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

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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.

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  • 13 March 2019

    Papers [1–6] have received funding from ERC under Grant Agreement No. 669655.


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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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Correspondence to David Kazhdan.

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Dedicated to A. Beilinson on the occasion of his 60th birthday.

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Kazhdan, D. A spectral decomposition of orbital integrals for PGL(2, F) (with an appendix by S. Debacker). Sel. Math. New Ser. 24, 473–497 (2018).

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