On the comparison of an infinitesimal variant of Guo-Jacquet trace formulae
Guo-Jacquet have proposed a comparison of two relative trace formulae in order to gerneralise Waldspurger’s well-known theorem relating toric periods to central values of automorphic L-functions for $GL(2)$. In our previous works, we have established an infinitesimal variant of Guo-Jacquet trace formulae and the weighted fundamental lemma in this case. In this talk, we shall explain several local results for the comparison of regular semi-simple terms. In particular, we shall talk about certain identities between Fourier transforms of local weighted orbital integrals, which are proved by Waldspurger’s global method on the endoscopic transfer. During the proof, we shall also need some results in local harmonic analysis such as local trace formulae for certain $p$-adic infinitesimal symmetric spaces.