If we consider ${\cal N}=1$ renormalizeable chiral gauge theories, specifically discussed in section 27.4 of Weinberg's Quantum Theory of Fields, Supersymmetry book, should the supercurrent be gauge invariant?
Any observable of a gauge invariant theory must also be gauge invariant. However, the supercurrent is a Grassmann variable, and thus is not observable.
I ask this because in order to calculate the supercurrent, Weinberg argues to calculate it in a specific gauge to make the analysis easier. This is only possible if the supercurrent itself is gauge invariant.