This preprint constructs a modular coherent Springer object S DCoh (Gₖ/ Gₖ) for a split finite reductive group G (Fq) at a prime q, assembled stratumwise from Lusztig–Jordan strata N_/H_ on the Langlands-dual parameter stack. The main result identifies the derived endomorphism algebra REnd (S) ₆ ₂₎₍₉ (₆) k\, CG (g), realizing the Morita reduction of the Drinfeld double D (kG) (each sector D (kG) e₆ Mat₍₆ (kCG (g) ) contributes its Morita-equivalent centralizer algebra). The proof rests on three structural inputs: (A) integral Z_-flatness/perfectness of the stratumwise Springer pushforward, (B) stratumwise endomorphism identification REnd (S_) kCG (g_), and (C) cross-stratum orthogonality across distinct semisimple classes. As a formal consequence (Rickard derived Morita), we obtain a derived equivalence Dᵇ (D (kG) -Mod) Dᵇ (IndCoh₍₈₋ (Gₖ/ Gₖ) ). Together with the companion ATB preprint (which forces the K₀-operator T₀₃₉=adj (C) /|D| for any such equivalence), this supplies the “existence” side of the modular Langlands dictionary while ATB supplies the K₀-forcing certificate.
Matthew Eltgroth (Wed,) studied this question.