An alternative form of the Kazhdan-Lusztig conjecture

Question

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.

Answer 1

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.)