Jacquet–Rallis fundamental lemma conjecture

Let $F/F_0$ be an unramified quadratic extension of $p$-adic fields for an odd prime $p$. Let $S_n$ be the Jacquet–Rallis symmetric space, let $G_i=\mathrm{U}(V_i^\sharp)$ and $H_i=\mathrm{U}(V_i)$, and let $K_0\subset H_0(F_0)$ and $\Lambda_0\subset V_0$ be the hyperspecial subgroup and self-dual lattice introduced above. Let $\mathcal{S}$ denote the relevant Schwartz-function spaces, and let “transfers” refer to the Jacquet–Rallis transfer relation. Jacquet–Rallis fundamental lemma conjecture. The following transfer identities should hold: the characteristic function $\mathbf{1}_{S_n(O_{F_0})}\in\mathcal{S}(S_n(F_0))$ transfers to $(\mathbf{1}_{K_0},0)\in\mathcal{S}(G_0(F_0))\times\mathcal{S}(G_1(F_0))$, and $\mathbf{1}_{(S_{n-1}\times V'_{n-1})(O_{F_0})}\in\mathcal{S}((S_{n-1}\times V'_{n-1})(F_0))$ transfers to $(\mathbf{1}_{K_0\times\Lambda_0},0)\in\mathcal{S}((H_0\times V_0)(F_0))\times\mathcal{S}((H_1\times V_1)(F_0))$. This is the local fundamental lemma for the group and semi-Lie algebra versions of the Jacquet–Rallis comparison; its status is not specified in the supplied source context.