Conjectural characterization of equivariant quantization by invariant differential operators

Let $\mathfrak{g}$ be an IFFT-algebra, let $\mathcal{V}$ and $\mathcal{W}$ be irreducible homogeneous bundles, and let $\mathcal{B}$ be a submodule of $\mathcal{S}^\delta_k(\mathcal{V},\mathcal{W})$. Consider the $\mathfrak{g}$-equivariant quantization

$$\mathcal{B}\otimes\bigl(\Gamma(\mathcal{V})\otimes\mathcal{F}^\lambda\bigr)\longrightarrow \Gamma(\mathcal{W})\otimes\mathcal{F}^\mu.$$

The equivariant quantization conjecture. The quantization has the following properties: (1) it does not exist or is not unique if and only if, for some $1\leq l\leq k$, there is a $\mathfrak{g}$-invariant differential operator $\mathcal{B}\rightarrow\mathcal{C}$, where $\mathcal{C}$ is a submodule of $\mathcal{S}^\delta_{k-l}(\mathcal{V},\mathcal{W})$; and (2) for such $\delta$, it exists if and only if there is a $\mathfrak{g}$-invariant differential operator on $\Gamma(\mathcal{V})\otimes\mathcal{F}^\lambda$ whose principal symbol lies in $\mathcal{A}$ and satisfies

$$\mathcal{D}_{\mathfrak{g}_{-1}\oplus\mathfrak{g}_0}(\mathcal{B},\mathcal{A})\neq 0, \qquad \mathcal{D}_{\mathfrak{g}_{-1}\oplus\mathfrak{g}_0}(\mathcal{C},\mathcal{A})=0.$$

The paper proves the corresponding first point as a theorem and presents the second point as the remaining conjectural characterization; it is intended to relate resonances to invariant operators on the source space.