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Conjugacy class asymptotics, orbital integrals, and the Bernstein center: the case of $SL(2)$

Author(s): Allen Moy; Marko Tadic
Journal: Represent. Theory 9 (2005), 327-353.
MSC (2000): Primary 22E50, 22E35
Posted: April 14, 2005
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Abstract: The Bernstein center of a reductive p-adic group is the algebra of conjugation invariant distributions on the group which are essentially compact, i.e., invariant distributions whose convolution against a locally constant compactly supported function is again locally constant complactly supported. In the case of $SL(2)$, we show that certain combinations of orbital integrals belong to the Bernstein center and reveal a geometric reason for this phenomenon.

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Bernstein, J., Le ``centre" de Bernstein (Edited by Deligne, P.) 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 0771671 (86e:22028)

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Dijk, G. van, Computation of certain induced characters of $\mathfrak{p}$-adic groups, Math. Ann. 199 (1972), 229-240. MR 0338277 (49 #3043)

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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 0579175 (58 #28313)

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Moy, A., Tadic, M., The Bernstein center in terms of invariant locally integrable functions., Represent. Theory 6 (2002), 313-329. MR 1979109 (2004f:22019)

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Sally, P. J. Jr., Character formulas for ${ SL}\sb {2}$., Harmonic analysis on homogeneous spaces Proc. Sympos. Pure Math., Williams Coll., Williamstown Mass., 1972, Vol. XXVI, (1973), 395-400. MR 0338281 (49 #3047)

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Sally, Paul J. Jr., Shalika, Joseph A., The Fourier transform of orbital integrals on ${SL}\sb {2}$ over a $p$-adic field., Lie group representations, II (College Park, Md., 1982/1983), Lecture Notes in Math., 1041, Springer 1041 (1984), 303-340,. MR 0748512 (86a:22017)

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Additional Information:

Allen Moy
Affiliation: Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
Email: amoy@ust.hk

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

DOI: 10.1090/S1088-4165-05-00274-8
PII: S 1088-4165(05)00274-8
Received by editor(s): September 17, 2004
Received by editor(s) in revised form: January 31, 2005
Posted: April 14, 2005
Additional Notes: The first author was partially supported by the National Science Foundation grant DMS--0100413 while at the University of Michigan, and also partially supported by Research Grants Council grant HKUST6112/02P
The second author was partially supported by Croatian Ministry of Science and Technology grant #37108
Copyright of article: Copyright 2005, American Mathematical Society

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