# Restriction of a representation of a reductive group to the derived subgroup

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

• You are asking quite a few questions, so it's unclear what might count as a correct "answer" here. (Also, it's desirable to clarify what you mean by the expression "reductive Lie group".) – Jim Humphreys Mar 2 '18 at 16:08
• I am sorry that the question was a bit vague. I've added a reformulation of the first two questions in, hopefully, a clearer manner. I admit that they are still relatively open... As for what I mean by "reductive Lie groups", I would say that what I am really interested at, at this point, are Levi subgroups of simple simply-connected groups. – Matht111101111 Mar 2 '18 at 17:50