Abstract: Let P be a parabolic upper-block subgroup of GLn(K). We look at the conjugation-action of P on the variety of nilpotent matrices in Lie(P). Our main question is motivated by the study of commuting varieties and reads as follows: "Does the mentioned P-action only admit a finite number of orbits?" We answer this question by developing a complete classification of finite actions. The proof makes use of methods from Representation Theory of finite-dimensional algebras and we sketch the main ideas and techniques. (Joint work with M. Bulois)