Status of Borho and Brylinski's irreducibility conjectures? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T05:50:45Z http://mathoverflow.net/feeds/question/53525 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53525/status-of-borho-and-brylinskis-irreducibility-conjectures Status of Borho and Brylinski's irreducibility conjectures? Ben Webster 2011-01-27T19:22:16Z 2011-01-30T00:05:01Z <p>In <a href="http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002101483" rel="nofollow">Differential Operators on Homogeneous Spaces, III</a>, Section 6.7, Borho and Brylinski make a long series of equivalent conjectures which they show are all equivalent. Perhaps the easiest statement is:</p> <blockquote> <p>If $J$ is a primitive ideal with principal central character in $U(\mathfrak{sl}(n))$, then the singular support of D-modules modulo $J$ on $SL_n/B$ is irreducible.</p> </blockquote> <p>Has any later work been done on this conjecture? Looking through the articles citing it on MathSciNet and Google Scholar, I didn't see anything, but there were a lot so I might have missed something. One thing that looks suspiciously like a counter-example, but which I'm fairly sure is not <a href="http://arxiv.org/abs/q-alg/9606009" rel="nofollow">Kashiwara and Saito</a>; the counter example there is to a much stronger irreducibility conjecture (for supports of simple modules, not ideals).</p> <p><strong>EDIT:</strong> At the request of David Ben-Zvi, let me explain why I don't think that Kashiwara and Saito resolved this question. Their theorem is that there are simple modules D-modules on $SL_n/B$ which are Schubert smooth and have non-irreducible characteristic varieties. What I'm asking about is the characteristic variety of $\mathcal D_{SL_n/B}/J$, which is a non-holonomic D-module; in fact, Borho and Brylinski prove that $SS(\mathcal D_{SL_n/B}/J)=G\cdot SS(\mathcal L_w)$ for any simple D-module $\mathcal L_w$ of which $J$ is the annihilator. Of course, lots of non-irreducible varieties have irreducible saturation; for example, any closed subvariety of $T^*G/B$ which contains a point whose image is a regular nilpotent (so the singular support of any Verma module).</p>