Waldspurger's endoscopic transfer and Fourier-transform compatibility conjecture

Let $F$ be a non-archimedean local field of characteristic zero, let $G$ be a reductive group over $F$, and let $H$ be its endoscopic group. Let

$$and$$

be the Lie algebras of $G$ and $H$, respectively.

Waldspurger's conjecture. (1) For every $f\in C_c^\infty(\mathfrak{g})$, there exists a transfer $f^H\in C_c^\infty(\mathfrak{h})$. (2) For compatible Fourier transforms $\mathcal{F}_G$ and $\mathcal{F}_H$ on $\mathfrak{g}$ and $\mathfrak{h}$, respectively, there is a constant $c$ such that whenever $f^H$ is a transfer of $f$, the function $\mathcal{F}_H(f^H)$ is a transfer of $c\mathcal{F}_G(f)$.

This conjecture asserts the existence of Langlands–Shelstad endoscopic transfer for Lie algebras and its compatibility with Fourier transforms. In the setting discussed in the paper, the corresponding endoscopic fundamental lemma follows from work of Kazhdan and Vashavsky; the general conjectural assertion is not resolved by the supplied text.