A general simple relative trace formula

Jayce R. Getz and Heekyoung Hahn

Vol. 277 (2015), No. 1, 99–118

DOI: 10.2140/pjm.2015.277.99

Abstract

In this paper we prove a relative trace formula for all pairs of connected algebraic groups $H\le G\times G$, with $G$ a reductive group and $H$ the direct product of a reductive group and a unipotent group, given that the test function satisfies simplifying hypotheses. As an application, we prove a relative analogue of the Weyl law, giving an asymptotic formula for the number of eigenfunctions of the Laplacian on a locally symmetric space associated to $G$ weighted by their $L^{\mathfrak{2}}$-restriction norm over a locally symmetric subspace associated to $H_{\mathfrak{0}}\le G$.

Keywords

relative trace formula, Weyl law

Mathematical Subject Classification 2010

Primary: 11F70

Secondary: 35P20

Milestones

Received: 30 October 2014

Revised: 17 February 2015

Accepted: 25 February 2015

Published: 6 August 2015