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In this paper we classify all irreducible linear representations of the simple Lie algebra An which admit a one-dimensional weight space with respect to some Cartan subalgebra H of An. We first show that the problem is equivalent to determining all algebra homomorphisms from the centralizer of the Cartan subalgebra H in the universal enveloping algebra of An to the base field. We construct all such algebra homomorphisms and provide conditions under which two such algebra homomorphisms providedoi:10.1090/s0002-9947-1982-0667158-4 fatcat:x3lp2tgvafcjrmdjfbzfuiapqu