# Distance between Verma modules in certain "strongly" standard filtrations

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

1. 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?

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 (?)