Level and the Virasoro action

Level and the Virasoro action

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 14 February 2010

Level and the Virasoro action

A 𝔤 -module M is level if c acts on L by l. id L .

A 𝔤 -module is restricted if it satisfies if  mM  then   𝔤 a m=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. e 1 ,, e n and f 1 ,, f n are locally nilpotent on M.
(See [Kac 3.6].)

If L λ is a simple moudle in 𝒪 int then char L λ 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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