The admissible-polarization conjecture for the character formula

Let $G$ be the connected group in the paper, let $\mathcal{O}$ be a semisimple coadjoint orbit, and let $\mathfrak{q}=\{\mathfrak{q}_{\lambda}\}_{\lambda\in\mathcal{O}}$ be an admissible polarization. Let $\mathcal{C}(\mathcal{O},\Gamma,\mathfrak{q},\sigma_c)$ denote the associated contour and let $\theta(\mathcal{O},\Gamma)$ denote the corresponding character distribution.

Admissible-polarization conjecture. The character formula

$$\mathcal{F}[\mathcal{C}(\mathcal{O},\Gamma,\mathfrak{q},\sigma_c)]=\theta(\mathcal{O},\Gamma)$$

holds for any admissible polarization $\mathfrak{q}=\{\mathfrak{q}_{\lambda}\}_{\lambda\in\mathcal{O}}$. In particular, the polarization need not be maximally real.

The result is proved in the paper for the chosen polarization under the hypotheses of the main theorem; extending it to every admissible polarization is presented as future work, presumably by generalizing the paper's techniques.