# Representations of parabolic subgroups of the general linear group over the complex numbers

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.

• It depends on what you mean by "easy", I think. Any semisimple representation of $B$ will of course be a direct sum of one dimensional subrepresentations. But arbitrary representations need not be. – Stephen Aug 26 '14 at 1:00
• Ohh, the representations of B are not semisimple? – Daniel Barter Aug 26 '14 at 1:03
• Definitely not! A simple example is the standard $n$-dimensional representation of $GL(n)$ restricted to $B$. – Steven Sam Aug 26 '14 at 2:03
• The problem is that $B$ is not a reductive group, so some of its finite-dimensional reps are not semisimple. – Peter Crooks Aug 26 '14 at 2:39
• that makes sense. – Daniel Barter Aug 26 '14 at 3:05

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.