Arbitrary-level Murphy Jucys–Murphy action is content triangular

Initial-row eigenvalues. Consider only a nonempty component $k$ and put $a=a_k$; empty components contribute no nodes or JM eigenvalues. Then [F1] gives $m_{\lambda }(X_{a+1}-Q_k)=x_{\lambda }{u}_b\in N^{>\lambda }$. The internal finite Hecke submodule generated by $m_{\lambda }$ modulo higher global shapes is spanned by standard internal component permutations; each such permutation fixes the smallest internal label $a+1$ (that label occupies the first node of its component), so its Hecke element has a reduced word in the internal generators with indices at least $a+2$ and commutes with $X_{a+1}$. Hence $X_{a+1}$ acts as the scalar $Q_k$ on that whole submodule, with errors in the two-sided ideal $N^{>\lambda }$. The $X$-recurrence then identifies $X_{a+i}$ on this submodule with $Q_k$ times the ordinary multiplicative Jucys–Murphy element $M_i$ of [F2], by induction from the finite quadratic relation and the additive-JM recurrence (a polynomial identity also at $q_H=1$). Ordinary JM triangularity gives diagonal $Q_k{q}_H^{c-r}$ at the node $(r,c,k)$ occupied by local entry $i$: the internal label $i$ is not a row index, and no extra subtraction of $i-1$ occurs. For the initial component tableau, which is maximal in tableau dominance, there are no strictly more dominant same-shape terms. Thus for the node $(r,c,k)$ carrying global entry $t$, $m_{\lambda }{X}_t\equiv Q_k{q}_H^{c-r}{m}_{\lambda }(\mathrm{m}\mathrm{o}\mathrm{d}{N}^{>\lambda })$.

JM triangularity for every standard multitableau. Inducting through a reduced word of $w$, the relations [F1] give $T_w{X}_r=X_{w(r)}{T}_w+{\sum }_{v<w}{\sum }_b{a}_{(v,b)}{X}_b{T}_v,$ where the corrections are supported on proper Bruhat subwords (at the first correction a generator is deleted, moving its linear $X$-coefficient through preceding generators deletes further generators, and finite quadratic multiplication preserves Bruhat support). Applying this with $w=d(t)$ and multiplying by $m_{\lambda }$ on the left, step 1.1 evaluates every surviving $X$ coefficient modulo higher shapes; a correction $m_{\lambda }{T}_v$ reduces first to a row-standard representative $d(u)$ of its Young-subgroup coset, the removed row subgroup contributing its row eigenvalue, with $d(u)\le v<d(t)$; the Bruhat/tableau-prefix relation on the ordered rows of all components then makes $u$ strictly dominate $t$, and componentwise straightening of $u$ keeps the shape or produces same-shape tableaux at least $u$ in prefix dominance, hence strictly dominating $t$. The leading eigenvalue is $Q_k{q}_H^{j-i}$ at the node carrying $r$, because $w(r)$ is its initial-row label under $t=t^{\lambda }\cdot w=w^{-1}\cdot t^{\lambda }$. Therefore $m_{(t^{\lambda },t)}{X}_r\equiv Q_k{q}_H^{j-i}{m}_{(t^{\lambda },t)}+{\sum }_{v⊲t}{a}_v{m}_{(t^{\lambda },v)}(\mathrm{m}\mathrm{o}\mathrm{d}{N}^{>\lambda }),$ the exact arbitrary-level JM content statement, independent of an affine center theorem, semisimplicity or block classification.

Residue words and boundary cases. On the resulting Specht module a total ordering extending tableau dominance triangularizes all the $X_r$ simultaneously; the one-dimensional successive quotients have exactly the content characters of step 2.1, so every simultaneous generalized eigenword is among the standard-tableau residue words $c_a+j-i(\mathrm{m}\mathrm{o}\mathrm{d}e)$ (The Hecke Jucys-Murphy action is dominance triangular for the ordinary part); the residue multiplicities of every such word equal the node-residue multiplicities of $\lambda$, since permuting the labels of a standard tableau does not change the multiset of node residues. For rank zero, $H=F$, the sole shape and tableau are empty and all assertions have their one-dimensional interpretation. The triangularity is polynomial in the parameters, so it specializes to the LM deformation DVR and to the generic fibre.