In Differential Operators on Homogeneous Spaces, III, Section 6.7, Borho and Brylinski make a long series of equivalent conjectures which they show are all equivalent. Perhaps the easiest statement is:

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.

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 Kashiwara and Saito; the counter example there is to a much stronger irreducibility conjecture (for supports of simple modules, not ideals).

**EDIT:** 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).

`$B_3, C_3$`

. This suggested to them a limitation of their conjecture to type A or special cases elsewhere. I also feel uncertain how this complicated story from 25 years ago turned out. If anyone knows for sure, it would probably be Tony Joseph. – Jim Humphreys Jan 27 '11 at 19:57