nLab
Langlands functoriality

Contents

Contents

Idea

Langlands functoriality is a generalization of Langlands conjectures that essentially tells us that a LL-homomorphism

LGā†’ LH{}^L G\to {}^L H

between the L-groups of two reductive groups over Q\Q should induce a transfer map from automorphic representations for GG to automorphic representations of HH.

Known results

Langlands functoriality is now known in the case of the natural embedding GāŠ‚GL nG\subset \GL_n of a classical (orthogonal, symplectic or unitary) group in the general linear group through its canonical representation, following the work of James Arthur on the Arthur-Selberg trace formula and on endoscopy for classical groups.

It is also known (starting from Jacquet and Langlandsā€™ work for GL 2\GL_2) in the case of inner forms of a group: there is an equivalence between automorphic representations on the group and automorphic representations on its inner form. This was used by Harris and Taylor to prove instances of the local Langlands correspondence for GL n\GL_n by using unitary Shimura varieties associated to some twisted (unitary) forms of GL n\GL_n.

References

Last revised on November 6, 2018 at 01:27:39. See the history of this page for a list of all contributions to it.