Parabolic induction of p-adic groups independent of the choice of parabolic.

Question

I noticed many papers concerning the theory of smooth representations of connected reductive p-adic groups, omit the mention of the specific parabolic subgroup $P\subseteq G$ used in defining the parabolic induction functor $Ind^G_{P,M}$, and write instead $Ind^G_M$ (where $M$ is some Levi subgroup). My question is why is this justified? are the functors $Ind^G_{P,M}$ and $Ind^G_{Q,M}$ naturally isomorphic for different prabolics $P\neq Q$? If so, how can one show this?

Answer 1

As I'm sure you're aware, there are only two choices for $P$: namely $P$ and $P^{op}$. The functors $Ind^G_P$ and $Ind^G_{P^{op}}$ (I'm assuming you mean normalized parabolic induction) are not the same but they do have the same Jordan--Holder factors. So they are the same in the Grothendieck group. I should also point out that the same is not true for Jacquet modules: $Jac^G_P$ and $Jac^G_{P^{op}}$ do not agree in the Grothendieck group. These issues once caused me a huge headache in a paper I was working on due to some typos I had discovered in the literature relating to this very question. So my challenge to you is to be careful about this in your own work!