most recent 30 from http://mathoverflow.net 2013-06-19T11:27:12Z http://mathoverflow.net/feeds/question/30223 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/30223/do-p-compact-groups-have-a-sufficiently-good-notion-of-flag-variety-and-inters Qiaochu Yuan 2010-07-01T19:24:29Z 2010-07-01T19:24:29Z

<p>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.</p> <h3>Background</h3> <p>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, <a href="http://linkinghub.elsevier.com/retrieve/pii/S0021869305005831" rel="nofollow">du Cloux</a>). However, a conceptual proof in this case is still lacking. </p> <p>In <a href="http://mathoverflow.net/questions/28422/does-the-poincare-series-of-a-coxeter-group-always-describe-a-flag-variety" rel="nofollow">another question I asked</a> 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 <a href="http://en.wikipedia.org/wiki/P-compact_group" rel="nofollow">$p$-compact group</a>. It is known (see, for example, <a href="http://www.nd.edu/~wgd/Dvi/Lie.Groups.pCompact.Groups.pdf" rel="nofollow">Dwyer</a>) 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.</p> <h3>Question</h3> <p>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.) </p>