This is a general question about representation theory of finite dimensional simple Lie algebras of Cartan type over algebraically closed fields of positive characteristic (vector fields on Frobenius neighborhood of a point on a smooth variety preserving an appropriate tensor field). Has the basic picture of representations been worked out? Classification, dimensions, character formulas?

Another questions is if their cohomology is understood.

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    $\begingroup$ The answer to your first question is "partly yes", which Sasha Premet and others could elaborate on further. But I think your combined questions are much too broad for this site. It would help if you started somewhere in the large literature available and narrowed your question accordingly. People like Dan Nakano at Georgia and Randy Holmes at Auburn could supply plenty of references to older work and maybe the recent work of Jantzen too. $\endgroup$ – Jim Humphreys Nov 29 '12 at 20:53
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    $\begingroup$ P.S. I don't intend to be dismissive about your question, but there are dozens of research papers on these subjects over many years. The subject itself is not in the mainstream of representation theory, partly because it doesn't arise naturally from group theory. But a lot of work has been done, much of it by Chinese or Russian authors. For one approach, see the AMS Memoir No. 98 (1992) by Dan Nakano based on his 1990 Yale thesis treating projective modules for the Lie algebras of Cartan type. Quite recently J.C. Jantzen has substantially improved older work (to appear). Etc. $\endgroup$ – Jim Humphreys Nov 30 '12 at 0:59

There are many types of finite dimensional Cartan type Lie algebras: to recover all finite dimensional simple Lie algebras in characteristic $p>3$ one has to define them over arbitrary finite dimensional divided power algebras (not just over functions on Frobenius neighbourhoods) and then also consider their natural filtered deformations. Since you are interested in the the special case of Frobenius neighbourhoods , the algebras in question will be restricted (in general, they are not). In this case one knows basically everything about irreducible restricted representations and almost nothing about the non-restricted ones (one notable exception is the Witt algebra where all irreducibles are described by Chang in 1941). The case of nontrival $p$-characters is extremely difficult already for $W(2,{\underline 1})$ (which is the full derivation algebra of the truncated polynomial ring in two variables).

The description of restricted representations resembles Rudakov's description of graded irreducible representations of Lie algebras of Cartan type over complex numbers: the majority of irreducibles are induced from irreducible modules over standard maximal subalgebras (vector fields without constant terms) and the non-induced ones appear as composition factors of modules arising from the de Rham complex. In characteristic $p$, complications arise in types $H$ and $K$ when $p$ is small. This is due to the fact that fundamental Weyl modules over $Sp_{2n}$ can be reducible when $p\le n$. But one knows dimensions and can write down character formulae (this will of course rely on the modular Lusztig conjecture for $SL_n$ and $Sp_{2n}$).

  • $\begingroup$ Thank you!! Is there a survey where some of this (representation theory, not the classification of Lie algebras) is explained? $\endgroup$ – Roman Nov 30 '12 at 20:31
  • $\begingroup$ I'm unaware of any surveys of this subject, but perhaps they exist in some hard-to-find conference proceedings. I think the best way to start is to follows Jim's suggestion and look at some original articles. Regarding the second part of your question, cohomology of induced modules are often computed by using a version of Shapiro's lemma, but one shouldn't expect a nice answer here. $\endgroup$ – Alexander Premet Dec 1 '12 at 11:30
  • $\begingroup$ @Roman: I'm also unaware of any survey on the representation theory, even a conference-type paper. The literature stretches back over many decades, but a lot of it is referenced in a 2012 preprint by J.C. Jantzen (Aarhus) Cartan matrices for restricted Lie algebras. Concerning cohomological aspects, probably the best informed person is Dan Nakano at Georgia, though his interests range more widely. Keep in mind that restricted Lie algebras of Cartan type and more general ones may require different techniques. $\endgroup$ – Jim Humphreys Dec 1 '12 at 14:23

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