Weyl character formula

For a dominant integral weight $\lambda$, the irreducible character is

The dimension obtained by evaluating the Weyl character formula at the identity is

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 $e^{\mu }=q^{\mu (h_{\mathrm{pr}})}$.