Generalized spherical functions on reductive -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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the group of rational points of a connected reductive -adic group and let be a maximal compact subgroup satisfying conditions of Theorem 5 from Harish-Chandra (1970). Generalized spherical functions on are eigenfunctions for the action of the Bernstein center, which satisfy a transformation property for the action of . 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 with fixed infinitesimal character belonging to this subset is semi-simple.

**[BD]**J. N. Bernstein,*Le “centre” de Bernstein*, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 1–32 (French). Edited by P. Deligne. MR**771671****[BDK]**J. Bernstein, P. Deligne, and D. Kazhdan,*Trace Paley-Wiener theorem for reductive 𝑝-adic groups*, J. Analyse Math.**47**(1986), 180–192. MR**874050**, 10.1007/BF02792538**[BR]**Bernstein, J. and Rumelhart, K.*Representations of p-adic groups*, Lectures by Joseph Bernstein, preprint.**[BZ]**I. N. Bernstein and A. V. Zelevinsky,*Induced representations of reductive 𝔭-adic groups. I*, Ann. Sci. École Norm. Sup. (4)**10**(1977), no. 4, 441–472. MR**0579172****[C]**Casselman, W.*Introduction to the theory of admissible representations of p-adic reductive groups*, preprint.**[HC1]**Harish-Chandra,*Harmonic analysis on reductive 𝑝-adic groups*, Lecture Notes in Mathematics, Vol. 162, Springer-Verlag, Berlin-New York, 1970. Notes by G. van Dijk. MR**0414797****[HC2]**Harish-Chandra,*Harmonic analysis on reductive 𝑝-adic groups*, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 167–192. MR**0340486****[HOW]**Jing-Song Huang, Toshio Oshima, and Nolan Wallach,*Dimensions of spaces of generalized spherical functions*, Amer. J. Math.**118**(1996), no. 3, 637–652. MR**1393264****[K]**Philip C. Kutzko,*Smooth representations of reductive 𝑝-adic groups: an introduction to the theory of types*, Geometry and representation theory of real and 𝑝-adic groups (Córdoba, 1995) Progr. Math., vol. 158, Birkhäuser Boston, Boston, MA, 1998, pp. 175–196. MR**1486141****[MT]**Moy, A. and Tadic, M.,*The Bernstein center in terms of invariant locally integrable functions*, Represent. Theory**6**(2002), 313-329.**[T]**Marko Tadić,*Geometry of dual spaces of reductive groups (non-Archimedean case)*, J. Analyse Math.**51**(1988), 139–181. MR**963153**, 10.1007/BF02791122**[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.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
22E50,
22E35

Retrieve articles in all journals with MSC (2000): 22E50, 22E35

Additional Information

**Jing-Song Huang**

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

Email:
mahuang@uxmail.ust.hk

**Marko Tadic**

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

Email:
tadic@math.hr

DOI:
https://doi.org/10.1090/S0002-9947-04-03604-9

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