Let $\mathfrak{g}(A)$ be a symmetrizable Kac-Moody Lie algebra over $\mathbb{C}$ and ($\mathfrak{h}$, $\Pi, \Pi^\vee)$ be a realization of the GCM $A$. Assume that $$\mathfrak{g}(A)=\mathfrak{h} \oplus \mathfrak{n}^+ \oplus \mathfrak{n}^-$$ be the standard triangular decomposition. $\Pi = \{\alpha_i, i=1,\cdots,n \}$ is the set of simple roots. Let $\Delta, \Delta^+, \Delta_{re}(\Delta^+_{re}), \Delta_{im}(\Delta^+_{im}) $ be the corresponding root system, the set of positive roots, the set of (positive) real roots, the set of (positive) imaginary roots respectively. Let $W$ be the Weyl group.
For each weight $\lambda \in \mathfrak{h}^*$, the Verma module with respect to $\lambda$ is denoted by $M(\lambda)$, which is defined by the quotient of the universal enveloping algebra $\mathrm{U}(\mathfrak{g})$ and its left ideal generated by $\mathfrak{n}^+$ and $h-\lambda(h)$, for each $h \in \mathfrak{h}^*$.
Let $(\cdot, \cdot)$ be the standard non-degenerated bilinear form of $\mathfrak{h}^*$. Fix $\rho \in \mathfrak{h}^*$, such that $2(\rho, \alpha_i)/(\alpha_i,\alpha_i)=1$, for each $i=1,\cdots,n$. Consider the set $\mathcal{C}$ of non-critical weights of $\mathfrak{g}(A)$ defined as following: $$\mathcal{C}:=\{\lambda \in \mathfrak{h}^* | 2(\lambda + \rho, \beta) \neq (\beta, \beta), \text{for any imaginary root } \beta \in \Delta_{im}\}.$$ It seems to be true that $$(*) \quad \mathrm{dim}_\mathbb{C}\mathrm{Hom}_{\mathfrak{g}(A)}(M(w \cdot \lambda), M(\lambda)) \leq 1,$$ for any $\lambda \in \mathcal{C} $ and any $w \in W$ no matter what the orbit $W(\lambda)\cdot \lambda$ contains a dominant weight (antidominant weight) or not, where $W(\lambda)$ is the integral Weyl group of $\lambda$, and $w\cdot \lambda = w(\lambda + \rho)-\rho$.
Questions:
(1) Is the above claim true? If the answer is yes, then how do we show it? If not, can you list any counterexample?
(2) In the critical case, i.e. $\lambda$ lies in $\mathfrak{h}^*-\mathcal{C}$, is $(*)$ true?
Thanks for your time. Please let me know if anything is unclear!
