A comparison of two complexes

We prove the conjecture of Lusztig in [5, Section 4]. Given a reductive group over F_q‾[ε]/(ε^r) for some r≥2, there is a notion of a character sheaf defined in [4, Section 8]. On the other hand, there is also a geometric analogue of the character constructed by Gérardin [2]. The conjecture in [5, Section 4] states that the two constructions are equivalent, which Lusztig also proved for r=2,3,4. Here we generalize his method to prove this conjecture for general r. As a corollary we prove that the characters derived from these two complexes are equal.