Do p-compact groups have a sufficiently good notion of "flag variety" and "intersection cohomology"? - MathOverflow 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 Do p-compact groups have a sufficiently good notion of "flag variety" and "intersection cohomology"? 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>