# An alternative form of the Kazhdan-Lusztig conjecture

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.

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.)