Recently I'm reading Humphrey's book "Introduction to Lie algebra and representation theory", section 21 on the finite dimensional module of a semisimple lie algebra, and I have a question here which I thought Humphrey's own proof might not be very clear.
Suppose $\mathfrak{g}$ is a finite dimensional complex semisimple lie algera, $V$ is a $\mathfrak{g}-$ module of highest weight $\lambda$ (of course $V$ is usually of infinite dimension). The assertion is that $V$ is finite dimension if and only if $\lambda$ is dominant integral, i.e. $(\lambda,\alpha_i)$ is a nonnegative integer for all simple root $\alpha_i$.
One key step in the proof of the above aseertion is that, suppose $V_\mu$ is a weight subspace of $V$ ( $V_\mu$ finite dimension), then $V_{s_i\cdot\mu}$ is also a weight subspace and $$\dim V_\mu=\dim V_{s_i\cdot\mu}.$$
Here $s_i$ is the reflection correspondeing to the simple root $\alpha_i$. This can be rephrased as the weight subspaces of $V$ are invariant under the action of the Weyl group of $\mathfrak{g}$.
The following argument is my own proof, I think it's very straightforward, but I haven't seen it on textbooks.
Let $L_i=\{x_i,y_i,h_i\}$ be a copy of $\mathfrak{sl}_2$ in $\mathfrak{g}$, $\phi(L_i)=\{\phi(x_i),\phi(y_i),\phi(h_i)\}$ are their corresponding linear transformations on $V$, then one can proof $\phi(x_i),\phi(y_i)$ are both locally nilpotent on $V$, so it make sense to define an invertible linear operator $\tau$ on $V$ by $$\tau = \exp(\phi(x_i))\exp(-\phi(y_i))\exp(\phi(x_i))$$ The operator $\tau$ has a nice property that it reverse any string(which form a basis of an irreducinle representation of $\phi(L_i)$): $$(v_{n},v_{n-2},\cdots,v_{2-n},v_{-n})\xrightarrow{\tau}(v_{-n},v_{2-n},\cdots,v_{n-2},v_n).$$
Now we can deduce that $V_{s_i\cdot\mu}$ has the same dimension with $V_\mu$. Consider the string of weight subspaces $$N=\bigoplus_{ k\in\mathbb{Z}}V_{\mu+k\alpha_i}$$ There are only finitely many summands, and $N$ is a $\phi(L_i)-$ module, so by Weyl's theorem $N$ is decomposed into a direct sum of some irreducible $\mathfrak{sl}_2$ modules. Since $\tau$ reverses every irreducible component of $N$, it must also reverse the string of $N$:because $$\mu(h_i)=(\mu,\alpha_i)$$ occurs as a weight of $\phi(h_i)$, then $$-(\mu,\alpha_i)=\left(\mu-\frac{2(\mu,\alpha_i)}{(\alpha_i,\alpha_i)}\alpha_i,\alpha_i\right)=(s_i\cdot\mu,\alpha_i)$$ must also be a weight of $\phi(h_i)$, this proofs that $V_{s_i\cdot\mu}$ is also a weight subspace and since $\tau$ is invertible so $\dim V_{s_i\cdot\mu}=\dim V_\mu$.
Is my proof above correct?
there are finitely many summands in $$N=\bigoplus_{ k\in\mathbb{Z}}V_{\mu+k\alpha_i}$$ is not obvious when $V$ is of infinite dimension. but one can proof that any $v\in V$ is contained in a finite $\mathfrak{sl}_2-$ submodule of $V$ and since $V_\mu$ is finite dimension so $V_\mu$ is contain is a finite dimension submodule of $V$. We are really writing the direct sum in the submodule.