The basic spectral decomposition conjecture for analytic Langlands eigenfunctions

Let $V$ be the space of smooth compactly supported sections of $\Omega^{1/2}$ on the stable locus $\operatorname{Bun}^c_G(X,S)$, let $\mathcal H$ be its Hilbert completion, and let $\widehat V$ be the space of smooth sections of $\Omega^{1/2}$ on $\operatorname{Bun}^c_G$. For a character $\chi:\mathcal A\to\mathbb C$, let $V_\chi\subset\widehat V$ be the corresponding eigenspace and set $V^0_\chi=V_\chi\cap\mathcal H$. Define

$$\Sigma=\{\lambda\in\operatorname{Op}_{{}^LG}(X)\mid\exists\mu:V^0_{(\lambda,\mu)}\ne0\}.$$

Basic spectral decomposition conjecture. The Hilbert space has an orthogonal decomposition

$$\mathcal H=\bigoplus_\chi V^0_\chi,$$

and $\Sigma$ is discrete in the complex topology on $\operatorname{Op}_{{}^LG}(X)$.

This reformulates the conjecture that the square-integrable analytic Langlands eigenfunctions form an orthogonal basis. It is proved directly in the abelian case and in the special non-abelian case treated by the paper; the general equality $V^0_\chi=V_\chi$ is explicitly left open.