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\begin{document}

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\section*{Langlands functoriality}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{known_results}{Known results}\dotfill \pageref*{known_results} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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

\begin{displaymath}
{}^L G\to {}^L H
\end{displaymath}
between the \href{http://en.wikipedia.org/wiki/Langlands_dual}{L-groups} of two reductive groups over $\Q$ should induce a transfer map from automorphic representations for $G$ to automorphic representations of $H$.

\hypertarget{known_results}{}\subsection*{{Known results}}\label{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 $\GL_n$.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Minhyong Kim]], \emph{A superficial introduction to Langlands functoriality}, \href{http://people.maths.ox.ac.uk/kimm/lectures/functoriality.pdf}{slides}

\end{itemize}


\end{document}