Let $W$ be a Coxeter group with simple reflections $S$ and let $J \subseteq S$. Let $P^J_{\tau, \sigma}$ be the parabolic Kazhdan-Lusztig polynomials in the case $u = q$ in the sense of On Some Geometric Aspects of Bruhat Orderings II. The Parabolic Analogue of Kazhdan-Lusztig Polynomials by Deodhar. These give the transition matrix between a canonical basis and standard basis $\{T_w\}$ of $M^J$, where $M^J \cong \text{Ind}_{W_J}^W \text{ triv}$. This canonical basis is like the $C_w$ basis, not the $C'_w$ basis --the trivial module is a cellular quotient, not a cellular submodule.

When is $P^J_{\tau, \sigma}$ nonzero? This question seems quite difficult, and I am wondering if there has been any work done on it or if it is equivalent to some well-known problem in Kazhdan-Lusztig theory that is known to be difficult.

Unlike ordinary Kazhdan-Lusztig polynomials which are nonzero if and only if $\tau \leq \sigma$, these are nonzero only if $\tau \leq \sigma$, but can often be $0$ when $\tau < \sigma$. For example, in the type A case and the case that $J$ is maximal parabolic, which $P^J_{\tau, \sigma}$ are nonzero is easily described in terms of the $sl_2$ graphical calculus (the number of nonzero $P^J_{\tau, \sigma}$ for fixed $\sigma$ is $2^k$ where $k$ is the number of arcs in the diagram corresponding to $\sigma$).