The Gross–Prasad parabolic-induction multiplicity reduction conjecture

Let $(W,V)$ be an admissible pair. Let $\pi_V\in \Pi_{\mathrm{CW}}(\operatorname{SO}(V))$ and $\pi_W\in \Pi_{\mathrm{CW}}(\operatorname{SO}(W))$, possibly reducible, and suppose they have parabolic-induction expressions

$$\pi_V=|\cdot|^{s_{V,1}}\rho_{V,1}\times\cdots\times|\cdot|^{s_{V,r_V}}\rho_{V,r_V}\rtimes\pi_{V_0},$$ $$\pi_W=|\cdot|^{s_{W,1}}\rho_{W,1}\times\cdots\times|\cdot|^{s_{W,r_W}}\times\rho_{W,r_W}\rtimes\pi_{W_0}.$$

Here $m(\pi_V\boxtimes\pi_W)$ denotes the Gross–Prasad multiplicity. Multiplicity reduction conjecture. One has

$$m(\pi_V\boxtimes\pi_W)=m(\pi_{V_0}\boxtimes\pi_{W_0}).$$

This is intended to reduce the multiplicity problem to the tempered inducing data. The paper proves the result in the real case, so the conjecture is resolved in the setting addressed there, while its formulation is broader than that proof context.