Let $W^I=\{w\in W: w^{-1}\Phi_I^+\subseteq \Phi^+\}$, $W_I$ be the Weyl group generated by $I$ and $w_I$ be the longest element in $W_I$

Let $M(\lambda)$ be the Verma module with highest weight $\lambda$, $M_I(\lambda)$ be the parabolic Verma module with highest weight $\lambda$ , $L(\lambda)$ be the simple module with highest weight $\lambda$.

For $y,w\in W^I$, define the Kazhdan-Lusztig Vogan polynomial $Q_{y,w}(q):=\sum_{i\ge 0} q^{\frac{\ell(w)-\ell(y)-i}{2}}\dim\text{Ext}_{\mathcal{O}^\mathfrak{p}}^i(M_I(w_Iy\cdot (-2\rho)),L(w_Iw\cdot (-2\rho)))$

and the Kazhdan-Lusztig polynomial $P_{y,w}(q):=\sum_{i\ge 0} q^{\frac{\ell(w)-\ell(y)-i}{2}}\dim\text{Ext}_{\mathcal{O}}^i(M(y\cdot (-2\rho)),L(w\cdot (-2\rho)))$.

This is well-known that for all $y,w\in W^I$, for all $r\in W_I$, $Q_{y,w}(q)=P_{ry,w_I w}(q)$.

Does anyone know how to prove it?


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