A parametrization of unipotent representations

TL;DR

This paper introduces a new parametrization method for unipotent representations of split semisimple groups over finite fields, linking them to pairs of left cell representations of the Weyl group within the same two-sided cell.

Contribution

It defines a novel map connecting unipotent representations to pairs of Weyl group representations, providing a new way to parametrize unipotent representations.

Findings

Established a map from unipotent representations to pairs of Weyl group representations.

Provided a parametrization framework for unipotent representations.

Linked unipotent representations to the structure of Weyl group cells.

Abstract

We define a map from the unipotent representations of a split semisimple group over a finite field to (essentially) the set of pairs of left cells representations of the Weyl group in the same two-sided cell. We use this map to parametrize the unipotent representations.