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 $$l.{\mathrm{id}}_L.$$

A $𝔤$-module is restricted if it satisfies $$\text{if }m\in M\text{ then }{𝔤}_{a}m=0\text{ for all but a finite number of }a\in R^{+}.$$

This condition makes ther action of the Casimir operator $\kappa$ 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. $e_1,\dots ,e_n$ and $f_1,\dots ,f_n$ are locally nilpotent on $M.$

(See [Kac 3.6].)

If $L\left(\lambda \right)$ is a simple moudle in ${𝒪}_{\mathrm{int}}$ then $\mathrm{char}\left(L\left(\lambda \right)\right)$ 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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