Compatibility properties of local Langlands

A local Langlands correspondence is expected to commute with the natural structures on its two sides. For a representation $\pi$ with parameter $\varphi_\pi$ , the standard compatibilities include:

unramified representations correspond to their Satake parameters;
tempered and essentially square-integrable representations correspond to tempered and discrete parameters;
central characters are obtained from the parameter through the center of the $L$ -group and local class field theory;
twisting $\pi$ by a character twists $\varphi_\pi$ by the corresponding one-dimensional parameter;
contragredients correspond to the dual, or Chevalley-transformed, parameter;
Langlands quotients of normalized parabolic induction correspond to the associated Levi parameters embedded in ${}^LG$ ;
local $L$ -, epsilon, and gamma factors agree whenever both sides have an intrinsic definition.

Theorem versus characterization

For $\operatorname{GL}_n$ , equality of Rankin–Selberg local factors, together with elementary compatibilities, characterizes the established correspondence. For a general reductive group, not all required factors have an intrinsic definition, and many entries in the list remain conjectural.

Refined compatibilities

The refined correspondence adds transformation laws for changing a Whittaker datum or rigidifying cocycle, compatibility with automorphisms and homomorphisms of groups, and endoscopic character identities. These data distinguish parametrizations that the basic finite-to-one map cannot distinguish.

Normalization discipline

The displayed principles do not remove normalization choices. Reciprocity maps, geometric versus arithmetic Frobenius, normalized induction, Whittaker data, additive characters, and Haar measures must be fixed consistently before an equality of factors or packet labels is meaningful.

References

Tasho Kaletha, “Representations of reductive groups over local fields,” §2.3, 2022. arXiv.
Michael Harris, “On the local Langlands correspondence,” 2003. arXiv.