L-packets in the local Langlands correspondence: why finite sets?

Question

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?

Answer 1

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.