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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).

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sorry for being dense, but could you explain the distinction between this conjecture and what Kashiwara-Saito disprove? i.e. isn't a primitive ideal precisely the annihilator of a simple module, and aren't B-B precisely looking at the characteristic varieties of these modules (which they think of as lifts of the associated varieties to the ideals in the dual of the Lie algebra)? – David Ben-Zvi Jan 27 '11 at 19:54
It does seem reasonable to consider only type A here, following the authors' note added in proof which cites counterexamples due to "Tanizaki" (Tanisaki) in types $B_3, C_3$. 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
I made the mistake above of using $\mathfrak{sl}_n$ at one spot, and G at another. This is why $SL_n$ should always be denoted $G$. – Ben Webster Jan 27 '11 at 20:18
This may or may not be relevant: ! – Geordie Williamson Apr 2 '15 at 19:24
@GeordieWilliamson By the time I saw you talk about this, I'd forgotten this question; I think the inverses of the elements you give should be a counterexample to the conjecture above. Do you feel like adding that as an answer? – Ben Webster Apr 3 '15 at 0:04

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