Kazhdan–Lusztig polynomials in terms of Ext groups

Question

Let $P_{x,w}$ be the Kazhdan–Lusztig polynomial, $\rho$ be the half sum of positive roots in $\Phi^+$, $M_x$ be the Verma module with highest weight $x\cdot(-2\rho)$ and $L_w$ be the simple highest weight module with highest weight $w\cdot(-2\rho)$.

It is well-known that Kazhdan–Lusztig Conjecture is equivalent to $P_{x,w}(q) = \sum_{i\ge 0}q^i \dim \mathrm{Ext}_\mathcal{O}^{\ell(x,w)−2i}(M_x, L_w)$ for all $x\le w$. And it is also well-known that Kazhdan–Lusztig Conjecture is true.

My question: Does $P_{x,w}(q) = \sum_{i\ge 0}q^i \dim \mathrm{Ext}_\mathcal{O}^{\ell(x,w)−2i}(M_x, L_w)$ even for all $x\not\le w$?

If the answer is yes, any reference about that would be appreciated. If the answer is no, I would like to know why.

Answer 1

The answer is yes, for fairly elementary reasons, though it's not easy to give a reference. The point is partly that the polynomials are undefined for two elements of the Weyl group not related by the Bruhat partial orderijg. More precisely, the "linkage principle"(or "Harish-Chandra principle") ensures that the Hom functor is zero on these pairs of highest weight modules. Thus the derived functors Ext$^n$ vanish too.

(By the way, I meant to comment on the original version of the question, pointing out that for $i=0$ and $q=0$ you'd get 1 as constant term of the polynomial; $i=0$ was missing then and has been supplied in the sum apparently without editing. On the other hand, the tags have heen edited. But 'coxeter-groups', and 'co.combinatorics' which includes algebraic methods, are already available and are appropriate here.)