# Recursive formula for inverse Kazhdan-Lusztig polynomials

Let $(W,S)$ be a Coxeter group. Then the Kazhdan-Lusztig polynomials $P_{x,w}$ are known to satisfy the following recursive identity:

For $x < w$ and $s \in S$ satisfying $\ell(sw) < \ell(w)$, define

$$c = c_s(x) := \begin{cases} 0, \,\,\,\text{ if }x < sx\\ 1, \,\,\,\text{ if }x > sx. \end{cases}$$

Then, $P_{x,w}$ satisfies $$P_{x,w} = q^{c−1}P_{sx,sw} + q^c P_{x,sw} −\sum_{sz<z<sw} \mu(z, sw)q^{(\ell(z)-\ell(w))/2} P_{x,z},$$ where $\mu(z, sw)$ is the coefficient of $q^{(\ell(sw)−\ell(z)−1)/2}$ in $P_{z,sw}$.

Is there a known identity similar to the above for the inverse Kazhdan-Lusztig polynomials $Q_{x,w}$?

In the finite case, we have $Q_{x,w}=P_{w_0w,w_0x}$ where $w_0\in W$ is the longest length element, but I am interested in the affine or general Coxeter case.

The case of an affine Weyl group is apparently the only one which has been looked at closely. But it may be hard to answer your specific question. As far as I know, there are two relevant papers, the first in 1986 by Andersen here (now accessible online) and the second in 1987 by Kaneda here. Lusztig's 1980 paper here inverted his previous conjecture in modular representaton theory, and Kato showed (following Andersen's paper in the same journal though published earlier) how the two conjectures are related; in particular, for large $p$ they are equivalent. In any case, Lusztig's paper gives a good sample of the rank 2 inverse polynomials, but computations in higher rank seem very difficult to carry out.
The work on Lusztig's conjectures has progressed beyond this, with some positive results by Andersen-Jantzen-Soergel for large enough $p$ and some cautionary results more recently by Williamson. But unlike the case of finite Weyl groups, the inverse polynomials are hard to get at for affine Weyl groups, not to mention other Coxeter groups.
Actually, the recursive formula for Inverse Weighted Parabolic Kazhdan Lusztig Polynomials of an arbitrary Coxeter group has been derived. To answer your question, you simply need to take $$J=\emptyset$$ and $$q_s=q$$ for all $$s\in S$$ in Theorem 5.6.