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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 $^LG$.

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?

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    $\begingroup$ For more refined versions of the local Langlands you can look at the paper "The Local Langlands Conjectures for Non-quasi split groups" by T. Kaletha. $\endgroup$ – ulrich Jan 9 at 16:10
  • $\begingroup$ It's not really an answer to your question, but I thought it interesting enough to record. Although I may be misquoting him, I understand Arthur to say at the IMSF 8 conference that "endoscopy is for quasi-split groups, and functoriality is for non-quasi-split groups"; that is, transfer among non-quasi-split forms should be viewed as part of functoriality. $\endgroup$ – LSpice Jan 9 at 20:23
  • $\begingroup$ Tasho's paper that @ulrich mentions: www-personal.umich.edu/~kaletha/llcnqs.pdf . $\endgroup$ – LSpice Jan 9 at 20:24
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    $\begingroup$ You realize that the definition of admissible homomorphism 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 at 22:58
  • $\begingroup$ @Aurel I totally forgot about that $\endgroup$ – D_S Jan 10 at 6:35
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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.

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  • $\begingroup$ You can define epsilon and $L$ factors for $G'$ directly as Godement and Jacquet did. This is then a feature of the local Jacquet-Langlands correspondence that local factors are preserved. $\endgroup$ – Paul Broussous Feb 18 at 9:39

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