Reference request: GL(n) There are many places, which describe the unitary irreducible representations of $GL(n, F)$ with $F = \mathbb{C}$ or $F =\mathbb{R}$. Basically, we obtain a bunch of parabolically induced representation indexed by irreducible reps of parabolic subgroups and complex parameters and construct various quotients.
However, I could not find how the restriction of a unitary irreducible representation to the maximal compact groups decomposes into irreducibles.
Does somebody have a reference for $GL(n, \mathbb{C})$?
For $SL(2, \mathbb{C})$, I found Barut-Raczka "The theory of group representations" pg.567 and for $GL(2, \mathbb{R})$, I found Knightly-Li "Hecke opeartors" and much more on $GL(n, \mathbb{R})$ in Goldfeld-Hundley's new book on automorphic forms.

For my own research I am intensively using the books by N. Ja. Vilenkin and A.U. Klimyk "Representation of Lie Groups and Special Functions". For $GL(n,\mathbb{C})$ I would recommend volumes 2 and 3.
