  We present necessary and sufficient conditions to have global hypoellipticity
for a class of complex-valued coefficient first order evolution equations
defined on $\mathbb{T}^1 \times G$, where $G$ is a compact Lie group. First, we
show that the global hypoellipticity of the constant coefficient operator
related to this operator is a necessary condition, but not a sufficient
condition. Under certain hypothesis, we show that the global hypoellipticity of
this class of operator is completely characterized by Nirenberg-Treves'
condition $(\mathcal{P})$.
