What is Riemann-Roch in arithmetic all about?
Have a look at Tate's thesis "Fourier analysis in number theory and Hecke's zeta function" (theorem 4.2.1), where he describes it as an equivariant Poisson summation formula.
In theorem 4.4.1 he proves the functional equation for $L$ functions with it.
It works equally well for function fields. I think this is elaborated in Bump "Automorphic representations" in chapter 3(?).