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The Bernstein center in terms of invariant locally integrable functions

Author(s): Allen Moy; Marko Tadic
Journal: Represent. Theory 6 (2002), 313-329.
MSC (2000): Primary 22E50, 22E35
Posted: November 19, 2002
Errata: Represent. Theory 9 (2005), 455-456.
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Abstract: We give a description of the Bernstein center of a reductive $p$-adic group $G$ in terms of invariant locally integrable functions and compute a basis of these functions for the group $SL(2)$.

References:

[BD]
Bernstein, J. (edited by Deligne, P.), Le ``centre" de Bernstein, in ``Représentations des Groupes Réductifs sur un Corps Local" written by J.-N. Bernstein, P. Deligne, D. Kazhdan, M.-F. and Vignéras,, Hermann, Paris, 1984. MR 86e:22028

[BDK]
Bernstein, J., Deligne, P. and Kazhdan, D., Trace Paley-Wiener theorem for reductive $p$-adic groups, J. Analyse Math 47 (1986), 180-192. MR 88g:22016

[vD]
Dijk, G. van, Computation of certain induced characters of $\mathfrak{p}$-adic groups, Math. Ann. 199 (1972), 229-240. MR 49:3043

[HC1]
Harish-Chandra, Harmonic analysis on reductive $p$-adic groups, Symp. Pure Math. 26,, Amer. Math. Soc., Providence, Rhone Island, 1973, pp. 167-192. MR 49:5238

[HC2]
Harish-Chandra, Admissible invariant distributions on reductive $p$-adic groups, Lie theories and their applications (Proc. Ann. Sem. Canad. Math. Congr., Queen's Univ., Kingston, Ont., 1977). Queen's Papers in Pure Appl. Math., No. 48, Queen's Univ., Kingston, Ont., 1978, pp. 281-347. MR 58:28313

[K]
Kazhdan, D., Cuspidal geometry of $p$-adic groups, J. Analyse Math. 47 (1986), 1-36. MR 88g:22017

[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, preprint.

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Additional Information:

Allen Moy
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109 - Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, SAR
Email: moy@math.lsa.umich.edu, amoy@math.ust.hk

Marko Tadic
Affiliation: Department of Mathematics, University of Zagreb, Bijenicka 30, 10000 Zagreb, Croatia
Email: tadic@math.hr

DOI: 10.1090/S1088-4165-02-00181-4
PII: S 1088-4165(02)00181-4
Received by editor(s): February 7, 2002
Received by editor(s) in revised form: August 27, 2002
Posted: November 19, 2002
Additional Notes: The first and second authors acknowledge partial support from the National Science Foundation grants DMS-9801264 and DMS-0100413
The second author acknowledges partial support from the Croatian Ministry of Science and Technology grant #37001
Copyright of article: Copyright 2002, American Mathematical Society

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