Equivalence Ho(H_GL2) ≃ Sing(X_q,GL2) recovers Grosse-Klönne's mod-p Langlands correspondence for Hecke modules, with explicit descriptions of Ho(H_G) for SL2(F) and PGL2(F).
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Pro-$p$ Iwahori-Hecke modules in semisimple rank one and singularity categories
Equivalence Ho(H_GL2) ≃ Sing(X_q,GL2) recovers Grosse-Klönne's mod-p Langlands correspondence for Hecke modules, with explicit descriptions of Ho(H_G) for SL2(F) and PGL2(F).