Hiraga–Ichino–Ikeda Plancherel density conjecture

Let $P=MN\subset G$ be a semi-standard $F$-parabolic subgroup, let $\mathcal{O}$ be an orbit of tempered essentially discrete series characters of $M$, and let $d\pi$ be the Haar measure on $\mathcal{O}$ with the normalization specified in the source. For $\pi\in\mathcal{O}$, let $\varphi$ be the corresponding parameter, let $\varphi_M$ be its parameter for $M$, let $\rho$ be the associated representation of $\mathcal{S}_{\varphi_M}$, and let $r_M$ be the adjoint representation of ${}^LM$ on $\operatorname{Lie}(G^\vee)/\operatorname{Lie}({}^LZ_M)$.

Hiraga–Ichino–Ikeda Plancherel density conjecture. Define

$$d\nu(\pi)=\frac{\operatorname{dim}(\rho)}{|\mathcal{S}_{\varphi_M}^\natural|}\left|\gamma(0,r_M\circ\varphi,\psi)\right|d\pi.$$

Then the Plancherel density at $\operatorname{Ind}_P^G(\pi)$ is $c_Md\nu(\pi)$ for some explicit constants $c_M\in\mathbb{R}_+$ independent of $F$ and $\mathcal{O}$.

This conjecture extends the formal-degree formula from essentially discrete series to general tempered representations through parabolic induction and is intended to describe the Plancherel measure in terms of Langlands parameters and adjoint gamma factors.