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

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?

• 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. – naf Jan 9 '19 at 16:10
• 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. – LSpice Jan 9 '19 at 20:23
• Tasho's paper that @ulrich mentions: www-personal.umich.edu/~kaletha/llcnqs.pdf . – LSpice Jan 9 '19 at 20:24
• 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) – Aurel Jan 9 '19 at 22:58
• @Aurel I totally forgot about that – D_S Jan 10 '19 at 6:35

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.

• 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. – Paul Broussous Feb 18 '19 at 9:39