Optimal left cohomological amplitude of the coarse Langlands functor LG,coarse

Determine the optimal left cohomological amplitude of the coarse Langlands functor LG,coarse: D-mod¹(Bun_G) → QCoh(LS_G); specifically, ascertain the minimal integer d (if any) such that the shifted functor LG,coarse[−d] is left t-exact, or equivalently, precisely characterize the left t-structure bounds of LG,coarse.

Background

The authors prove that LG,coarse has bounded cohomological amplitude on the left (Theorem 2.1.2), giving an explicit bound depending on the dimension of LS_G and the dimensions of Bun_G and Bun_{N,ρ(ω_X)}. They note this bound may not be sharp and provide the example G = T (a torus), where LG,coarse is left t‑exact without any shift.

They therefore pose the problem of identifying the exact left cohomological amplitude (i.e., the sharp bound) for LG,coarse beyond current estimates.