Properties of maximums of the multiplicity function

We redefine the multiplicity function $M(\nu,p)$ for the tensor power $(L^{\omega})^{\otimes p}$ decomposition as a smooth function on the weight space $P_{\mathfrak{g}}$ of a Lie algebra $\mathfrak{g}$ and study the behavior of its maximums. As a result, the submodule with the maximum multiplicity can be easily found for any fixed power $p$.