Step 1: Existence of highest weight

Let $V$ be a finite-dimensional irreducible representation. Fix a Cartan subalgebra $\mathfrak{h}$ and decompose: $V = \bigoplus_{\lambda \in \mathfrak{h}^*} V_\lambda$

Since $V$ is finite-dimensional, there are finitely many weights. Choose a maximal weight $\lambda$ (with respect to some fixed ordering).

For any positive root $\alpha$, the weight space $V_{\lambda + \alpha}$ must be zero (otherwise $\lambda$ wouldn't be maximal).

Therefore $E_\alpha \cdot V_\lambda = 0$ for all $\alpha \in \Phi^+$, so any non-zero $v \in V_\lambda$ is a highest weight vector.

Step 2: Highest weight is unique

If $V$ had two linearly independent highest weight vectors $v_{\lambda_1}, v_{\lambda_2}$ with weights $\lambda_1 \neq \lambda_2$, then the submodule generated by $v_{\lambda_1}$ would be a proper non-zero submodule, contradicting irreducibility.

Step 3: Highest weight is dominant and integral

For any simple root $\alpha_i$, consider the $\mathfrak{sl}_2$-triple $\{H_i, E_i, F_i\}$ corresponding to $\alpha_i$.

The highest weight vector $v_\lambda$ generates a finite-dimensional $\mathfrak{sl}_2$-representation, which must be irreducible of dimension $n_i + 1$ for some $n_i \geq 0$.

From $\mathfrak{sl}_2$ representation theory: $H_i \cdot v_\lambda = \langle \lambda, \alpha_i^\vee \rangle v_\lambda = n_i v_\lambda$.

Thus $\langle \lambda, \alpha_i^\vee \rangle \in \mathbb{Z}_{\geq 0}$ (integrality and dominance).

Step 4: Construction via Verma modules

For $\lambda \in P^+$, construct the Verma module $M(\lambda) = U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} \mathbb{C}_\lambda$.

$M(\lambda)$ has a unique maximal proper submodule $N(\lambda)$. Define: $L(\lambda) = M(\lambda) / N(\lambda)$

This quotient is irreducible with highest weight $\lambda$.

Step 5: Finite-dimensionality

To show $L(\lambda)$ is finite-dimensional when $\lambda \in P^+$, use:

• Weight multiplicities: Bounded by Weyl group orbit size
• Freudenthal's formula: Recursive formula shows finitely many weights
• Casimir action: The Casimir element acts as a scalar, constraining the module

The key is that integrality and dominance ensure the $\mathfrak{sl}_2$-subalgebras act with finite-dimensional irreducibles, forcing global finite-dimensionality.

Conclusion: The map $\lambda \mapsto L(\lambda)$ establishes a bijection $P^+ \leftrightarrow \{\text{f.d. irreducibles}\}$.