A spectral decomposition of orbital integrals for $PGL(2,F)$

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 $$I_{\Omega }(f)={\int }_{G}f(g^{-1}hg)dg$$ This paper provides the spectral decomposition of functionals $I_{\Omega }$ in the case $G=PGL(2,F)$ and in the last section first steps of such an analysis for the general case.

article by D. Kazhdan with an Appendix by S. DeBacker