Relative fundamental lemma for U(n) × U(n)

Let $F$ be a nonarchimedean local field, let $E/F$ be quadratic, and let $\mathrm{V}_n^+$ and $\mathrm{V}_n^-$ be hermitian spaces of rank $n$ with determinant valuation respectively even and odd. Let regular semisimple orbits in $\mathrm{S}_n(F)\times\mathrm{M}_n(F)$ be matched with the corresponding orbits in $\mathrm{U}(\mathrm{V}_n^\pm)(F)\times\mathrm{V}_n^\pm(E)$, and let $\omega(\zeta,y)\operatorname{Orb}(0;f,\phi;\zeta,y)$ be the normalized orbital integral.

Relative fundamental lemma. For every regular semisimple orbit $(\zeta,y)$, the normalized orbital integral with the characteristic functions of $\mathrm{S}_n(O_F)$ and $\mathrm{M}_n(O_F)$ is zero on the minus set. On the plus set it equals the unitary orbital integral

$$\int_{\mathrm{U}(\mathrm{V}_n^+)}\mathbbm{1}_{K_n}(g^{-1}\xi g)\mathbbm{1}_{\Lambda_n}(g^{-1}x)\,\mathrm{d}g,$$

where $(\xi,x)$ is the matching orbit, $\Lambda_n$ is self-dual, and $K_n$ stabilizes $\Lambda_n$.

This is the local fundamental lemma underlying the relative trace formula comparison for Fourier–Jacobi periods. It is attributed in the source to Liu and is presented as a recalled conjecture; its status is not resolved in the supplied text.