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

1. M is semisimple as an 𝔥-module,
2. 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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