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

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?

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.
• 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. – Cooler Panda Jan 9 at 16:42
• @CoolerPanda, it is a finite 2-group for classical groups. I think that $H_\rho$ can be $\operatorname S_3$ in type $\mathsf G_2$. – LSpice Mar 31 at 13:40