We prove a unified modular singular-support theory for the adjoint quotient stack by two independent constructions, identify the nilpotent spectral category by Springer generation, establish formality and derived Morita equivalence on the modular spectral side, and give an explicit tame depth-zero (GLₙ) finite-field realization together with motivic Euler-product and fiber-factorization results. The paper also includes extensions to bad primes, selected quasi-smooth derived stacks, and the exotic nilpotent cone in type (C) at (=2).
Matthew Eltgroth (Fri,) studied this question.