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

## TL;DR

This paper provides a spectral decomposition of orbital integrals for PGL(2,F), advancing understanding of harmonic analysis on p-adic groups and initiating steps towards generalization.

## Contribution

It offers the first spectral decomposition of orbital integrals for PGL(2,F) and begins exploring similar decompositions for broader classes of groups.

## Key findings

- Spectral decomposition of orbital integrals for PGL(2,F) achieved.
- Methodology for spectral analysis of orbital integrals established.
- Initial steps toward generalization to other groups outlined.

## 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.