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We establish an irreducibility property for the characters of finitedimensional, irreducible representations of simple Lie algebras (or simplealgebraic groups) over the complex numbers, i.e., that the characters ofirreducible representations are irreducible after dividing out by (generalized)Weyl denominator type factors.  For $SL(r)$ the irreducibility result is the following: let $\lambda=(a_1\geqa_2\geq ... a_{r-1}\geq 0)$ be the highest weight of an irreducible rationalrepresentation $V_{\lambda}$ of $SL(r)$.  Assume that the integers $a_1+r-1, ~a_2+r-2,..., a_{r-1}+1$ are relativelyprime. Then the character $\chi_{\lambda}$ of $V_{\lambda}$ is stronglyirreducible in the following sense: for any natural number $d$, the function$\chi_{\lambda}(g^d), ~g\in SL(r,\C)$ is irreducible in the ring of regularfunctions of $SL(r,\C)$.

On the irreducibility of irreducible characters of simple Lie algebras