[proofplan]
We prove invariance under the simple reflections and then use that the simple reflections generate the Weyl group. For a fixed simple root $\alpha$, the root vectors $e_\alpha,f_\alpha$ and the coroot $h_\alpha$ form an $\mathfrak{sl}_2(\mathbb C)$-subalgebra of $\mathfrak g$. To preserve the full $\mathfrak h$-weight information, we split $V$ into subspaces indexed by cosets of $P/\mathbb Z\alpha$; each such subspace is stable under this $\mathfrak{sl}_2$-subalgebra. Inside each coset, the $h_\alpha$-eigenvalue determines the $\mathfrak h$-weight, so the usual $\mathfrak{sl}_2$ symmetry of finite-dimensional irreducibles gives $\dim V_\lambda=\dim V_{s_\alpha\lambda}$.
[/proofplan]
[step:Write the character as a finite formal sum over weights]
Because $V$ is finite-dimensional, only finitely many weight spaces $V_\lambda$ are nonzero. For each $\lambda\in P$, define the weight multiplicity
\begin{align*}
m_\lambda:=\dim_{\mathbb C}(V_\lambda)\in\mathbb Z_{\ge 0}.
\end{align*}
Then
\begin{align*}
\operatorname{ch}V=\sum_{\lambda\in P}m_\lambda e^\lambda
\end{align*}
is a finite element of the group algebra $\mathbb C[P]$.
For $w\in W$, the natural action on $\mathbb C[P]$ is the $\mathbb C$-linear extension of $w(e^\lambda)=e^{w\lambda}$. Thus $w(\operatorname{ch}V)=\operatorname{ch}V$ is equivalent to
\begin{align*}
m_{w\lambda}=m_\lambda
\end{align*}
for every $\lambda\in P$.
[/step]
[step:Reduce the invariance statement to a single simple reflection]
Choose a positive root system $\Phi^+\subset\Phi$, and let $\Delta\subset\Phi^+$ denote the corresponding set of simple roots. For $\alpha\in\Delta$, let
\begin{align*}
s_\alpha:P\to P
\end{align*}
be the corresponding simple reflection, given by
\begin{align*}
s_\alpha(\lambda)=\lambda-\langle\lambda,\alpha^\vee\rangle\alpha,
\end{align*}
where $\alpha^\vee$ is the coroot of $\alpha$ and $\langle\lambda,\alpha^\vee\rangle=\lambda(h_\alpha)$.
The standard generation theorem for Weyl groups says that $W$ is generated by the simple reflections $s_\alpha$ with $\alpha\in\Delta$. Hence it is enough to prove
\begin{align*}
s_\alpha(\operatorname{ch}V)=\operatorname{ch}V
\end{align*}
for each $\alpha\in\Delta$.
[/step]
[step:Restrict the module to the $\mathfrak{sl}_2$-subalgebra attached to a simple root]
Fix a simple root $\alpha\in\Delta$. For each $\beta\in\Phi$, let
\begin{align*}
\mathfrak g_\beta=\{x\in\mathfrak g:[h,x]=\beta(h)x\text{ for every }h\in\mathfrak h\}
\end{align*}
denote the corresponding root space. Choose root vectors
\begin{align*}
e_\alpha\in\mathfrak g_\alpha
\end{align*}
and
\begin{align*}
f_\alpha\in\mathfrak g_{-\alpha}
\end{align*}
normalized so that, together with the coroot $h_\alpha\in\mathfrak h$, they form an $\mathfrak{sl}_2(\mathbb C)$-triple:
\begin{align*}
[h_\alpha,e_\alpha]=2e_\alpha,
[h_\alpha,f_\alpha]=-2f_\alpha,
[e_\alpha,f_\alpha]=h_\alpha.
\end{align*}
Let
\begin{align*}
\mathfrak s_\alpha:=\operatorname{span}_{\mathbb C}\{e_\alpha,h_\alpha,f_\alpha\}\subset\mathfrak g.
\end{align*}
Then $\mathfrak s_\alpha\cong\mathfrak{sl}_2(\mathbb C)$ as a [Lie algebra](/page/Lie%20Algebra), by the standard root-string construction of the $\mathfrak{sl}_2$-subalgebra attached to a root.
Restrict the given $\mathfrak g$-module structure on $V$ to $\mathfrak s_\alpha$. Since $V$ is finite-dimensional, it is a finite-dimensional $\mathfrak s_\alpha$-module. By the complete reducibility theorem for finite-dimensional $\mathfrak{sl}_2(\mathbb C)$-modules, applied through the isomorphism $\mathfrak s_\alpha\cong\mathfrak{sl}_2(\mathbb C)$, $V$ is a [direct sum](/page/Direct%20Sum) of irreducible finite-dimensional $\mathfrak s_\alpha$-modules.
[guided]
We isolate one simple root because the Weyl group is generated by the corresponding simple reflections. Fix $\alpha\in\Delta$. The root-space construction gives elements
\begin{align*}
e_\alpha\in\mathfrak g_\alpha,
f_\alpha\in\mathfrak g_{-\alpha},
h_\alpha=\alpha^\vee\in\mathfrak h
\end{align*}
satisfying
\begin{align*}
[h_\alpha,e_\alpha]=2e_\alpha,
[h_\alpha,f_\alpha]=-2f_\alpha,
[e_\alpha,f_\alpha]=h_\alpha.
\end{align*}
These are exactly the defining commutation relations of the standard generators $e,h,f$ of $\mathfrak{sl}_2(\mathbb C)$. Therefore
\begin{align*}
\mathfrak s_\alpha:=\operatorname{span}_{\mathbb C}\{e_\alpha,h_\alpha,f_\alpha\}
\end{align*}
is a Lie subalgebra of $\mathfrak g$ isomorphic to $\mathfrak{sl}_2(\mathbb C)$.
Now regard $V$ only as a module over this smaller Lie algebra $\mathfrak s_\alpha$. This restriction does not change the underlying [vector space](/page/Vector%20Space); it only forgets the action of elements of $\mathfrak g$ outside $\mathfrak s_\alpha$. Because $V$ is finite-dimensional over $\mathbb C$, it is a finite-dimensional $\mathfrak s_\alpha$-module. The complete reducibility theorem for finite-dimensional $\mathfrak{sl}_2(\mathbb C)$-modules, applied through the isomorphism $\mathfrak s_\alpha\cong\mathfrak{sl}_2(\mathbb C)$, then gives a direct sum decomposition of $V$ into irreducible finite-dimensional $\mathfrak s_\alpha$-modules. This is the point of passing to $\mathfrak s_\alpha$: irreducible finite-dimensional $\mathfrak{sl}_2$-modules have completely explicit symmetric weight strings, as in [citetheorem:9363].
[/guided]
[/step]
[step:Split the module into cosets modulo the root direction]
Let $P/\mathbb Z\alpha$ denote the set of additive cosets of the subgroup $\mathbb Z\alpha\subset P$. For a coset $C\in P/\mathbb Z\alpha$, define
\begin{align*}
V_C:=\bigoplus_{\mu\in C}V_\mu.
\end{align*}
The sum is finite because $V$ is finite-dimensional. The element $h_\alpha$ preserves each $V_\mu$, while [citetheorem:9362] gives
\begin{align*}
e_\alpha V_\mu\subset V_{\mu+\alpha},\qquad f_\alpha V_\mu\subset V_{\mu-\alpha}
\end{align*}
for every $\mu\in P$. Since $\mu+\alpha$ and $\mu-\alpha$ lie in the same coset $C$ as $\mu$, each $V_C$ is an $\mathfrak s_\alpha$-submodule of $V$.
[guided]
The equality of total $h_\alpha$-eigenspace dimensions is not enough, because several different $\mathfrak h$-weights can have the same value on $h_\alpha$. We therefore retain the missing information by separating weights according to their cosets modulo $\mathbb Z\alpha$. For a coset $C\in P/\mathbb Z\alpha$, define
\begin{align*}
V_C:=\bigoplus_{\mu\in C}V_\mu.
\end{align*}
This is a finite direct sum because only finitely many $\mathfrak h$-weight spaces of $V$ are nonzero.
We now check that $V_C$ is stable under the three generators of $\mathfrak s_\alpha$. The element $h_\alpha\in\mathfrak h$ acts on $V_\mu$ by the scalar $\mu(h_\alpha)$, so it preserves each summand $V_\mu$. For $e_\alpha\in\mathfrak g_\alpha$, [citetheorem:9362] gives $e_\alpha V_\mu\subset V_{\mu+\alpha}$; for $f_\alpha\in\mathfrak g_{-\alpha}$, the same theorem applied to the root $-\alpha$ gives $f_\alpha V_\mu\subset V_{\mu-\alpha}$. Both $\mu+\alpha$ and $\mu-\alpha$ remain in the coset $C$. Hence $e_\alpha$, $h_\alpha$, and $f_\alpha$ preserve $V_C$, so $V_C$ is an $\mathfrak s_\alpha$-submodule.
[/guided]
[/step]
[step:Apply $\mathfrak{sl}_2$ symmetry inside each coset]
Fix a coset $C\in P/\mathbb Z\alpha$. Since $V_C$ is a finite-dimensional $\mathfrak s_\alpha$-module, complete reducibility for finite-dimensional $\mathfrak{sl}_2(\mathbb C)$-modules decomposes $V_C$ as a direct sum of irreducible finite-dimensional $\mathfrak s_\alpha$-modules. Let $L(m)$ denote an irreducible summand in this decomposition, where $m\in\mathbb Z_{\ge 0}$ is its highest $h_\alpha$-weight. Under the chosen isomorphism $\mathfrak s_\alpha\cong\mathfrak{sl}_2(\mathbb C)$ sending $h_\alpha$ to the standard element $h$, [citetheorem:9363] gives a basis
\begin{align*}
\{v_i:0\le i\le m\}
\end{align*}
such that
\begin{align*}
h_\alpha v_i=(m-2i)v_i.
\end{align*}
Therefore the $h_\alpha$-eigenvalues in $L(m)$ are $m,m-2,\dots,-m$, each with multiplicity $1$. Pairing the basis vector $v_i$ with $v_{m-i}$ shows that the multiplicity of the $h_\alpha$-eigenvalue $r$ in $L(m)$ equals the multiplicity of the $h_\alpha$-eigenvalue $-r$ in $L(m)$. Summing over all irreducible summands in the complete reducibility decomposition of $V_C$, the multiplicity of the $h_\alpha$-eigenvalue $r$ in $V_C$ equals the multiplicity of the $h_\alpha$-eigenvalue $-r$ in $V_C$ for every $r\in\mathbb Z$.
[/step]
[step:Identify the cosetwise $\mathfrak{sl}_2$ symmetry with the Weyl reflection]
Fix $\lambda\in P$, and let $C:=\lambda+\mathbb Z\alpha$ be its coset. For any $\mu\in C$, write $\mu=\lambda+k\alpha$ with $k\in\mathbb Z$. Then
\begin{align*}
\langle\mu,\alpha^\vee\rangle=\langle\lambda,\alpha^\vee\rangle+2k.
\end{align*}
Hence the map $\mu\mapsto\langle\mu,\alpha^\vee\rangle$ is injective on $C$. Therefore, inside $V_C$, the $h_\alpha$-eigenspace with eigenvalue $\langle\lambda,\alpha^\vee\rangle$ is exactly $V_\lambda$.
Set
\begin{align*}
r:=\langle\lambda,\alpha^\vee\rangle.
\end{align*}
The unique element of $C$ with $h_\alpha$-eigenvalue $-r$ is
\begin{align*}
\lambda-r\alpha,
\end{align*}
because
\begin{align*}
\langle\lambda-r\alpha,\alpha^\vee\rangle=r-2r=-r.
\end{align*}
By the formula for the simple reflection,
\begin{align*}
\lambda-r\alpha
=
\lambda-\langle\lambda,\alpha^\vee\rangle\alpha
=
s_\alpha(\lambda).
\end{align*}
The cosetwise $\mathfrak{sl}_2$ symmetry from the previous step gives
\begin{align*}
\dim_{\mathbb C}(V_\lambda)=\dim_{\mathbb C}(V_{s_\alpha\lambda})
\end{align*}
for every $\lambda\in P$.
[/step]
[step:Conclude invariance under every Weyl group element]
Using the equality of multiplicities from the previous step,
\begin{align*}
s_\alpha(\operatorname{ch}V)=s_\alpha\left(\sum_{\lambda\in P}m_\lambda e^\lambda\right)=\sum_{\lambda\in P}m_\lambda e^{s_\alpha\lambda}.
\end{align*}
Since $s_\alpha:P\to P$ is a bijection and $m_{s_\alpha\lambda}=m_\lambda$, reindexing the finite sum gives
\begin{align*}
\sum_{\lambda\in P}m_\lambda e^{s_\alpha\lambda}=\sum_{\mu\in P}m_\mu e^\mu=\operatorname{ch}V.
\end{align*}
Thus $\operatorname{ch}V$ is fixed by every simple reflection $s_\alpha$. Since the simple reflections generate $W$, every $w\in W$ is a product of simple reflections, and repeated application of the preceding equality gives
\begin{align*}
w(\operatorname{ch}V)=\operatorname{ch}V.
\end{align*}
This proves Weyl invariance of the finite-dimensional character.
[/step]
