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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.

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2 Answers 2

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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.

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  • $\begingroup$ Thanks so much for your answer. I'm surprised there has been such a small amount of attention given to inverse Kazhdan-Lusztig polynomials in the literature since it seems to me that inverse Kazhdan-Lusztig polynomials are equally as natural as the ordinary Kazhdan-Lusztig polynomials. The formulas in Andersen's paper, for example, seem less approachable than the identity stated in my question. $\endgroup$
    – Seth
    Mar 3, 2018 at 13:53
  • $\begingroup$ @Seth: I guess the emphasis on Kazhdan-Lusztig polynomials came at first from the natural question of generalizing the Weyl/Kostant approach to finite dimensional simplre modules with a dominant integral highest weight. But I agree that in the affine Weyl group case most relevant to modular representations there has been some neglect of the inverse polynomials. (An unpublished note on my webpage raises for example the natural question of organizing the composition factors of a Weyl module in prime characteristic when the highest weight is large, after one finds the simple modules.) $\endgroup$ Mar 4, 2018 at 0:13
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The recursive formula for inverse Kazhdan-Lusztig polynomials of an arbitrary Coxeter group has been derived in the paper: Some Properties of Inverse Weighted Parabolic Kazhdan Lusztig Polynomials by Hiroyuki Tagawa.

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.

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