Evidence points to the assertion (unproven, in general) that unipotent orbits in the Langlands dual group \(\widehat{G}\) depend functorially with respect to morphisms \(H\rightarrow G\) of semisimple groups. The object of this paper is to investigate this conjectural relationship.
The paper has a total of 10 conjectures (7 of his own, 3 due to Kazhdan, Venkatesh, and Waldspurger) explaining this connection, discussion making the relationship more explicit, and proofs of relationships between the various conjectures.
The work is primarily based on many deep results in the theory of automorphic forms, for example, J. Arthur, “Unipotent automorphic representations: conjectures,” Astérisque 171–172, 13–71 (1989; Zbl 0728.22014); M. Burger and P. Sarnak, “Ramanujan duals. II”, Invent. Math. 106, 1–11 (1991; Zbl 0774.11021); and two preprints of the author [L. Clozel, “Spectral theory of automorphic forms” (2002) and the second has now been published L. Clozel and E. Ullmo, Hida, Haruzo (ed.) et al., Contributions to automorphic forms, geometry, and number theory. Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birthday, Johns Hopkins University, Baltimore, MD, USA, May 14–17, 2002. Baltimore, MD: Johns Hopkins University Press, 193–254 (2004; Zbl 1068.11042)].