A local relative trace formula for $\mathop{\rm PGL}(2)$

Patrick Delorme and Pascale Harinck

Vol. 291 (2017), No. 1, 121–147

DOI: 10.2140/pjm.2017.291.121

Abstract

Following a scheme inspired by recent results of B. Feigon, who obtained what she called a local relative trace formula for ${PGL}_2$ and a local Kuznetsov trace formula for $U\left(2\right)$, we describe the spectral side of a local relative trace formula for $G:=PGL\left(2,E\right)$ relative to the symmetric subgroup $H:=PGL\left(2,F\right)$ where $E∕F$ is an unramified quadratic extension of local nonarchimedean fields of characteristic $0$. The spectral side is given in terms of regularized normalized periods and normalized $C$-functions of Harish-Chandra. Using the geometric side of the local relative trace formula obtained in a more general setting by the authors and S. Souaifi, we deduce a local relative trace formula for $G$ relative to $H$. We apply our result to invert some orbital integrals.

Keywords

$p$-adic reductive groups, symmetric spaces, local relative trace formula, truncated kernel, regularized periods

Mathematical Subject Classification 2010

Primary: 11F72, 22E50

Milestones

Received: 19 September 2016

Revised: 24 March 2017

Accepted: 27 March 2017

Published: 20 August 2017