ON THE IRREDUCIBILITY OF IRREDUCIBLE CHARACTERS OF SIMPLE LIE ALGEBRAS

Abstract. We establish an irreducibility property for the characters of finite di-mensional, irreducible representations of simple Lie algebras (or simple algebraic groups) over the complex numbers, i.e., that the characters of irreducible represen-tations are irreducible after dividing out by (generalized) Weyl denominator type factors. For SL(r) the irreducibility result is the following: let λ = (a1 ≥ a2 ≥ · · · ar−1 ≥ 0) be the highest weight of an irreducible rational representation Vλ of SL(r). As-sume that the integers a1 + r − 1, a2 + r − 2, · · · , ar−1 + 1 are relatively prime. Then the character χλ of Vλ is strongly irreducible in the following sense: for any natural number d, the function χλ(g d), g ∈ SL(r,C) is irreducible in the ring of regular functions of SL(r,C). 1