Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165

 
 

 

On certain elements in the Bernstein center of $ \mathbf{GL}_2$


Author: Sandeep Varma
Journal: Represent. Theory 17 (2013), 99-119
MSC (2010): Primary 22E50, 22E35
DOI: https://doi.org/10.1090/S1088-4165-2013-00427-2
Published electronically: February 6, 2013
MathSciNet review: 3017263
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ F$ be a nonarchimedean local field of residue characteristic $ p$, and let $ r$ be an odd natural number less than $ p$. Using the work of Moy and Tadić, we find an element $ z$ of the Bernstein center of $ G = \mathbf {GL}_2(F)$ that acts on any representation $ \pi $ of $ G$ by the scalar $ z(\pi ) = \operatorname {tr} \left (\operatorname {Frob} ; \left ( \operatorname {Sym}^r \circ \varphi _{\pi }\right )^{I_F} \right )$, the trace of any geometric Frobenius element $ \operatorname {Frob}$ of the absolute Weil group $ W_F$ of $ F$, acting on the inertia-fixed points of the representation $ \operatorname {Sym}^r \circ \varphi _{\pi }$ of $ W_F$, where $ \varphi _{\pi } : W_F \rightarrow \hat {G}$ is the restriction to $ W_F$ of the Langlands parameter of $ \pi $. This element $ z$ is specified by giving the functions obtained by convolving it with the characteristic functions of a large class of compact open subgroups of $ G$, that includes all the groups of both the congruence and the Iwahori filtrations of $ G$ having depth at least one.


References [Enhancements On Off] (What's this?)

  • [1] Anne-Marie Aubert and Roger Plymen, Explicit Plancherel formula for the $ p$-adic group $ {\rm GL}(n)$, C. R. Math. Acad. Sci. Paris 338 (2004), no. 11, 843-848 (English, with English and French summaries). MR 2059659 (2005a:22010), https://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. 1-32 (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, 117-150. MR 1923579 (2003h:11065), https://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. 583-642. MR 2192017 (2006m:11085)
  • [5] Robert E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), no. 2, 373-444. MR 1124982 (93a:11053), https://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), 313-329 (electronic). MR 1979109 (2004f:22019), https://doi.org/10.1090/S1088-4165-02-00181-4
  • [7] Allen Moy and Marko Tadić, Erratum to: The Bernstein center in terms of invariant locally integrable functions, Represent. Theory 9 (2005), 455-456 (electronic). MR 2167901
  • [8] Michael Rapoport, A guide to the reduction modulo $ p$ of Shimura varieties, Astérisque 298 (2005), 271-318 (English, with English and French summaries). Automorphic forms. I. MR 2141705 (2006c:11071)
  • [9] Peter Scholze, The Langlands-Kottwitz approach for the modular curve, Int. Math. Res. Not. IMRN 15 (2011), 3368-3425. MR 2822177 (2012k:11090), https://doi.org/10.1093/imrn/rnq225
  • [10] G. van Dijk, Computation of certain induced characters of $ {\mathfrak{p}}$-adic groups, Math. Ann. 199 (1972), 229-240. MR 0338277 (49 #3043)
  • [11] J.-L. Waldspurger, La formule de Plancherel pour les groupes $ p$-adiques (d'après Harish-Chandra), J. Inst. Math. Jussieu 2 (2003), no. 2, 235-333 (French, with French summary). MR 1989693 (2004d:22009), https://doi.org/10.1017/S1474748003000082

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 22E50, 22E35

Retrieve articles in all journals with MSC (2010): 22E50, 22E35


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: https://doi.org/10.1090/S1088-4165-2013-00427-2
Keywords: Bernstein center, stable Bernstein center
Received by editor(s): December 23, 2011
Received by editor(s) in revised form: August 19, 2012
Published electronically: February 6, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society