Weight multiplicity

The multiplicity of a weight $\mu$ is the dimension of its weight space.

Thus it is a dimension attached to a Weight and its weight space.

Multiply the Weyl character formula by the generating series of the Kostant partition function. In each numerator summand, the coefficient of $e^{\mu }$ occurs when $\nu =w(\lambda +\rho )-(\mu +\rho )$. Summing those coefficients with the Weyl group signs proves the displayed weight multiplicity formula. Although $\rho$ need not itself be an integral group weight, $w\rho -\rho$ is in the root lattice, so each partition-function argument is an integral weight.

For linear maps $E:U\to W$ and $F:W\to U$ between finite-dimensional vector spaces, ${\mathrm{tr}}_U(FE)={\mathrm{tr}}_W(EF)$ even when their dimensions differ; expanding rectangular matrices proves it. For root vectors normalized by $[E_{\alpha },F_{\alpha }]=t_{\alpha }$, with $B(t_{\alpha },h)=\alpha (h)$, this gives $T_{\alpha }(\mu )=T_{\alpha }(\mu +\alpha )+(\mu +\alpha ,\alpha )\dim V_{\mu +\alpha }$. Iteration along a finite weight string is the trace step in Freudenthal multiplicity formula.

For a finite-dimensional Irreducible Lie algebra representation of a complex semisimple Lie algebra with highest weight $\lambda$, this recursion computes its weight multiplicities from $m_{\lambda }(\lambda )=1$. Here $\rho$ is the half-sum of positive roots and the inner product is induced by the Killing form. Write the Casimir operator on the weight-$\mu$ space as $(\mu ,\mu +2\rho )I+2{\sum }_{\alpha >0}{F}_{\alpha }{E}_{\alpha }$. The cyclic trace identity between adjacent weight spaces gives $\mathrm{tr}(F_{\alpha }{E}_{\alpha }{|}_{V_{\mu }})={\sum }_{j\ge 1}(\mu +j\alpha ,\alpha )m_{\lambda }(\mu +j\alpha )$. Taking the trace and using the Casimir eigenvalue proves the formula. Only finitely many terms are nonzero.