Salamanca-Riba–Vogan unitarity conjecture for Hermitian modules

Let $G$ be a real reductive Lie group with complexified Lie algebra $fmathfrak{g}$ and maximal compact subgroup $K$. For a real parameter $fmathfrak{u}^*$, let $\Pi_h^{\lambda_u}(G)$ denote the irreducible Hermitian $(\mathfrak{g},K)$-modules having a lowest $K$-type $\delta$ with $\lambda_u(\delta)=\lambda_u$, and let $G(\lambda_u)$ be the corresponding subgroup. The theorem preceding this conjecture gives a bijection

$$\Pi_h^{\lambda_u}(G(\lambda_u))\longrightarrow \Pi_h^{\lambda_u}(G).$$

Salamanca-Riba–Vogan's unitarity conjecture. The above bijection preserves unitarity.

This conjecture would reduce the study of the unitary dual to representations whose lowest $K$-types are unitarily small. The source states that there had been nearly no progress toward proving it, but the paper explains that the subsequent convex-hull conjecture implies it.