Distance between Verma modules in certain "strongly" standard filtrations

Question

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?

Answer 1

It's worthwhile to explain something of the background, since Patrick Delorme's preprint never got published in full. It's a 23 page typed double-spaced document with symbols inserted by hand, distributed in October 1978 after the appearance of the BGG paper and its English translation (1976) but just before the publication of the Kazhdan-Lusztig paper in 1979. A very short version of the preprint then appeared in 1980 in the Russian journal Funcktion. Anal i Prilozen., with author listed as "M. Delorm". There is a published author named Marianne Delorme, but email with the author P. Delorme suggests that the M. stands for "Monsieur". The article is freely available here in Russian but not so easily accessible in the English language translation journal.

Anyway, it's not a good idea at this site to formulate two unreleated questions, since it complicates upvoting.

Having said that, I can confirm that there is a definition of the symbol $n:=\ell(\lambda, \mu)$ on page 16 of the preprint. The symbol just means that the module $M$ has a filtration with subquotients which are Verma modules while these have natural embeddings and $n$ is the number of them.

Here the notion of "standard $p$-filtration" (which the Brazilian mathematician Alvany Rocha introduced in her 1978 Rutgers thesis under the direction of N. Wallach) just refers to a module in $\mathcal{O}$ which has a filtration with subquotients which are Verma modules. This was simplified in my book to "standard filtration" but strengthened by Delorme to require that the highest weights of the Verma modules increase.

In your formulation, the weights are dot-images of $-2\rho$ (for some reason), using Weyl group elements which are presumably unrelated by the Bruhat ordering (?)