Advertisement

Inverse Satake Transforms

  • Yiannis Sakellaridis
Conference paper
Part of the Simons Symposia book series (SISY)

Abstract

Let H be a split reductive group over a local non-Archimedean field, and let \(\check {H}\) denote its Langlands dual group. We present an explicit formula for the generating function of an unramified L-function associated to a highest weight representation of the dual group, considered as a series of elements in the Hecke algebra of H. This offers an alternative approach to a solution of the same problem by Wen-Wei Li. Moreover, we generalize the notion of “Satake transform” and perform the analogous calculation for a large class of spherical varieties.

Keywords

Satake transform Spherical varieties Spherical functions L-functions 

References

  1. [BFGM02]
    A. Braverman, M. Finkelberg, D. Gaitsgory, and I. Mirković. Intersection cohomology of Drinfeld’s compactifications. Selecta Math. (N.S.), 8(3):381–418, 2002. https://doi.org/10.1007/s00029-002-8111-5.MathSciNetCrossRefGoogle Scholar
  2. [BNS16]
    A. Bouthier, B. C. Ngô, and Y. Sakellaridis. On the formal arc space of a reductive monoid. Amer. J. Math., 138(1):81–108, 2016.  https://doi.org/10.1353/ajm.2016.0004.MathSciNetCrossRefGoogle Scholar
  3. [Del]
    Patrick Delorme. Neighborhoods at infinity and the Plancherel formula for a reductive p-adic symmetric space. Math. Ann., 370(3–4):1177–1229, 2018. https://doi.org/10.1007/s00208-017-1554-y.MathSciNetCrossRefGoogle Scholar
  4. [GJ72]
    Roger Godement and Hervé Jacquet. Zeta functions of simple algebras. Lecture Notes in Mathematics, Vol. 260. Springer-Verlag, Berlin-New York, 1972.Google Scholar
  5. [Gro98]
    Benedict H. Gross. On the Satake isomorphism. In Galois representations in arithmetic algebraic geometry (Durham, 1996), volume 254 of London Math. Soc. Lecture Note Ser., pages 223–237. Cambridge Univ. Press, Cambridge, 1998.  https://doi.org/10.1017/CBO9780511662010.006.
  6. [Li17]
    Wen-Wei Li. Basic functions and unramified local L-factors for split groups. Sci. China Math., 60(5):777–812, 2017. https://doi.org/10.1007/s11425-015-0730-4.MathSciNetCrossRefGoogle Scholar
  7. [Lun01]
    D. Luna. Variétés sphériques de type A. Publ. Math. Inst. Hautes Études Sci., (94):161–226, 2001. https://doi.org/10.1007/s10240-001-8194-0.CrossRefGoogle Scholar
  8. [Ngô]
    Bao Châu Ngô. On a certain sum of automorphic L-functions. In Automorphic forms and related geometry: assessing the legacy of I. I. Piatetski-Shapiro, volume 614 of Contemp. Math., pages 337–343. Amer. Math. Soc., Providence, RI, 2014.  https://doi.org/10.1090/conm/614/12270.
  9. [Sak08]
    Yiannis Sakellaridis. On the unramified spectrum of spherical varieties over p-adic fields. Compos. Math., 144(4):978–1016, 2008. https://doi.org/10.1112/S0010437X08003485.MathSciNetCrossRefGoogle Scholar
  10. [Sak12]
    Yiannis Sakellaridis. Spherical varieties and integral representations of L-functions. Algebra & Number Theory, 6(4):611–667, 2012.  https://doi.org/10.2140/ant.2012.6.611.MathSciNetCrossRefGoogle Scholar
  11. [Sak13]
    Yiannis Sakellaridis. Spherical functions on spherical varieties. Amer. J. Math., 135(5):1291–1381, 2013.  https://doi.org/10.1353/ajm.2013.0046.MathSciNetCrossRefGoogle Scholar
  12. [SV]
    Yiannis Sakellaridis and Akshay Venkatesh. Periods and harmonic analysis on spherical varieties. Astérisque, 396:viii+360, 2017.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics & Computer ScienceRutgers University - NewarkNewarkUSA

Personalised recommendations