This question might be elementary and standard.

Standard Notions: Let $g$ be a semisimple Lie algebra. Let $\pi=({\alpha_{1},....\alpha_{n}})$ be simple roots

$P^{+}(\pi)=\Sigma\mathbb{N}\alpha _{i}$

Suppose $\lambda,\nu\in P^{+}(\pi)$ and $e_{w\lambda}, e_{w\nu}$ are extreme vector. ($\lambda,\nu$ are highest weight and $w\in W$, Weyl group).

Claim 1: $e_{w\lambda}\otimes e_{w\nu}$ is an extreme vector of weight vector $e_{w(\lambda+\nu)}$.(This has been done, easily follows from the definition of tensor products of representations)

Claim 2: $e_{w\lambda}\otimes e_{w\nu}=A^{*}e_{w(\lambda+\nu)}$, where $A^{*}$ is multiplicative set of invertible elements

There is a paper by A.Joseph talking about this observation. He claimed that it follows from **Weyl Character formula**. I think he might talk about the decomposition of tensor product of irreducible representations. However, I can not find a proof.

Any comments are welcome.

Edit: The paper I talked about is "Faithfully flat embeddings for minimal primitive quotients of quantized enveloping algebras" by Anthony Joseph. There is another paper by A.Rosenberg and V.Lunts on "localization for quantum group" page 138

Section 2: Ore localization of rings $R_{A}$. They claimed

$e_{w\lambda}\otimes e_{w\nu}=A^{*}e_{w(\lambda+\nu)}$, where $A^{*}$ is multiplicative set of invertible elements by using Weyl character formula,but how?