Is the unramified local Langlands conjecture true?

Question

I'm working on the unramified local Langlands conjecture and there is something that I don't understand if it is true or not. I want to start by saying that I don't care about endoscopic transfer or orbital integrals, but I just care about the "easiest" conjecture. By this I mean that I'm only trying to understand the fiber of the conjectured map.

Let me start by stating the setting. Let $G$ be an unramified group (so a quasi-split group that splits over an unramified extension) over a non-archimedean local field $F$.

We know that, thanks to the Satake isomorphism, we can define a map $$LL_{ur}: \Pi_{ur}(G)\to \Phi_{ur}(G)$$ from the set of smooth irreducible unramified representation of $G(F)$ to the set of unramified $L$-paramters of $G$. The map is defined as follows: given an unramified representation $\pi$ of $G(F)$, we can use the Satake isomorphism to see this representation as an unramified representation of some maximal torus $T(F)$ of $G(F)$, and then we can use the bijection in the case of tori to get an $L$-parameter $\varphi_{\pi}:W_F'\to {}^LT$ from the Weil-Delinge group of $F$ to the Langlands dual of $T$. Finally, we can embed ${}^LT$ in ${}^LG$ to get an unramified $L$-paramater of $G$.

The natural question to ask now is: "Is this map good? Does it have the property that one wants on the fibers? In particular, after picking an unramified $L$-parameter $\varphi$ of $G$ is it true that the fiber of $\varphi$ is in bijection with the set of irredcuible representation of $\overline{S}_{\varphi}:=\pi_0(Cent(Im(\varphi)))$?".

The answer should be yes. The tool that should be used to prove this, is the unramified principal series of a character: given an $L$-parameter $\varphi$ of $G$, we can see this as an unramified character of some torus $T$, and we can consider the associated unramified representation $\chi_{\varphi}$ of $T(F)$ via the local Langlands map for tori. Then, the unramified principal series $I(\chi_{\varphi})$ of $\chi_{\varphi}$ is the normalized parabolic induction from $T(F)$ to $G(F)$ of $\chi_{\varphi}$.

And now there is my question: is it true that the irreducible components of $I(\chi_{\varphi})$ are in bijection with the fiber of $\varphi$ via $LL_{ur}$? I think Borel in his paper "Automorphic $L$-functions" says that this is just a conjecture, but this paper is from 40 years ago. In more recent papers, it seems to me that they assume that this is true (at least for unitary characters) but no one actually states it clearly, so I'm assuming this might still be unknown. Even if you can just point me to some reference, I would really appreciate it.

Even if you can just point me to some reference, I would really appreciate it.

Answer 1

I made this into an answer as it was too long for a comment.

We know in fact much more, and have done since 1987: We know Local Langlands not just for unramified principal series but for the entire principal block (representations with nonzero Iwahori-fixed vector) aka the Deligne-Langlands conjecture, thanks to Kazhdan-Lusztig (who strictly speaking dealt with split adjoint groups). (See also Ginzburg's approach in Chriss-Ginzburg.) In fact, thanks to Lusztig and others, we know even more still: we know the Local Langlands Correspondence for unipotent representations. I think the current state of the art here is this paper of Solleveld's.

It is true that given a Deligne-Langlands parameter $(s,N)$, the corresponding packet is parameterized by enhanced parameters $(s,N, \rho)$, where $\rho$ is a representation of the component group of the simultaneous centralizer of $(s,N)$. Not every $\rho$ labels a representation in the principal block; this is equivalent to $\rho$ appearing in the Borel-Moore homology of $\mathcal{B}_N^s$, the $s$-fixed points of the Springer fibre $\mathcal{B}_N$. The trivial representation always labels something (consider the sum of fundamental lasses of components of $\mathcal{B}_N^s$) in the principal block, and in fact for $N=0$ labels the unique unramified constituent of the packet, see e.g. Aubert-Baum-Plymen-Solleveld, Prop. 10.1.