q-character of a highest-weight representation
= q-character of a highest-weight representation
{title2=$\operatorname{ch}_qL_\lambda$}
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 is
$$
\operatorname{ch}_qL_\lambda=
\sum_\mu(\dim L_\lambda[\mu])q^{\mu(h_{\mathrm{pr}})}.
$$
It is obtained from the <Weyl character formula> by substituting $e^\mu=q^{\mu(h_{\mathrm{pr}})}$.