Fix a complex semisimple Lie algebra $\mathfrak{g}$. Denote by $W$ the corresponding Weyl group, with length function $\ell$ and Bruhat order $\leq$. Let $\lambda$ be an integral anti-dominant weight. In the paper TOWARDS THE KAZHDAN-LUSZTIG CONJECTURE by Gabber and Joseph, they claim that there is a formula, due to D Vogan,
$$P\_{w,w'}(q)=\sum\_{k=0}^{\infty}q^{\frac{\ell(w')-\ell(w)-k}{2}}\dim\operatorname{Ext}\_{\mathcal{O}}^k(M(w\cdot \lambda),L(w'\cdot \lambda))$$
that is equivalent to the Kazhdan-Lusztig conjecture. Here $w'\leq w$ are elements in the Weyl group,$P\_{w,w'}$ is the Kazhdan-Lusztig polynomial, $M(w\lambda)$ is the Verma module of highest weight $w\lambda$ and $L(w'\lambda)$ is the simple module of highest weight $w'\lambda$.


I want to know how to derive this formula from Kazhdan-Lusztig conjecture. Moreover, I want to ask if there is a clear reference (say, by Vogan) on this.



An alternative form of the Kazhdan-Lusztig conjecture
Let $G$ be the simply-connected Lie group with Lie algebra $\mathfrak g$, let $B\subset G$ be the Borel subgroup, and let $X=G/B$ be the flag variety. There are equivalences $$\mathrm{Mod}\_f(\mathfrak g,B,\chi\_\lambda)\xrightarrow{-\otimes\_{U(\mathfrak g)}D\_X}\mathrm{Mod}\_c(D\_X,B)\xrightarrow{DR\_X}\mathrm{Perv}(\mathbb C\_X,B),$$
where the first equivalence is the Beilinson-Bernstein correspondence, and the second equivalence is the Riemann-Hilbert correspondence.


Note that $X$ has a stratification by $B$-orbits, indexed by the Weyl group: for $w\in W$, let $X\_w:=BwB/B$, of dimension $\ell(w)$. Then under the equivalence $M(w\lambda)$ corresponds to $\mathbb C\_{X\_w}[\ell(w)]$ and $L(w\lambda)$ corresponds to $IC(\overline X\_w,\mathbb C\_{X\_w})$.


Moreover, Kazhdan-Lusztig "Schubert varieties and Poincaré duality" shows
$$P\_{y,w}(q)=\sum\_{i}(\dim H^{i-\ell(w)}(IC(\overline X\_w,\mathbb C\_{X\_w}))\_{yB})q^{i/2}.$$
Here,
$$
\begin{align\*}
H^{i-\ell(w)}(IC(\overline X\_w,\mathbb C\_{X\_w})\_{yB})&=\mathrm{RHom}\_{\mathrm{Perv}(\mathbb C\_X,B)}(\mathbb C\_{X\_y},IC(\overline X\_w,\mathbb C\_{X\_w})[i-\ell(w)])\\
&=\mathrm{RHom}\_{\mathcal O}(M(y\lambda),L(w\lambda)[i+\ell(y)-\ell(w)])\\
&=\mathrm{Ext}\_{\mathcal O}^{-i-\ell(y)+\ell(w)}(M(y\lambda),L(w\lambda)).
\end{align\*}$$
(most of this is in Hotta-Takeuchi-Tanisaki *D-modules, Perverse Sheaves, and Representation Theory*.)


