q-character of a highest-weight representation (${\mathrm{ch}}_q{L}_{\lambda }$)

For the principal specialization, let $h_{\mathrm{pr}}=2{\rho }^{\vee }$, so every simple root takes value two on $h_{\mathrm{pr}}$. The q-character isIt is obtained from the Weyl character formula by substituting $e^{\mu }=q^{\mu (h_{\mathrm{pr}})}$.