Characters of finite-dimensional modules are Weyl-invariant

By [F1] the character is the finite sum $\mathrm{ch}V={\sum }_{\mu }(\dim V_{\mu })e^{\mu }$, so [F2] gives $w\cdot \mathrm{ch}V={\sum }_{\mu }(\dim V_{\mu })e^{w\mu }$ for every $w\in W$.

Reindexing the finite sum of step 1.1 by $\nu =w\mu$, so that $\mu =w^{-1}\nu$, gives $w\cdot \mathrm{ch}V={\sum }_{\nu }(\dim V_{w^{-1}\nu })e^{\nu }$; since $w^{-1}$ is a product of simple reflections by [F3] and each simple reflection preserves the multiplicities by [F3], applying the invariance one reflection at a time yields $\dim V_{w^{-1}\nu }=\dim V_{\nu }$ for every $\nu \in {\mathfrak{h}}^{\ast }$.

Substituting into step 2.1 gives $w\cdot \mathrm{ch}V={\sum }_{\nu }(\dim V_{\nu })e^{\nu }=\mathrm{ch}V$, which is the first assertion; comparing the coefficient of $e^{\nu }$ in the two displayed expressions for $w\cdot \mathrm{ch}V$ in steps 1.1 and 2.1 gives $\dim V_{\nu }=\dim V_{w^{-1}\nu }$, that is, $\dim V_{w\mu }=\dim V_{\mu }$ after replacing $w$ by $w^{-1}$, and then $\mathrm{ch}V={\sum }_{\mu }(\dim V_{w\mu })e^{\mu }$; applying the result to a finite-dimensional simple module $V=L(\lambda )$ gives the final assertion.