nLab Langlands functoriality

Idea

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

$${}^L G\to {}^L H$$

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

Known results

Langlands functoriality is now known in the case of the natural embedding $G\subset \GL_n$ 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 Langlands’ work for $\GL_2$) 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 $\GL_n$ by using unitary Shimura varieties associated to some twisted (unitary) forms of $\GL_n$.

James Newton and Jack Thorne proved symmetric power functoriality for holomorphic modular forms in NewtonThorne19 and NewtonThorne20.

References

• Minhyong Kim, A superficial introduction to Langlands functoriality, slides

• Jae-Hyun Yang, Langlands Functoriality Conjecture, Kyungpook Math. J. vol. 49, no. 2 (June 2009), 355-387 (arXiv:0808.0917)

• James Newton and Jack Thorne, Symmetric power functoriality for holomorphic modular forms, arxiv:1912.11261

• James Newton and Jack Thorne, Symmetric power functoriality for holomorphic modular forms, II, arxiv:2009.07180