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Antichains in weight posets associated with gradings of simple Lie algebras

For a reductive Lie algebra $\mathfrak h$ and a simple finite-dimensional $\mathfrak h$-module $V$, the set of weights of $V$, $P(V)$, has a natural poset structure. We consider antichains in the weight poset $P(V)$ and a certain operator $\mathfrak X$ acting on antichains. Eventually, we impose stronger constraints on $(\mathfrak h,V)$ and stick to the case in which $\mathfrak h$ and $V$ are associated with a $Z$-grading of a simple Lie algebra $\mathfrak g$. Then $V$ is a weight multiplicity free $\mathfrak h$-module and $P(V)$ can be regarded as a subposet of $Δ^+$, where $Δ$ is the root system of $\mathfrak g$. Our goal is to demonstrate that antichains in the weight posets associated with $Z$-gradings of $\mathfrak g$ exhibit many good properties similar to those of $Δ^+$ that are observed earlier in arXiv: math.CO 0711.3353 (=Ref. [14] in the text).