Germ expansion in arbitrary characteristics

Let F F be a local field of characteristic p p and G G be a connected reductive F F -group. Shalika’s germ expansion of orbital integrals of regular semisimple elements near the identity, when it exists, is a sum indexed by the set U G U^G of unipotent conjugacy classes in G ( F ) G(F) . For G = S L ( 2 ) G=SL(2) the set U G U^G is finite if p ≠ 2 p≠ 2 while it is uncountable if p = 2 p= 2 . As a consequence, Shalika’s germ expansion for elliptic elements does not make sense for S L ( 2 , F ) SL(2,F) if p = 2 p=2 . On the other hand endoscopic expansion of elliptic orbital integrals exists in arbitrary characteristics, at least for S L ( 2 ) SL(2) , and yields a germ expansion, equivalent if p ≠ 2