@SamHopkins I cannot find one either. Maybe you are right, they both span the whole space.

@SamHopkins Actually one of $\text{Span}_{\mathbb{R}}(\Phi_{[\lambda]})$ and $\text{Span}_{\mathbb{R}}(\Phi_{[\mu]})$ must be the whole $\text{Span}_{\mathbb{R}}(\Phi)$.

I'm sorry I don't get how to write Ch$(L(\lambda))$ as a rational function in reduced form. Is there a generalization of Weyl character formula to $\lambda\notin \Lambda^+$ or it is a consequence of the Kazhdan-Lusztig conjecture?

Moreover, by "For $L(\lambda)$ and $L(\mu)$ to be finite dimensional, we must have that x and y are not the identity." I think you mean "For $L(\lambda)$ and $L(\mu)$ to be infinite dimensional, we must have that x and y are not the identity.". Isn't it?

By "If $L(\lambda)\otimes L(\mu)$ is infinite dimensional, then $T_x\cup T_y$ is all positive roots by looking at the character." I think you mean "If $L(\lambda)\otimes L(\mu)$ is in $\mathcal{O}$, then $T_x\cup T_y$ is all positive roots by looking at the character." Isn't it?

Sorry, I mean $v_{-2\alpha-\beta}\otimes w_{-2\alpha-3\beta}$, $v_{-3\alpha-2\beta}\otimes w_{-\alpha-2\beta}$, and $v_{-3\alpha-\beta}\otimes w_{-\alpha-3\beta}$. I think $v_{-3\alpha-\beta}$ is non-zero in $L(st\cdot 0)$ and $w_{-\alpha-3\beta}$ is non-zero in $L(ts\cdot 0)$.

It seems that $\text{M}\neq \text{ch}\Delta(-3\rho)$. For example, in the Verma module $\Delta(-3\rho)$ the multiplicity of weight $-4\rho$ is $2$. However in $M=L(st\cdot 0)\otimes L(ts\cdot 0)$ the multiplicity of weight $-4\rho$ is $3$: it contains $v_{-2\alpha-\beta}\otimes w_{-2\alpha-3\beta}$, $v_{-3\alpha-2\beta}\otimes w_{-\alpha-2\beta}$, $v_{-2\alpha-2\beta}\otimes w_{-2\alpha-2\beta}$, which are linearly independent.