Determine the unresolved higher-order intersection for a scalar representation in the symplectic pair

Determine whether, for the pair $(\\mathfrak{sp}(n+1),\\mathfrak{sp}(n)+\\mathfrak{sp}(1))$ and a scalar holomorphic discrete series representation $(\\tau,W)$, the intersection $\\mathcal U(\\mathfrak u(n,1))W\\cap\\mathcal L_{W,\\mathfrak{sp}(n)+\\mathfrak{sp}(1)}\\cap\\mathcal V^{(k)}$ is zero for every $k\\geq2$.

Background

The paper relates normal derivative symmetry breaking operators of order kk to the intersection of the homogeneous degree-kk component V(k)\mathcal V^{(k)} with U(h0)W\mathcal U(\mathfrak h_0)W and the space LW,Hc\mathcal L_{W,H}^c generated by lowest LL-types occurring in the restriction to HH. Vanishing of this intersection would imply nonexistence of normal derivative operators of that order.

For the symplectic pair with g=sp(n+1)g=\mathfrak{sp}(n+1), h=sp(n)+sp(1)h=\mathfrak{sp}(n)+\mathfrak{sp}(1), and h0=u(n,1)h_0=\mathfrak u(n,1), the authors state the expected vanishing for scalar representations but explicitly do not establish it. This is a concrete unresolved higher-order normal-operator question.

References

$g=\mathfrak{sp}(n+1,)$, $h=\mathfrak{sp}(n, )+\mathfrak{sp}(1, )$ $h_0=\mathfrak u(n,1)$. $(\tau, W)$ is a scalar representation, then, for $k\geq 2$, we believe, it holds \begin{equation} \mathcal U( \mathfrak u(n,1))W \cap \mathcal L_{W,\mathfrak{sp}(n, )+\mathfrak{sp}(1, )} \cap \mathcal V{(k)} ={0} \end{equation}

— Symmetry breaking differential operators and Discrete Series  (2609.19082 - Ørsted et al., 16 Sep 2026) in Section 4, subsection \ref{sub:interequalceronosigma}, near the discussion of $g=\\mathfrak{sp}(n+1)$