Let $G$ be a reductive p-adic or Lie group and let $G'$ be its derived subgroup. The main examples I am interested at are $GL_n(F)$ and $SL_n(F)$, similitude groups and classical groups, $GSpin_{n}(F)$ and $Spin_{n}(F)$ etc. (here $F$ is $\mathbb{R}$, $\mathbb{C}$ or a $p$-adic field) To be more precise, I am interested at Levi subgroups of simple simply-connected groups.
Given an irreducible representation $\pi$ of $G$, the restriction of $\pi$ to $G'$ is not necessarily irreducible. For example, taking $G=GL_2(F)$, $G'=SL_2(F)$ and $\pi$ to be the (normalized) parabolic induction of quadratic character on the torus of $G'$ (extended trivially to the torus of $G$). In this case, the restriction to $\pi$ decompose into a direct sum of two (inequivalent) irreducible representations.
My questions are:
1) Can an irreducible parabolic induction from non-unitary data decompose when restricted to $G'$?
2) More generally, assume that $\pi$ is an irreducible subquotient of a parabolic induction. If the restriction of $\pi$ to $G'$ is not irreducible, what can be said about the inducing data? Also, what can be said about the restriction of other irreducible subquotients of this induction?
In other words, let $\pi$ be an irreducible subquotient of $Ind_P^G \sigma\otimes\lambda$, where $P=MAN$, $\sigma$ is an irreducible representation of $M$ (possibly trivial) and $\lambda$ is a character of $A$. If the restriction of $\pi$ to $G'$ is reducible, is it true that $\sigma$ and $\lambda$ are necessarily unitary? Is there anything else/further that can be said about $\sigma$ and $\lambda$?
3) When the restriction decomposes, is there a natural way to describe the irreducible pieces and bound their number? I assume that there should be a nice description in terms of characters of the abelianization.
Thanks
