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, $$P_{w,w'}(q)=\sum_{k=0}^{\infty}q^{\frac{\ell(w')-\ell(w)-k}{2}}\dim\operatorname{Ext}_{\mathcal{O}}^k(M(w\lambda),L(w'\lambda))$$ 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.

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.