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Generalized spherical functions on reductive $p$-adic groups

Authors: Jing-Song Huang and Marko Tadic
Journal: Trans. Amer. Math. Soc. 357 (2005), 2081-2117
MSC (2000): Primary 22E50, 22E35
Published electronically: December 28, 2004
MathSciNet review: 2115092
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be the group of rational points of a connected reductive $p$-adic group and let $K$ be a maximal compact subgroup satisfying conditions of Theorem 5 from Harish-Chandra (1970). Generalized spherical functions on $G$ are eigenfunctions for the action of the Bernstein center, which satisfy a transformation property for the action of $K$. In this paper we show that spaces of generalized spherical functions are finite dimensional. We compute dimensions of spaces of generalized spherical functions on a Zariski open dense set of infinitesimal characters. As a consequence, we get that on that Zariski open dense set of infinitesimal characters, the dimension of the space of generalized spherical functions is constant on each connected component of infinitesimal characters. We also obtain the formula for the generalized spherical functions by integrals of Eisenstein type. On the Zariski open dense set of infinitesimal characters that we have mentioned above, these integrals then give the formula for all the generalized spherical functions. At the end, let as mention that among others we prove that there exists a Zariski open dense subset of infinitesimal characters such that the category of smooth representations of $G$ with fixed infinitesimal character belonging to this subset is semi-simple.

References [Enhancements On Off] (What's this?)

  • [BD] Bernstein, J. and Deligne, P., Le ``center'' de Bernstein, Représentations des groupes reductifs sur un corps local (1985), 1-32, Hermann, Paris. MR 86e:22028
  • [BDK] Bernstein, J., Deligne, P. and Kazhdan, D., Trace Paley-Wiener theorem for reductive p-adic groups, Journal D'analyse Mathématique 47 (1986), 180-192. MR 88g:22016
  • [BR] Bernstein, J. and Rumelhart, K. Representations of p-adic groups, Lectures by Joseph Bernstein, preprint.
  • [BZ] Bernstein, I. N. and Zelevinsky, A.V., Induced representations of reductive $p$-adic groups I, Ann. Sci. École Norm Sup. 10 (1977), 441-472. MR 58:28310
  • [C] Casselman, W. Introduction to the theory of admissible representations of p-adic reductive groups, preprint.
  • [HC1] Harish-Chandra (notes by G. van Dijk), Harmonic Analysis on Reductive $p$-adic Groups, Lecture Notes in Math. 162, Springer-Verlag, Berlin, 1970. MR 54:2889
  • [HC2] Harish-Chandra, Harmonic analysis on reductive p-adic groups, Proc. Sympos. Pure Math. XXVI, Amer. Math. Soc., Providence, 1973, 167-192.MR 49:5238
  • [HOW] Huang, J.-S., Oshima, T. and Wallach, N., Dimensions of spaces of generalized spherical functions, Amer. J. Math. 118 (1996), 637-652.MR 97d:22015
  • [K] Kutzko, P. (J. Tiran, D. Vogan and J. Wolf, eds.), Smooth representations of reductive p-adic groups: An introduction to the theory of types, Geometry and Representation Theory of Real and p-adic Groups, Birkhauser, Boston, 1997, 175-196.MR 98k:22072
  • [MT] Moy, A. and Tadic, M., The Bernstein center in terms of invariant locally integrable functions, Represent. Theory 6 (2002), 313-329.
  • [T] Tadic, M., Geometry of dual spaces of reductive groups (non-archimedean case), J. Analyse Math. 51 (1988), 139-181. MR 90c:22057
  • [W] Waldspurger, J.-L., La formule de Plancherel pour les groupes p-adiques, d'après Harish-Chandra, Journal de l'Institut de Math. de Jussieu 2 2, (2003), 235-333.

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Additional Information

Jing-Song Huang
Affiliation: Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

Marko Tadic
Affiliation: Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia

Keywords: Reductive $p$-adic group, generalized spherical function, Bernstein center, infinitesimal character
Received by editor(s): March 31, 2003
Received by editor(s) in revised form: January 2, 2004
Published electronically: December 28, 2004
Additional Notes: The first author was partially supported by Hong Kong Research Grant Council Competitive Earmarked Research Grant. The second author was partly supported by Croatian Ministry of Science and Technology grant # 37108
Article copyright: © Copyright 2004 American Mathematical Society

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