A Comparison of Two Complexes

TL;DR

This paper proves Lusztig's conjecture that two different geometric constructions of character sheaves on certain algebraic groups are equivalent for any integer r ≥ 2, extending previous results for small r.

Contribution

The paper generalizes Lusztig's method to prove the equivalence of two character sheaf constructions for all r ≥ 2, confirming their equality.

Findings

Proved Lusztig's conjecture for all r ≥ 2.

Established the equality of characters derived from the two complexes.

Extended previous proofs from small r to general r.

Abstract

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.