Freudenthal's weight multiplicity recursion

Given: The Axiom of Choice, a dominant integral weight $\lambda \in {\mathrm{\Lambda }}^{+}$, an element $\mu \in {\mathfrak{h}}^{\ast }$, the positive system ${\mathrm{\Phi }}^{+}$, the Weyl vector $\rho$, and the multiplicities $m_{\lambda }(\nu )=\dim L(\lambda {)}_{\nu }$ of the finite-dimensional simple module of highest weight $\lambda$.

The Axiom of Choice is assumed; it is inherited from the published Casimir and classification suppliers used in [F1] and [F2] (The Axiom of Choice).

The Casimir comparison on the weight space $L(\lambda {)}_{\mu }$ reads $((\lambda ,\lambda +2\rho )-(\mu ,\mu ))m_{\lambda }(\mu )={\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}{\mathrm{tr}}_{L(\lambda {)}_{\mu }}(e_{\alpha }{f}_{\alpha }+f_{\alpha }{e}_{\alpha })$ (The Casimir comparison on a weight space).

Each positive root contributes its string trace ${\mathrm{tr}}_{L(\lambda {)}_{\mu }}(e_{\alpha }{f}_{\alpha }+f_{\alpha }{e}_{\alpha })=m_{\lambda }(\mu )(\mu ,\alpha )+2{\sum }_{j\ge 1}{m}_{\lambda }(\mu +j\alpha )(\mu +j\alpha ,\alpha ),$ the sum being finite and the coefficients vanishing off the weights of $L(\lambda )$ (Positive root strings sum the Freudenthal correction).

$\rho =\frac{1}{2}{\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}\alpha$, so the bilinear form gives ${\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}(\mu ,\alpha )=(\mu ,2\rho )=2(\mu ,\rho )$, and $m_{\lambda }(\nu )=\dim L(\lambda {)}_{\nu }=0$ for every $\nu$ that is not a weight (The Weyl vector rho for a chosen positive system, The formal character of a finite-dimensional weight module).

Expanding the shifted squares with bilinearity and symmetry of $(,)$ gives $(\lambda +\rho ,\lambda +\rho )-(\mu +\rho ,\mu +\rho )=(\lambda ,\lambda +2\rho )-(\mu ,\mu )-2(\mu ,\rho )\mathrm{.}$