Let G be a connected simple real Lie group with maximal compact subgroup K, and let \((\pi,V)\) be a finite dimensional representation of G of highest weight \(\lambda\). Let \(K_ 1\) be the semisimple part of K. Necessary and sufficient conditions on \(\lambda\) are given in order that \(\pi\) contains one-dimensional K-types, or equivalently that the set \(V^{K_ 1}\) of \(K_ 1\)-invariant vectors in V is non-zero. Moreover, the representations of K on \(V^{K_ 1}\) is identified (Theorem 7.2). This generalizes Helgason's description of the sperical finite dimensional representations [\textit{S. Helgason}, Groups and Geometric Analysis: (1984), Chapter V, Theorem 4.1].