On the $\mathrm{\ell }$-adic Fourier transform and the determinant of the middle convolution

We study the relation of the middle convolution to the $\mathrm{\ell }$-adic Fourier transformation in the 'etale context. Using Katz’ work and Laumon’s theory of local Fourier transformations we obtain a detailed description of the local monodromy and the determinant of Katz’ middle convolution functor ${\text{MC}}_{\chi }$ in the tame case. The theory of local $ϵ$-constants then implies that the property of an 'etale sheaf of having an at most quadratic determinant is often preserved under ${\text{MC}}_{\chi }$ if $\chi$ is quadratic.