In Lie algebra book by Humphreys, he has used Weyl Character Formula to find the dimension of $V(\lambda)$ in the examples followed by the proof of this formula. But how to find the dimensions of the weight spaces $M_\lambda(\mu)$ using the Weyl character formula?
There are other formulas to find the dimension of the weight spaces, but I want to do it using the Weyl Character Formula.
thanks in advance.