I have two reducible representations of a finite group $G$ of Lie type, $\rho, \pi$. They both have multiplicity one, and I know that they share exactly one irreducible subrepresentation.
Is there a method to explicitly obtain the character of the common subrepresentation using their characters? (so, not using character tables)
Using Mackey theory I have found an intertwining operator $M:\rho\rightarrow\pi$, so the subrepresentation is isomorphic to the image of $M$ in $\pi$. But the dimension of $\pi$ is quite large, "objectively" - not just compared to that of $\rho$, so it seems tedious to compute the image.