Level and the Virasoro action
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu
Last updates: 14 February 2010
Level and the Virasoro action
A 𝔤-module M is level if c acts on L by
l.idL.
A 𝔤-module is restricted if it satisfies
if m∈M then 𝔤am=0 for all but a finite number of a∈R+.
This condition makes ther action of the Casimir operator κ on M well defined.
For the Virasoro action see [Kac Ex 12.11 and Ex 12.12].
An integrable 𝔤-module is a 𝔤-module M such that
- M is semisimple as an 𝔥-module,
- e1,…,en and f1,…,fn are locally nilpotent on M.
(See [Kac 3.6].)
If Lλ is a simple moudle in 𝒪int then charLλ is given by Weyl's character formula.
References [PLACEHOLDER]
[BG]
A. Braverman and
D. Gaitsgory,
Crystals via the affine Grassmanian,
Duke Math. J.
107 no. 3, (2001), 561-575;
arXiv:math/9909077v2,
MR1828302 (2002e:20083)
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