

22E46 22E47 

Notes

Acknowledgments

We would like to thank Wee Teck Gan and Jiu-Kang Yu for their valuable comments. Hung Yean Loke is supported by a MOE-NUS AcRF Tier 1 Grant R-146-000-208-112. Jia-Jun Ma is partially supported by ISF Grant 1138/10 during his postdoctoral Fellowship at Ben Gurion University and HKRGC Grant CUHK 405213 during his postdoctoral fellowship in IMS of CUHK. Gordan Savin is supported by an NSF Grant DMS-1359774.

References

  1. 1.
    Adams, J.: L-functoriality for dual pairs. Asterisque 171–172, 85–129 (1989)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Adler, J.D., Roche, A.: An intertwining result for \(p\)-adic groups. Can. J. Math. 52(3), 449–467 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Broussous, P., Stevens, S.: Buildings of classical groups and centralizers of Lie algebra elements. J. Lie Theory 19, 55–78 (2009)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bruhat, F., Tits, J.: Schémas en groupes et immeubles des groupes classiques sur un corps local. Bull. Math. Soc. Fr. 112, 259–301 (1984)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bruhat, F., Tits, J.: Schémas en groupes et immeubles des groupes classiques sur un corps local, II. Groupes unitaires. Bull. Math. Soc. Fr. 115, 141–195 (1987)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gan, W.T., Takeda, S.: A proof of the Howe duality conjecture. J. AMS (to appear). arXiv:1407.1995 (2014)
  7. 7.
    Goodman, R., Wallach, N.R.: Symmetry, Representations, and Invariants, GTM, vol. 255. Springer, Berlin (2009)Google Scholar
  8. 8.
    Gross, B., Levy, P., Reeder, M., Yu, J.-K.: Gradings of positive rank on simple Lie algebras. Transform. Groups 17(4), 1123–1190 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Howe, R.: \(\theta \)-series and invariant theory. Proc. Symp. Pure Math. 33(Part 1), 275-285 (1979)Google Scholar
  10. 10.
    Howe, R.: Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond. In: Piatetski-Shapiro, I., et al. (ed.) The Schur Lectures (1992). Ramat-Gan: Bar-Ilan University, Isr. Math. Conf. Proc. 8, 1-182 (1995)Google Scholar
  11. 11.
    Kim, J.-L.: Supercuspidal representations: an exhaustion theorem. J. Am. Math. Soc. 20(2), 273–320 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kim, J.-L., Murnaghan, F.: Character expansions and unrefined minimal \(K\)-types. Am. J. Math. 125(6), 1199–1234 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lemaire, B.: Comparison of lattice filtrations and Moy-Prasad filtrations for classical groups. J. Lie Theory 19(1), 029–054 (2008)MathSciNetGoogle Scholar
  14. 14.
    Levy, P.: Vinbergs \(\theta \)-groups in positive characteristic and Kostant–Weierstrass slices. Transform. Groups 14(2), 417–461 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    McNinch, G.J.: Levi factors of the special fiber of a parahoric group scheme and tame ramification. Algebr. Represent. Theory 17(2), 469–479 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Moeglin, C., Vigneras, M.F., Waldspurger, J.-L.: Correspondances de Howe sur un Corps \(p\)-Adique, Lecture Notes in Mathematics, vol. 1291. Springer, Berlin (1987)Google Scholar
  17. 17.
    Moen, C.: The dual pair \((\text{ U }(1),\text{ U }(1))\) over a \(p\)-adic field. Pac. J. Math. 158(2), 365–386 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Moy, A., Prasad, G.: Unrefined minimal \(K\)-types for p-adic groups. Invent. Math. 116, 393–408 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Prasad, G., Yu, J.-K.: On finite group actions on reductive groups and buildings. Invent. Math. 147(3), 545–560 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pan, S.-Y.: Splittings of the metaplectic covers of some reductive dual pairs. Pac. J. Math. 199(1), 163–226 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pan, S.-Y.: Depth preservation in local theta correspondence. Duke Math. J. 113(3), 531–592 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pan, S.-Y.: Local theta correspondence and minimal \(K\)-types of positive depth. Isr. J. Math. 138, 317–352 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Reeder, M., Yu, J.-K.: Epipelagic representations and invariant theory. J. Am. Math. Soc. 27, 437–477 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sun, B., Zhu, C.-B.: Conservation relations for local theta correspondence. J. Am. Math. Soc. 28, 939–983 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tits, J.: Reductive Groups Over Local Fields, Automorphic Forms, Representations and L-Functions I. American Mathematical Society, Providence (1979)Google Scholar
  26. 26.
    Vinberg, E.B.: The Weyl group of a graded Lie algebra. Izv. Akad. Nauk SSR 40(3), 463–495 (1976)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Waldspurger, J.-L.: Démonstration Dune Conjecture de Dualité de Howe dans le cas \(p\)-Adique, \(p \ne 2\) in Festschrift in honor of I. I. Piatetski-Shapiro on the Occasion of his Sixtieth Birthday, Israel Mathematical Conference Proceedings, vol. 2, pp. 267-324. Weizmann, Jerusalem (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsNational University of SingaporeSingaporeSingapore
  2. 2.The Institute of Mathematical SciencesThe Chinese University of Hong KongShatinHong Kong
  3. 3.Department of MathematicsUniversity of UtahSalt Lake CityUSA

Personalised recommendations