Langlands' automorphic transfer from reductive groups to linear groups is equivalent to the existence of local and global Fourier transform operators induced by representations of the dual groups, of local and global functional spaces that should be stabilized by these operators and of a Poisson linear form on the global functional spaces that should be fixed by Fourier transform.

Looking for a definition of non-additive Fourier transform operators leads to the crucial question of determining the Fourier transform of the operator of point-wise multiplication of functions. It has to be a generalisation of ordinary additive convolution operators.