# Kazhdan–Lusztig polynomials in terms of Ext groups

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.

• Is it not the case that then both sides are zero? LHS is trivial by definition, while for the RHS this follows from BB localization (but probably there is a more direct way to see it). – dhy May 25 '19 at 17:36
• I agree with you. I would like to know how to show the RHS is zero for $x\not\le w$ by BB localization or any other methods. – James Cheung May 26 '19 at 8:34
• The BB localization argument goes as follows: $M_x$ is sent to $j_{x,!}\omega_x,$ where $j_x$ is the inclusion of the Bruhat cell corresponding to $x$. Taking Hom from this object is equivalent to taking the !-fiber on this cell. On the other hand, $L_w$ is supported on cells $\leq w$, so this !-fiber is zero if $x\not\leq w.$ – dhy May 26 '19 at 14:03

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.)