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 $Ω=h^G\subset G$ we denote by $I_Ω$ the $G$-invariant functional on $\mathcal S (G)$ given by $$I_Ω(f)=\int_G f(g^{-1}hg)dg$$ This paper provides the spectral decomposition of functionals $I_Ω$ in the case $G=PGL(2,F)$ and in the last section first steps of such an analysis for the general case.