On certain elements in the Bernstein center of
Author:
Sandeep Varma
Journal:
Represent. Theory 17 (2013), 99119
MSC (2010):
Primary 22E50, 22E35
Posted:
February 6, 2013
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Abstract: Let be a nonarchimedean local field of residue characteristic , and let be an odd natural number less than . Using the work of Moy and Tadić, we find an element of the Bernstein center of that acts on any representation of by the scalar , the trace of any geometric Frobenius element of the absolute Weil group of , acting on the inertiafixed points of the representation of , where is the restriction to of the Langlands parameter of . This element is specified by giving the functions obtained by convolving it with the characteristic functions of a large class of compact open subgroups of , that includes all the groups of both the congruence and the Iwahori filtrations of having depth at least one.
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 AnneMarie Aubert and Roger Plymen, Explicit Plancherel formula for the adic group , C. R. Math. Acad. Sci. Paris 338 (2004), no. 11, 843848 (English, with English and French summaries). MR 2059659 (2005a:22010), http://dx.doi.org/10.1016/j.crma.2004.03.026
 [2]
 J. N. Bernstein, Le ``centre'' de Bernstein, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 132 (French). Edited by P. Deligne. MR 771671 (86e:22028)
 [3]
 T. Haines and B. C. Ngô, Nearby cycles for local models of some Shimura varieties, Compositio Math. 133 (2002), no. 2, 117150. MR 1923579 (2003h:11065), http://dx.doi.org/10.1023/A:1019666710051
 [4]
 Thomas J. Haines, Introduction to Shimura varieties with bad reduction of parahoric type, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 583642. MR 2192017 (2006m:11085)
 [5]
 Robert E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), no. 2, 373444. MR 1124982 (93a:11053), http://dx.doi.org/10.2307/2152772
 [6]
 Allen Moy and Marko Tadić, The Bernstein center in terms of invariant locally integrable functions, Represent. Theory 6 (2002), 313329 (electronic). MR 1979109 (2004f:22019), http://dx.doi.org/10.1090/S1088416502001814
 [7]
 Allen Moy and Marko Tadić, Erratum to: The Bernstein center in terms of invariant locally integrable functions, Represent. Theory 9 (2005), 455456 (electronic). MR 2167901
 [8]
 Michael Rapoport, A guide to the reduction modulo of Shimura varieties, Astérisque 298 (2005), 271318 (English, with English and French summaries). Automorphic forms. I. MR 2141705 (2006c:11071)
 [9]
 Peter Scholze, The LanglandsKottwitz approach for the modular curve, Int. Math. Res. Not. IMRN 15 (2011), 33683425. MR 2822177 (2012k:11090), http://dx.doi.org/10.1093/imrn/rnq225
 [10]
 G. van Dijk, Computation of certain induced characters of adic groups, Math. Ann. 199 (1972), 229240. MR 0338277 (49 #3043)
 [11]
 J.L. Waldspurger, La formule de Plancherel pour les groupes adiques (d'après HarishChandra), J. Inst. Math. Jussieu 2 (2003), no. 2, 235333 (French, with French summary). MR 1989693 (2004d:22009), http://dx.doi.org/10.1017/S1474748003000082
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Additional Information
Sandeep Varma
Affiliation:
Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, Illinois 60637
Address at time of publication:
School of Mathematics, Tata Institute of Fundamental Research, Colaba, Mumbai  400 005, India
Email:
sandeepv@math.tifr.res.in
DOI:
http://dx.doi.org/10.1090/S108841652013004272
PII:
S 10884165(2013)004272
Keywords:
Bernstein center,
stable Bernstein center
Received by editor(s):
December 23, 2011
Received by editor(s) in revised form:
August 19, 2012
Posted:
February 6, 2013
Article copyright:
© Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
