The local Gan–Gross–Prasad multiplicity conjecture for special orthogonal groups

Let $(W,V)$ be a pair of quadratic spaces over a local field $F$ of characteristic zero, with $W$ a subspace of $V$ such that $W^{\perp}$ is odd-dimensional and split. Let $G={\mathrm{SO}}(W)\times {\mathrm{SO}}(V)$, and let $H$ be the subgroup formed by the semidirect product of ${\mathrm{SO}}(W)$ with the unipotent subgroup of ${\mathrm{SO}}(V)$ attached to the full isotropic flag determined by $W^{\perp}$. For each $\alpha\in H^{1}(F,H)$, let $(W_{\alpha},V_{\alpha})$ be the corresponding pair of quadratic spaces and set $G_{\alpha}={\mathrm{SO}}(W_{\alpha})\times {\mathrm{SO}}(V_{\alpha})$. For a generic $L$-parameter $\varphi:{\mathcal W}_{F}\to {}^{L}G$, let $\Pi^{G_{\alpha}}(\varphi)$ be the associated $L$-packet, and write $m(\pi)=\dim {\mathrm{Hom}}_{H}(\pi,\xi)$. Local Gan–Gross–Prasad conjecture. For every generic $L$-parameter $\varphi:{\mathcal W}_{F}\to {}^{L}G$,

$$\sum_{\alpha\in H^{1}(F,H)}\sum_{\pi\in\Pi^{G_{\alpha}}(\varphi)}m(\pi)=1.$$

This refines the multiplicity-one result $m(\pi)\leq 1$ by predicting that exactly one representation across all relevant pure inner forms has a nonzero $H$-equivariant multiplicity for each generic parameter. The statement is part of the local Gan–Gross–Prasad conjecture; the supplied text gives no resolution status.