Relative Aspects in Representation Theory, Langlands Functoriality and Automorphic Forms Aspects relatifs en théorie de la représentation, ​fonctorialité de Langlands et formes automorphes

January-June 2016

Representation theory of real and $p$-adic groups, Automorphic representations, and associated $L$-functions have been among the most important topics of study in Number theory, attracting a lot of attention through many spectacular successes in the recent decades. The proof of the Shimura-Taniyama conjecture, and the Sato-Tate conjecture are two of the highlights besides the Local Langlands conjecture for GL(n) and now through the work of Arthur, the Local Langlands Conjecture for all classical groups. Thus it can be said that, at least for classical groups, representation theory and automorphic forms are reasonably well understood. If one takes this to mean the understanding of $L^{2}(G)$, or of $L^{2}(G(Q) \backslash G(A))$, then the next natural question in this optic is to understand $L^{2}(H \backslash G)$ locally, or $H(Q) \backslash H(A)$ period integrals on $G(Q) \backslash G(A)$ for $H$ a closed subgroup of G. A good framework to understand these questions in 'relative representation theory' is already there, with distinct inputs:

1. The work of Gan-Gross-Prasad, and that of Ichino-Ikeda sets the stage for some of the natural questions and their proposed answers.

2. The work of Sakellaridis and Venkatesh which sets up an ideal stage, that of Spherical subgroups where these questions could be considered.

3. The work of Jacquet and collaborators on Relative Trace formula as a tool to prove global theorems.

Our project hopes to contribute to this theme, using the expertise in Marseille on symmetric varieties.