MathOverflow

Let $\mathfrak g$ be a Lie algebra over $\mathbb Q_p$ acting semisimply on afinite-dimensional $\mathbb Q_p$-vector space $V$, that is, $V$ is a semi-simple $\mathfrak g$-module.
Does every element of the center $Z(\mathfrak g)$ acts by a scalar on each irreducible $\mathfrak g$-submodule of $V_i$?...

$\def\C{\mathcal{C}}\def\D{\mathcal{D}}\def\set{\mathbf{Set}}\def\op{\mathrm{op}}\DeclareMathOperator{\hom}{Hom}\def\K{\mathbb{K}}\def\mank{\mathbf{Man}_\K}\def\M{\mathcal{M}}\def\cas{\mathbf{CAS}}\def\N{\mathcal{N}}$Let $\C$ be a category. A subcategory...

Background
Consider the $(n \times n)$ Hessenberg matrix
$$ A_{n} := \begin{pmatrix} 1/2 & 1/3 & 1/4 & 1/5 & \dots & \dots & 1/(n+1) & \dots \\ 1 & 1/2 & 1/3 & 1/4 & 1/5 & \dots & 1/n \\ 0 & 1 & 1/2 & 1/3 & 1/4 & 1/5 & \dots \\ 0 ...

Let $\pi: [1,..,n] \rightarrow [1,..,n]$ be a permutation. $\pi$ is called $\textbf{SIF}$ if there is no proper interval $[a,b]$ in $[1,..,n]$ that is fixed by $\pi$, see for example https://arxiv.org/abs/math/0310157 .
Now let $L$ be a finite distributive lattice with $n$ vertices, given as the set of order ideals of a finite poset $P$. Choose a linear extension for $L$ labeled by $[1,..,n]$ and let $\pi: [1,..,n] \rightarrow [1,..,n]$ denote the rowmotion bijection...