A Comparison of Two Complexes

In this paper we prove the conjecture of Lusztig in "Generic character sheaves on groups over $\mathbf{k}[\epsilon]/(\epsilon^r)$." Given a reductive group over $\mathbb{F}_q$ for some $r\geq 2$, there is a notion of a character sheaf defined in "Character sheaves and generalizations" by Lusztig. On the other hand, there is also a geometric analogue of the character constructed by G\'erardin. The conjecture 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.