# What is Extreme/Extremal vector according to some weights

I know this might be a very elementary question. But I could not find the original definition of Extreme(or Extremal)vectors of some weights $\lambda$(fixed the $w\in W$,where $W$ is Weyl group)

I am looking for definition for Extreme vector for finite dimensional Lie algebra and Affine Lie algebra. I found a paper saying :"Extreme vector verifies Weyl Character formula" What does it mean?

I am looking for reference talking about this concept. Thanks in advance

EDIT: I guess it is just highest weight vector, but I am not sure

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## 1 Answer

Usually "extremal weight" means a weight in the Weyl group orbit of the highest, and I would interpret "extremal vector" as an element of said weight space.

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Yes, this is the standard notion developed originally in the setting of finite dimensional irreducible representations of complex semisimple Lie algebras (or groups); extremal weights have multiplicity 1. This can to some extent be carried over to "integrable" representations of affine Kac-Moody algebras, where Kac found a good analogue of the Weyl character formula. Also, the theory of Demazure modules (geometrically motivated) involves study of the subspace obtained by fixing an extremal weight space and applying to it all negative root vectors in the Lie algebra. – Jim Humphreys Apr 17 '10 at 13:21
References? Besides textbooks, it's interesting to look at some of the relevant literature such as MR943925 (89j:17009) 17B10 (22E46) Kumar, Shrawan (6-TIFR), Proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture. Invent. Math. 93 (1988), no. 1, 117–130. – Jim Humphreys Apr 17 '10 at 13:34