Determine the unresolved parameter range in Table B

Determine whether the intersection $\\mathcal V^{(k)}\\cap\\mathcal U(\\mathfrak h_0)W\\cap\\mathcal L_{W,H}^c$ vanishes for the scalar representations and parameter range $n-2\\geq m>1$ identified by the marker $\\natural$ in Table B.

Background

Table B records symmetric triples for which the authors could not construct the nilpotent subalgebra needed to prove the existence of first-order normal derivative operators. The marker ♮\natural singles out a parameter range not resolved by the cited rank-one results.

The unresolved issue concerns whether the relevant higher-degree intersection, and hence the corresponding normal derivative operators, vanishes in the remaining cases with scalar representation and n−2≥m>1n-2\geq m>1.

References

The sign $\natural$ points that for a scalar $\tau$, and $m=1$ has shown: $k\geq 2$ forces $\mathcal V{(k)}\cap \mathcal U(h_0)W \cap \mathcal L_{W,H}c={0}$. We have not been able to achieve a conclusion for $m : n-2 \geq m>1$.

— Symmetry breaking differential operators and Discrete Series  (2609.19082 - Ørsted et al., 16 Sep 2026) in Section 4, immediately following Table B in subsection \ref{sub:prop:nu1nonzero}