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 Extn O which are more general that 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:

1. 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 $u\cdot(-2\rho)$.

2. 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?

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:

1. 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 $u\cdot(-2\rho)$.

2. 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?

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 Extn O which are more general that 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:

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

2. Does anyone have the papers of Delorme about this result? I cannot find the references mentions in Humphreys' book:

[75] P. Delorme, Extensions dans la categorie O de Bernstein–Gelfand–Gelfand. Applications, preprint, 1978.

[76] P. Delorme, Extensions in the Bernstein–Gelfand–Gelfand category O. Applications, Funktsional. Anal. i Prilozhen 14 (1980), no 3, 77–78; English transl., Funct. Anal. Appl. 14 (1980), 228–229.

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 $u\cdot(-2\rho)$.

3. 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?

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 Extn O which are more general that 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:

1. 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 $u\cdot(-2\rho)$.

2. 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?