Kostka numbers and Fourier duality

TL;DR

This paper explores the connection between Fourier transforms of certain sheaves and dualities in representation theory, extending known relations to general linear groups over finite fields and arbitrary Coxeter groups.

Contribution

It establishes a new link between Fourier duality of perverse sheaves and Deligne-Lusztig duality of unipotent representations, generalizing to all finite Coxeter groups.

Findings

Relates Fourier transforms of sheaves to representation dualities

Extends duality relations to arbitrary finite Coxeter groups

Provides a new perspective on the interplay between geometry and algebra

Abstract

We relate the Fourier transform of perverse sheaves smooth along the coordinate hyperplane configuration in a complex vector space to the Deligne-Lusztig duality of unipotent representations of a general linear group over a finite field. A similar relation is established for arbitrary finite Coxeter groups.