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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: 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.


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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
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