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This is mostly an idle question, since I don't think I'd be able to do anything with a positive answer. But a positive answer would still be interesting, I think.

Background

An outstanding problem in algebraic combinatorics is to prove that, for any Coxeter group $W$, the coefficients of the Kazhdan-Lusztig polynomials $P_{x,w}(q)$ are always non-negative. Kazhdan and Lusztig showed this was the case when $W$ is the Weyl group of a semisimple Lie group by relating the coefficients to intersection cohomology of the corresponding Schubert varieties. This covers all of the cases when $W$ is finite except the non-crystallographic dihedral groups and the exceptional groups $H_3, H_4$. In the dihedral case all of the Kazhdan-Lusztig polynomials are equal to $1$, and in the exceptional cases non-negativity is known through computer calculations (see, for example, du Cloux). However, a conceptual proof in this case is still lacking.

In another question I asked about Coxeter groups that aren't Weyl groups, Stephen Griffeth suggested that when $W$ is a $p$-adic reflection group for some $p$, the correct analogue of an associated Lie group is a $p$-compact group. It is known (see, for example, Dwyer) that any complex reflection group - in particular, any finite Coxeter group - occurs as a $p$-adic reflection group for an appropriate choice of $p$, and I believe it is known that every $p$-adic reflection group occurs as the Weyl group of some $p$-compact group.

Question

Do $p$-compact groups have a good enough notion of "flag variety" and "intersection cohomology" that the Kazhdan-Lusztig polynomials for $H_3$ and $H_4$ can be interpreted in terms of them? Is there a simple description of the $p$-compact groups associated to $H_3$ and $H_4$, say, when $p = 11$? (This is the smallest prime such that $H_3$ and $H_4$ occur as a $p$-adic reflection group.)

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    $\begingroup$ I'm no expert on p-compact groups. But there is an analogue of "flag varieties". If K is a p-compact group, then (by a theorem of Dwyer-Wilkinson) it has a "maximal torus" T. The "quotient space" K/T might not be a finite CW-complex, but it is a space with finite mod-p homology (and non-0 Euler characteristic). I don't know what there is to say about the cohomology of such things. $\endgroup$
    – Charles Rezk
    Commented Jul 1, 2010 at 20:50
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    $\begingroup$ In between the homotopy type of the flag variety and the intersection homology are the Schubert cells. I don't think anyone has pinned them down up to homotopy, let alone with the kind of geometry that would be needed for IH. $\endgroup$
    – Ben Wieland
    Commented Jul 1, 2010 at 21:37
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    $\begingroup$ If you haven't left for the UK yet, I suggest you ask Haynes Miller, because he is the local expert. One can construct a p-profinite homotopy type analogous to the flag variety, but I don't think anyone has cooked up a suitable analog of Schubert variety or perversity. $\endgroup$
    – S. Carnahan ♦
    Commented Jul 3, 2010 at 4:22
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    $\begingroup$ You could set as a preliminary goal to model the stack B\G/B for a complex Lie group in terms of the homotopy types K,T of its maximal compact and torus (ie things that translate well to the p-compact setting). From there to Kazhdan-Lusztig theory the main additional jump would be the theory of weights/mixed sheaves, but at least you'd be halfway there. $\endgroup$
    – David Ben-Zvi
    Commented Apr 10, 2011 at 18:02

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