Fourier coefficients and a filtration on Shv(Bun_G)

We define a filtration by DG-subcategories on the DG-category Shv(Bun_G) of sheaves on the moduli of G-torsors on a curve, which is stable under the action of Hecke functors. We formulate a conjecture relating this filtration with another filtration on the spectral side of the categorical geometric Langlands conjecture. We also formulate a conjectural compatibility with the parabolic induction.