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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.

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    $\begingroup$ 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). $\endgroup$ – dhy May 25 '19 at 17:36
  • $\begingroup$ 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. $\endgroup$ – James Cheung May 26 '19 at 8:34
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    $\begingroup$ 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 $x\not\leq w.$ $\endgroup$ – dhy May 26 '19 at 14:03
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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.)

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