On the definition of unipotent representations

TL;DR

This paper provides an alternative non-cohomological definition for a map linking irreducible representations of a finite field group to semisimple classes in the dual group, valid for large q.

Contribution

It introduces a new non-cohomological approach to define the map originally established by Deligne and the author in 1976.

Findings

The new definition applies when q is sufficiently large.

It offers a different perspective on unipotent representations.

The approach simplifies understanding of the representation map.

Abstract

Let G be a connected reductive group over a finite field F_q. A map from irreducible representations of G(F_q) to semisimple classes in the dual group was defined by a cohomological method by Deligne and the author in 1976. Here we show that when q is large enough this map has an alternative (non-cohomological) definition.