In all that follows, we are working over $\mathbb{C}$. Let $B \subseteq P \subseteq {\rm GL}(n)$ be a parabolic subgroup. Can you say anything in general about the representations of $P$? I suspect the answer is no because I couldn't find anything about this in the standard books or using google.


If $P = B$ then everything is easy: any rep breaks up as a sum of 1 dimensional subreps on which the torus acts by some character and the unipotent radical acts as the identity.



Representations of parabolic subgroups of the general linear group over the complex numbers
I think the idea is to write $P$ as a semidirect product of its unipotent radical $N$ and its (maximal reductive) Levi subgroup $M$. (If $P$ is a Borel, then $M$ is a maximal torus.) You can then restrict a complex representation of $P$ to $M$ and decompose it into irreducibles. 


