On p. 128 of the book: **Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$**.

**I quote:** "......Delorme arrives at
vanishing criteria for Ext$^n(\mathcal{O})$ which are more general than those in Theorem
6.11 above. These involve a kind of length function $\ell(\mu, \lambda)$ expressing the distance between $M(\mu)$ and $M(\lambda)$ in certain “strongly” standard filtrations:

$\mathrm{Ext}^n_\mathcal{O}(M(\mu), M(\lambda)) = 0$ for all $n > \ell(\mu, \lambda)$,

$\mathrm{Ext}^n_\mathcal{O}(M(\mu), L(\lambda)) = 0$ for all $n > \ell(\mu, \lambda)$,......"

**My question:**

- Does anyone know what is the definition of length function $\ell(\mu, \lambda)$?

Let $\rho$ be the half sum of positive roots in $\Phi^+$, $M(u\cdot(-2\rho))$ be the Verma module with highest weight $u\cdot(-2\rho)$ and $L(u\cdot(-2\rho))$ be the simple highest weight module with highest weight $u\cdot(-2\rho)$.

- I want to show that for all $x\not\le w$, we have $\mathrm{Ext}^n_\mathcal{O}(M(x\cdot(-2\rho)), L(w\cdot(-2\rho))) = 0$ for all $n\in\mathbb{N}$. Does anyone know how to show this fact?