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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

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2 Answers 2

up vote 14 down vote accepted

The result of that book is that the conjecture is true for sufficiently large, but unspecified characteristic. (First fix a Dynkin type.) More recently Peter Fiebig has given actual bounds. See

An upper bound on the exceptional characteristics for Lusztig's character formula by Peter Fiebig arXiv:0811.1674v2 at http://arxiv.org/pdf/0811.1674v2

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4  
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. –  Jim Humphreys Mar 28 '10 at 11:00
1  
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. –  Jim Humphreys Apr 4 '10 at 16:10
    
Update: This final version of a survey article by Peter Fiebig is an updated version of the last v4 posted on arXiv and is now freely available in PDF format from Bull. London Math. Soc.: blms.oxfordjournals.org/cgi/content/abstract/ –  Jim Humphreys Aug 30 '10 at 18:57

Geordie Williamson has found counterexamples to Lusztig's conjecture: http://people.mpim-bonn.mpg.de/geordie/Torsion.pdf

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.

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