A local-global principle for unipotent characters

TL;DR

This paper extends local-global principles to unipotent characters in finite reductive groups, providing formulas for counting characters of each defect and establishing geometric connections in representation theory.

Contribution

It adapts Dade's and Sp"ath's conjectures to unipotent characters of specific finite groups, introducing compatible parametrizations and a geometric perspective.

Findings

Derived formulas for unipotent character counts in blocks

Established a geometric local-global principle

Constructed compatible parametrizations of unipotent series

Abstract

We obtain an adaptation of Dade's Conjecture and Sp\"ath's Character Triple Conjecture to unipotent characters of simple, simply connected finite reductive groups of type $\bf{A}$, $\bf{B}$ and $\bf{C}$. In particular, this gives a precise formula for counting the number of unipotent characters of each defect $d$ in any Brauer $\ell$-block $B$ in terms of local invariants associated to $e$-local structures. This provides a geometric version of the local-global principle in representation theory of finite groups. A key ingredient in our proof is the construction of certain parametrisations of unipotent generalised Harish-Chandra series that are compatible with isomorphisms of character triples.