Generalized spherical functions on reductive adic groups
Authors:
JingSong Huang and Marko Tadic
Journal:
Trans. Amer. Math. Soc. 357 (2005), 20812117
MSC (2000):
Primary 22E50, 22E35
Published electronically:
December 28, 2004
MathSciNet review:
2115092
Fulltext 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 HarishChandra (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 semisimple.
 [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
(86e:22028)
 [BDK]
J.
Bernstein, P.
Deligne, and D.
Kazhdan, Trace PaleyWiener theorem for reductive 𝑝adic
groups, J. Analyse Math. 47 (1986), 180–192. MR 874050
(88g:22016), http://dx.doi.org/10.1007/BF02792538
 [BR]
Bernstein, J. and Rumelhart, K. Representations of padic groups, Lectures by Joseph Bernstein, preprint.
 [BZ]
I.
N. Bernstein and A.
V. Zelevinsky, Induced representations of reductive 𝔭adic
groups. I, Ann. Sci. École Norm. Sup. (4) 10
(1977), no. 4, 441–472. MR 0579172
(58 #28310)
 [C]
Casselman, W. Introduction to the theory of admissible representations of padic reductive groups, preprint.
 [HC1]
HarishChandra,
Harmonic analysis on reductive 𝑝adic groups, Lecture
Notes in Mathematics, Vol. 162, SpringerVerlag, Berlin, 1970. Notes by G.
van Dijk. MR
0414797 (54 #2889)
 [HC2]
HarishChandra,
Harmonic analysis on reductive 𝑝adic 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
(49 #5238)
 [HOW]
JingSong
Huang, Toshio
Oshima, and Nolan
Wallach, Dimensions of spaces of generalized spherical
functions, Amer. J. Math. 118 (1996), no. 3,
637–652. MR 1393264
(97d:22015)
 [K]
Philip
C. Kutzko, Smooth representations of reductive 𝑝adic
groups: an introduction to the theory of types, (Córdoba,
1995) Progr. Math., vol. 158, Birkhäuser Boston, Boston, MA,
1998, pp. 175–196. MR 1486141
(98k:22072)
 [MT]
Moy, A. and Tadic, M., The Bernstein center in terms of invariant locally integrable functions, Represent. Theory 6 (2002), 313329.
 [T]
Marko
Tadić, Geometry of dual spaces of reductive groups
(nonArchimedean case), J. Analyse Math. 51 (1988),
139–181. MR
963153 (90c:22057), http://dx.doi.org/10.1007/BF02791122
 [W]
Waldspurger, J.L., La formule de Plancherel pour les groupes padiques, d'après HarishChandra, Journal de l'Institut de Math. de Jussieu 2 2, (2003), 235333.
 [BD]
 Bernstein, J. and Deligne, P., Le ``center'' de Bernstein, Représentations des groupes reductifs sur un corps local (1985), 132, Hermann, Paris. MR 86e:22028
 [BDK]
 Bernstein, J., Deligne, P. and Kazhdan, D., Trace PaleyWiener theorem for reductive padic groups, Journal D'analyse Mathématique 47 (1986), 180192. MR 88g:22016
 [BR]
 Bernstein, J. and Rumelhart, K. Representations of padic groups, Lectures by Joseph Bernstein, preprint.
 [BZ]
 Bernstein, I. N. and Zelevinsky, A.V., Induced representations of reductive adic groups I, Ann. Sci. École Norm Sup. 10 (1977), 441472. MR 58:28310
 [C]
 Casselman, W. Introduction to the theory of admissible representations of padic reductive groups, preprint.
 [HC1]
 HarishChandra (notes by G. van Dijk), Harmonic Analysis on Reductive adic Groups, Lecture Notes in Math. 162, SpringerVerlag, Berlin, 1970. MR 54:2889
 [HC2]
 HarishChandra, Harmonic analysis on reductive padic groups, Proc. Sympos. Pure Math. XXVI, Amer. Math. Soc., Providence, 1973, 167192.MR 49:5238
 [HOW]
 Huang, J.S., Oshima, T. and Wallach, N., Dimensions of spaces of generalized spherical functions, Amer. J. Math. 118 (1996), 637652.MR 97d:22015
 [K]
 Kutzko, P. (J. Tiran, D. Vogan and J. Wolf, eds.), Smooth representations of reductive padic groups: An introduction to the theory of types, Geometry and Representation Theory of Real and padic Groups, Birkhauser, Boston, 1997, 175196.MR 98k:22072
 [MT]
 Moy, A. and Tadic, M., The Bernstein center in terms of invariant locally integrable functions, Represent. Theory 6 (2002), 313329.
 [T]
 Tadic, M., Geometry of dual spaces of reductive groups (nonarchimedean case), J. Analyse Math. 51 (1988), 139181. MR 90c:22057
 [W]
 Waldspurger, J.L., La formule de Plancherel pour les groupes padiques, d'après HarishChandra, Journal de l'Institut de Math. de Jussieu 2 2, (2003), 235333.
Similar Articles
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
JingSong 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:
http://dx.doi.org/10.1090/S0002994704036049
PII:
S 00029947(04)036049
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
