 Principal q-character of the G2 adjoint representation

Use ${\mathrm{ch}}_{q}V={\mathrm{tr}}_V{q}^{2{\rho }^{\vee }}$. The positive roots of a root system of the G2 root system have simple-root heights $1,1,2,3,4,5$. Each root space has dimension one, and the zero weight space has dimension two. ConsequentlyBy the classification of finite-dimensional sl2 representations, restriction to the Principal sl2 subalgebra is $L_{10}\oplus L_2$, of dimensions eleven and three. If the grading uses ${\rho }^{\vee }$ instead, divide every exponent by two.