Strong irreducibility from non-scalar local inertia in general dimension

Prove, for general dimension \(n\), that a strongly irreducible Galois representation \(\rho_f:G_{F,S}\to\mathrm{GL}_n(C_p)\) cannot have scalar inertia Lie algebras at every \(p\)-adic place, or equivalently establish that the support of \(E^{\mathrm{plec},(r)}/A^{(r)}\) is contained in the non-strongly-irreducible locus under the relevant hypotheses.

Background

For two-dimensional representations, the paper proves that points in the support of Eplec,(r)/A(r)E^{\mathrm{plec},(r)}/A^{(r)} are either not strongly irreducible or have scalar inertia Lie algebras at every pp-adic place. The authors further note that distinct Hodge–Tate–Sen weights rule out the scalar-inertia alternative.

For arbitrary dimension, the paper does not prove the analogous elimination of the scalar-inertia alternative. The conjectural expectation is that support outside the strongly irreducible locus should be the only obstruction to identifying the full plectic Cayley–Hamilton algebra with the algebra generated by the constructed symmetries.

References

For general $n$, a similar argument shows that conjecturally $\ker f$ belongs to the support of the $R_{\bar{D}{(r)}}$-module $E{plec,(r)}/A{(r)}$ only if $\rho_f$ is not strongly irreducible.

— Plectic Lie Algebra Action on Hilbert modular varieties  (2609.34465 - Jiang et al., 28 Sep 2026) in Remark following Proposition \ref{prop-suppA(r)}, Section 3.2

Conjecturally, $\rho_f$ cannot be strongly irreducible if all $\Lie \rho_f(I_{F_v})$ are scalars.

— Plectic Lie Algebra Action on Hilbert modular varieties  (2609.34465 - Jiang et al., 28 Sep 2026) in Remark following Proposition \ref{prop-suppA(r)}, Section 3.2

If we are able to construct the full plectic symmetry of on $\tilde{H}{d,}$ i.e. extend the action of $R_{\bar{D}[G_{Q}]$ to an action of $R_{\bar{D}[G_{F}{plec}]$, the same argument would show that a power of the ideal of reducible locus of $T$ annihilates $\tilde{H}{<d}(Kp)_{Q_p}$.

— Plectic Lie Algebra Action on Hilbert modular varieties  (2609.34465 - Jiang et al., 28 Sep 2026) in Remark immediately following the proof of Theorem \ref{thm-belmidcnsi}, Section 4.4