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Let $G$ be a connected, reductive group over a local field $k$, and let $^LG$ be the Langlands dual group. As explained by Borel in his article in the Corvallis proceedings, the general local Langlands correspondence should give (1) a partition of the classes of irreducible admissible representations of $G(k)$ into finite sets, called L-packets, and (2) a bijection between the L-packets and the equivalence classes of admissible homomorphisms of the Weil-Deligne group $W_k'$ into $^LG$.

When $G = \operatorname{GL}_n$, the L-packets are just singleton sets. I believe that only the local Langlands conjectures for $\operatorname{GL}_2$ were proved at the time Borel's article was written. There were no worked out examples of L-packets with more than one element at the time, as far as I know.

Why did Borel and others in the 1970s expect the L-packets to be finite? Why do we still expect this today?

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The fiber $\mathcal{L}^{-1}(\rho)$ of an L-parameter $\rho:W_k'\rightarrow ^LG$ is expected to be in bijection with the set of irreducible representations of a certain finite group attached to $\rho$. In greater detail, let $Z(\rho)$ denote the centralizer in $\hat{G}$ of the image of $\rho$ in $ ^LG$ and $Z$ the center of $\hat{G}$. Clearly, $Z^{\text{Gal}(\bar{k}/k)}$ is contained in $Z(\rho)$. Let $\iota: Z(\rho)\rightarrow \pi_0(Z(\rho))$ denote the natural map, let $H_{\rho}:=\pi_0(Z(\rho))/ \iota(Z^{\text{Gal}(\bar{k}/k)})$. It is expected that there is a natural bijection of $\mathcal{L}^{-1}(\rho)$ with the irreducible representations of $H_{\rho}$. See for instance, https://ims.nus.edu.sg/events/2018/lang/files/gan.pdf for further details.

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    $\begingroup$ In the following book, math.nus.edu.sg/~matgwt/work8-3.pdf, chapter 4 together with theorem 8.1 explain why $H_{\rho}$ is a finite 2-group. $\endgroup$ – Cooler Panda Jan 9 at 16:42
  • $\begingroup$ @ Kimball I'm not a specialist in this area so correct me if I'm wrong but as far as I can recall the relationship is for L-packets as stated in slide 7 of ims.nus.edu.sg/events/2018/lang/files/gan.pdf "Parametrization of L-Packets" $\endgroup$ – Anwesh Ray Jan 10 at 3:14
  • $\begingroup$ Okay, that is what those slides say. Though I am still not sure if this answers the question---did Borel and Langlands have this precise expectation in the 1970's, or was it only later? $\endgroup$ – Kimball Jan 11 at 7:25
  • $\begingroup$ @CoolerPanda, it is a finite 2-group for classical groups. I think that $H_\rho$ can be $\operatorname S_3$ in type $\mathsf G_2$. $\endgroup$ – LSpice Mar 31 at 13:40

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