A local trace formula for $p$-adic infinitesimal symmetric spaces: the case of Guo-Jacquet

TL;DR

This paper develops an invariant local trace formula for the tangent space of symmetric spaces over non-archimedean fields, advancing the understanding of Guo-Jacquet trace formulae and related orbital integrals.

Contribution

It introduces a new invariant local trace formula for symmetric spaces, extending previous work and providing tools for comparing semi-simple terms in trace formulas.

Findings

Established an invariant local trace formula for symmetric spaces.

Derived a noninvariant local trace formula and proved Howe's finiteness.

Showed the representability of Fourier transforms of weighted orbital integrals.

Abstract

We establish an invariant local trace formula for the tangent space of some symmetric spaces over a non-archimedean local field of characteristic zero. These symmetric spaces are studied in Guo-Jacquet trace formulae and our methods are inspired by works of Waldspurger and Arthur. Some other results are given during the proof including a noninvariant local trace formula, Howe's finiteness for weighted orbital integrals and the representability of the Fourier transform of weighted orbital integrals. These local results are prepared for the comparison of regular semi-simple terms, which are weighted orbital integrals, of an infinitesimal variant of Guo-Jacquet trace formulae.