Let $\mathfrak{g}$ be a semisimple Lie algebra with a Cartan subalgebra $\mathfrak{h}^*$ and Weyl group $W$. For $\mu \in \mathfrak{h}^*$ denote by {$\mu$} the unique dominant weight which is $W$-conjugate to $\mu$. Also, let $\rho$ be the half-sum of positive roots and $P(V(\lambda))$ be the set of weights of the irreducible representation defined by a dominant weight $\lambda.$

Now consider two dominant weights $\lambda\neq\lambda'$ and let $\mu \in P(V(\lambda))$. When is the weight $\{\mu + \lambda' + \rho\}- \rho$ not of the form $ k \lambda' + \rho $, for $k$ an integer?

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    I fixed a few typos, but the question remains unclear; some notation is introduced and not used, and some notation is used but not introduced... – YCor Sep 14 at 12:11
  • I work in the theory of representation of Lie algebras, i can't introduce all the basic notion of this theory. i hope that the experts in this domain could help me. – user123423 Sep 14 at 12:59
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    I guess the confusion is that you use $\rho$ for both the representation and the half-sum of positive roots (which is a standard notation, indeed, but is not compatible with your choice for the representation). – Victor Petrov Sep 14 at 14:59
  • You're right Mr. Victor Petrov. Thank you. – user123423 Sep 14 at 15:56
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    This is indeed the minimal effort you should have done by yourself. I'd not really call Vit's edits "introduce all basic notions of this theory". Writing a meaningful question with coherent notation, and replying to such requests, is the minimal respect you should have for your readers. – YCor Sep 15 at 13:50

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