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Denote by $P^{I,y}_{x,w}$ be the parabolic Kazhdan-Lusztig polynomial of ${}^IW$ of type $y$.

I have heard that the polynomials $P^{I,q}_{x,w}$ give the transition matrix between a canonical basis and standard basis $\{T_w\}$ of $M^I$, where $M^I \cong \text{Ind}_{W_I}^W (\text{triv})$. Note that $T_s$ act as multiplication by $q$ on the trivial module. See here.

My questions:

  1. Is the above statement correct?

  2. What is about the polynomials $P^{I,-1}_{x,w}$?

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  • $\begingroup$ Probably you should give a cross-reference to the earlier question (with the same title and overlapping content): mathoverflow.net/questions/325436/about- parabolic-kazhdan-lusztig-polynomials $\endgroup$
    – Jim Humphreys
    Commented Jun 3, 2019 at 17:00

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