Given a highest weight irreducible representation of a simple Lie algebra, the question whether it is finite-dimensional has an affirmative answer if and only if its highest weight is dominant. In the case of a lowest weight simple module of a rational Cherednik algebra for a given parameter, the above, natural, question does not have an answer in general yet. In this talk, I will discuss joint work with S. Griffeth, A. Gusenbauer and D. Juteau providing a necessary condition for finite dimensionality of such modules.