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Selecta Mathematica

, Volume 24, Issue 1, pp 473–497 | Cite as

A spectral decomposition of orbital integrals for PGL(2, F) (with an appendix by S. Debacker)

  • David KazhdanEmail author
Article

Abstract

Let F be a local non-archimedian field, G a semisimple F-group, dg a Haar measure on G and \(\mathcal {S}(G)\) be the space of locally constant complex valued functions f on G with compact support. For any regular elliptic congugacy class \(\Omega =h^G\subset G\) we denote by \(I_\Omega \) the G-invariant functional on \(\mathcal {S}(G)\) given by
$$\begin{aligned} I_\Omega (f)=\int _G f(g^{-1}hg)dg \end{aligned}$$
This paper provides the spectral decomposition of functionals \(I_\Omega \) in the case \(G={\text {PGL}}(2,F)\) and in the last section first steps of such an analysis for the general case.

Mathematics Subject Classification

22E50 

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Notes

Acknowledgements

Many thanks for J.Bernstein, S. Debacker and Y. Flicker who corrected a number of imprecisions in the original draft and S. Debacker for writing an Appendix. I am partially supported by the ERC grant 669655-HAS.

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© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Einstein Institute of Mathematics, Edmond J. Safra CampusThe Hebrew University of JerusalemGivat Ram. JerusalemIsrael

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