# Is Lusztig's conjecture solved?

What I said is Lusztig's conjecture about representation of quantum group at root of unity and representation of Lie algebra at positive characters.

It seems that Andersen-Jantzen-Soergel ever wrote a book on this conjecture.

Is it solved? Any recent development? I am looking for reference talking about it.

Thank you

• To expand Wilberd's comment, the ongoing work of R. Bezrukavnikov and I. Mirkovic (following up their joint work with D. Rumynin) takes a more geometric viewpoint. This preprint, now in version 3, addresses the closely related conjectures of Lusztig in 1997-99 on bases in equivariant K-theory: front.math.ucdavis.edu/1001.2562 As in AJS, applications to characteristic $p$ are (so far) dependent on $p$ being "large enough". But the conjectures go beyond restricted Lie algebra representations to those attached to arbitrary nilpotent orbits. Mar 28 '10 at 11:00
• A little more detail about [AJS], which is a 300+ page paper (Asterisque 220, 1994) but has a readable introduction. They work in a graded setting with Lie algebra modules, getting for large enough $p$ and suitably bounded weights a precise comparison with the quantum enveloping algebra of Lusztig at a $p$th root of unity. In the latter case, one combines work of Kashiwara-Tanisaki on the analogue of the Kazhdan-Lusztig Conjecture for affine Lie algebras with work of K-L passing from there to quantum groups. [AJS] relies more on combinatorics than on geometry. Apr 4 '10 at 16:10
Apparently, any bound on the largest prime that breaks Lusztig's conjecture must be at least proportional to $n\log n$, and this is probably still far too optimistic; I think it's now believed that you'll need an exponential bound.