This is a reference request. You can answer the question yourself without referencing anything, but i`ll only look at such answers in december (I want to try to figure things out for myself first)
I've been studying the Iwasawa decomposition ( $\mathfrak{g} \equiv \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$) and the principal series representations of semi-simple lie algebras that come from parabolic induction, i.e., induction from the subalgebra $\mathfrak{m} \oplus \mathfrak{a} \oplus \mathfrak{n}$, the standard parabolic subalgebra, where $\mathfrak{m} = \mathfrak{z}_\mathfrak{k}(\mathfrak{a}) $ is centralizer of $\mathfrak{a}$ in $\mathfrak{k}$.
For instance, in Dixmier`s book, Envelopping Algebras, using this parabolic subalgebra and his concept of coinduction (which is in some sense dual to induction: under very mild and reasonable hypotheses $(Ind_\mathfrak{h}^\mathfrak{g}\pi)^* \equiv (CoInd_\mathfrak{h}^\mathfrak{g}\pi^*))$, you obtain all the representations of the principal series (c.f. J. Dixmier, Envelopping algebras, 9.3).
But for reasons too complex to state here, I cannot use this parabolic subalgebra, but am instead supposed to use $\mathfrak{k} \oplus \mathfrak{n}$.
My Request: What is there in the literature about the representations you get from inducing or coinducing from this subalgebra $\mathfrak{k} \oplus \mathfrak{n}$?