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Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra, and $M(\Lambda)$ the Verma module with dominant integral highest weight $\Lambda$. The singular vectors in $M(\Lambda)$ are well known (they are something like $f_i^{\Lambda(h_i^\vee)+1} v_\Lambda$). A similar fact is true for the affine Lie algebra $\widehat{\mathfrak{g}}$, with integrable highest weight.

I know of a paper by Feigin, Fuchs and Malikov (1986) where an explicit formula for singular vectors is given for Verma modules over $\widehat{\mathfrak{sl}_2}$ with more general highest weight. The answer uses a notation involving Lie algebra elements raised to complex powers!

My question is about how much is known for more general Lie algebras, and where can I find information.

For instance, have formulas for singular vectors in $M(\Lambda)$ been given for every highest weight $\Lambda$ over any affine Lie algebra $\widehat{\mathfrak{g}}$? Or has little been done since Feigin, Fuchs and Malikov.

Also, I wonder if the answer to my question exists already but in some different (perhaps more abstract) language?

Thanks, Jethro

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    $\begingroup$ It's not clear exactly what you mean by "singular vector", since in the classical case of a finite dimensional simple Lie algebra your parenthetic description "something like" only applies to the most easily constructed ones. $\endgroup$
    – Jim Humphreys
    Oct 15, 2011 at 17:00
  • $\begingroup$ P.S. By now there are a number of module categories which feature some analogue of classical Verma modules, but the literature is not very unified (Kac-Moody, Lie superalgebras, quantum groups, modular versions). Aside from that, locating "singular vectors" is usually far from enough to pin down the module structure, which helped to motivate the more subtle Kazhdan-Lusztig theory in the classical case. $\endgroup$
    – Jim Humphreys
    Oct 18, 2011 at 12:43
  • $\begingroup$ Ah I see, thanks for the comments. I suppose what I'm looking for is an explicit set of vectors that generate the maximal submodule. $\endgroup$
    – Jethro
    Oct 21, 2011 at 15:38

1 Answer 1

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This was supposed to be a comment but grew too much. Some progress using the lines of work of the article you cite is given in [1] where the author gives explicit formulas (he gives an algorithm), he uses rational powers of nilpotent elements and he sketches that his algorithm works in the affine case as well. Also Feigin and Semikhatov had extended the results of $\hat{sl}_2$ to $N=2$ and $sl(2|1)$, this is not the article I had in mind, but the one I can find now [2]. Malikov himself worked out quantum group versions of their formulas (1991).

As for your actual question, I do not know of any explicit formulas but if you're the Jethro I think you are the right person to ask would be Kac.

[1] de Graaf, Willem A. Constructing homomorphisms between Verma modules. J. Lie Theory 15 (2005), no. 2, 415–428.
[2] Theoret. and Math. Phys. 112 (1997), no. 2, 949–987 (1998)

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  • $\begingroup$ Yes, I'm the same Jethro :) Thanks for the references, I'll look those up. Eventually I want to understand the Lie superalgebra case. $\endgroup$
    – Jethro
    Oct 21, 2011 at 15:44

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