Unitary Adams conjecture for the Plancherel decomposition of Howe duality

Let ${\boldsymbol{\omega}}_v$ be the local unitary representation attached to a Howe dual pair with small group $G_{2,v}$, and set ${^LG}_{\boldsymbol{\omega}}={^LG}_2$. Let $[\phi]$ range over isomorphism classes of local tempered Langlands parameters

$$\phi:\mathcal W_{k_v}'\to {^LG}_{\boldsymbol{\omega}}.$$

Let $\mu_{2,v}$ be a measure on these parameters, and let the canonical morphism be

$${^LG}_{\boldsymbol{\omega}}\times\operatorname{SL}_2\to {^L(G_1\times G_2)}.$$

Unitary Adams conjecture. There is a direct integral decomposition

$${\boldsymbol{\omega}}_v=\int_{[\phi]}\mathcal H_\phi\,\mu_{2,v}(\phi),$$

where $\mathcal H_\phi$ is isomorphic to a possibly empty direct sum of irreducible representations in the Arthur packet associated to

$$\mathcal W_{k_v}'\times\operatorname{SL}_2\xrightarrow{\phi\times\operatorname{Id}}{^LG}_{\boldsymbol{\omega}}\times\operatorname{SL}_2\to {^L(G_1\times G_2)}.$$

This is the unitary analogue of the relative local Langlands conjecture for the $L^2$-space of a spherical variety. It refines the ordinary Plancherel decomposition by describing its spectral support through tempered Langlands parameters and their Arthur packets.