How should the local Langlands correspondence for general reductive groups take into account different inner forms?

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 $^LG$.

The bijection should be such that if $r$ is a finite dimensional complex representation of $^LG$ whose restriction to $^LG^{\circ}$ is complex analytic, and $\pi$ is a member of an L-packet corresponding to an admissible homomorphism $\rho$, then we should have an equality of L and epsilon factors

$$L(s,r \circ\rho) = L(s,\pi,r)$$

$$\epsilon(s,r \circ \rho,\psi) = \epsilon(s, \pi,r,\psi)$$

whenever the objects on the analytic side can be defined.

The conjectural correspondence $\{\pi\} \rightarrow \rho$ seems unsatisfactory to me for the following reason: the L-group $^LG$ does not change if we replace $G$ by an inner form. This is on account of the fact that the group of inner automorphisms of $G_{\overline{k}}$ act trivially as automorphisms of the based root datum of $G_{\overline{k}}$.

If $G'$ is an inner form of $G$, the groups $G(k)$ and $G'(k)$ and their representations should look completely different. However, if we take the local Langlands correspondence as I stated it above at face value, there should be correspondences between these representations giving them the same L-functions and epsilon factors.

Do we really expect the representations of different forms of algebraic groups to be give the same L functions and epsilon factors? Or do we expect there will eventually be more complicated formulation of the local Langlands correspondence which takes into account different inner forms?

Answer 1

Sort of. You are asking about a (generalized local) Jacquet-Langlands correspondence. Roughly what this says is that there is a correspondence between discrete series representations of $G$ and $G'$. Consider the simplest case of $G=PGL(2)$ and $G'$ is a compact inner form, i.e. $G=PD^\times$ for $D$ a quaternion division algebra over $k$.

Crudely, is how the correspondence goes:

1-dimensional representations of $G'$ correspond to (twisted) Steinberg representations of $G$

higher-dimensional irreducible representations of $G'$ correspond to supercuspidal representations of $G$

This correspondence preserves $L$- and $\epsilon$-factors, essentially because you define the factors on $G'$ so they match. However, there are more representations of $G$: 1-dimensionals and irreducible principal series. While in some sense the 1-dimensionals of $G$ should also correspond with the 1-dimensionals of $G'$ (if one thinks about A-parameters rather than L-parameters), the irreducible principal series do not correspond to anything on $G'$.

What happens is that some $L$-parameters will be relevant for a given inner form $G'$ and some will not (if $G'$ is not quasi-split). So you don't see exactly the same $L$-parameters for all inner forms, but when you do you can talk about a correspondence of packets.