$L$-parameters and parabolic induction

Question

I apologize in advance if the answer to this question is well-known to experts.

So let $F$ be a $p$-adic field, and $G$ a reductive group over $F$. Let $P$ be an $F$-parabolic subgroup of $G$ and $M$ the Levi of $P$. (For convenience, I just use the same notations for an algebraic group and a group of rational points.) We know that roughly, irreducible admissible (smooth) representations of $G$ come from (normalised) parabolic induction: if $\pi$ is a non-cuspidal irreducible representation of $G$, then $\pi$ can be embedded into some $I_Q^G(\sigma)$ for some representation $\sigma$ of Levi of some proper parabolic $Q$.

Meanwhile, we know what is local Langlands correspondence. Let $WD(F):=W(F)\times\operatorname{SL}_2(\mathbb{C})$ be the Weil-Deligne group, then an $L$-parameter of $G$ is a conjugacy class of homomorphism $$\phi:WD(F)\rightarrow {^LG}.$$ Suppose now we have an irreducible representation $\sigma$ of $M$ (setting as above), and then we consider $I_P^G(\sigma)$, and take some irreducible composition factor $\pi$, which is an irreducible representation of $G$. Suppose $\phi_\sigma:WD(F)\rightarrow {^LM}$ is the $L$-parameter of $\sigma$. We know that $M\subset G$ provides us with ${^LM}\subset {^LG}$, so we get a parameter of $G$: $WD(F)\rightarrow {^LM}\subset {^LG}$.

Question: some expert told me that this new parameter is just the parameter for the Langlands quotient of $I_P^G(\sigma)$. This is related to the Langlands classification of $p$-adic groups (for which I'm a total beginner). I want to ask where can I find the proof of this fact in the literature? And I also want to ask: in general, is there any systematic relation between the parameter of $\sigma$ and the parameter of $\pi$ (a composition factor of the parabolic induction)? How to characterise tempered/discrete series/cuspidal representations via parameters? (I know such facts about $\operatorname{GL}_n$, but I'm curious about whether there's a general result for general reductive groups, which so far I didn't find in any paper, but seems to be folklore to experts...)

Again, I want to apologize if this question is standard for veterans/experts. Thanks a lot in advance for any explanation!

Answer 1

I think what you say is a part of the local Langlands Conjecture. See Conjecture 4.1 (7)(8)(10)in Kaletha and Taibi's Lecture notes on LLC for IHES 2022. The local Langlands conjectures for different groups should be compatible with parabolic inductions. As a special case, you can consider spherical representations and calculate their Satake parameters.

For your other problem, here is my understanding (these can also be found in the lecture mentioned above):

• A Langlands parameter is tempered if its image under the projection to $\widehat{G}(\mathbb{C})$ is bounded （Thanks to Kenta's comment, here "the image of the Weil group is bounded" should be more precise）. By the LLC for quasi-split groups, there should be a surjective map from the set of isomorphism classes of tempered irreducible admissible representations to the set of equivalence classes of tempered parameters.
• We should have: the image of a tempered Langlands parameter is not contained in any proper parabolic subgroup of the L-group (equivalently, its centralizer in $\widehat{G}(\mathbb{C})$ is finite modulo $Z(\widehat{G})^{G_{F}}$) if and only if one element (equivalently, every element) in its L-packet is essentially square-integrable. We call this kind of parameter essentially discrete.
• Here we just assume the coefficient field $C=\mathbb{C}$ or $\mathbb{Q}_{l},l\neq p$ so that we can view cuspidal and supercuspidal as the same thing. We should have: a Langlands parameter is essentially discrete and trivial on Deligne's $\mathrm{SL}_{2}(\mathbb{C})$ if and only if every member in its L-packet is supercuspidal. However, even for $\mathrm{GL}_{n}$ it can happen that for an essentially discrete Langlands parameter nontrivial on $\mathrm{SL}_{2}(\mathbb{C})$ its L-packet contains a supercuspidal representation. Of course you can try to describe the correspondence in terms of enhanced Langlands parameters.
• For discrete series, it should correspond to some enhanced Langlands parameter consisting of an essentially discrete Langlands parameter and a character on its component group satisfying some extra condition. Notice that here we need the refined LLC.

I am also not so familiar with the representation theory of $p$-adic groups so my answer may contain some errors and comments are welcome!