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?

admissiblehomomorphism depends on the inner form and not just on the $L$-group, right? (cf 8.2 (ii) in Borel's article) $\endgroup$ – Aurel Jan 9 '19 at 22:58