Trying to construct explicitly kernels for Langlands' functoriality 3
Seminar Room 1, Newton Institute
I want to explain part of my present research work on Langlands' functoriality principle. The validity of Langlands' transfer of automorphic representations from an arbitrary reductive group G to a linear group GL(r) is equivalent to the existence of a huge family of "kernel functions" on the product of the adelic groups associated to G and GL(r).
The purpose of these lectures is to propose explicit formulas for these kernel functions (in the everywhere unramified case over function fields F, for the sake of simplicity, even though it could be done as well in the case of number fields, with some extra work at archimidean places). It can be done by combining explicit local constructions and summations over some discrete groups of rational points.
These constructions only make use of function theory on local groups. They can be phrased without any use of global automorphic representations theory. Langlands' functoriality principle is equivalent to the fact that these explicit kernel functions are left invariant by the discrete arithmetic group GL(r,F).