Sheaves on nilpotent cones, Fourier transform, and a geometric Ringel duality

TL;DR

This paper explores the interplay between sheaves on nilpotent cones, Fourier transform, and a geometric form of Ringel duality, revealing autoequivalences in derived categories relevant to Springer theory and character sheaves.

Contribution

It establishes that the Fourier--Sato transform induces an autoequivalence of the derived category of sheaves on the nilpotent cone, connecting to geometric Ringel duality in type A.

Findings

Fourier--Sato transform restricts to an autoequivalence of the nilpotent cone's derived category.

In type A, this autoequivalence corresponds to geometric Ringel duality.

Enhances understanding of symmetries in Springer theory and character sheaves.

Abstract

Given the nilpotent cone of a complex reductive Lie algebra, we consider its equivariant constructible derived category of sheaves with coefficients in an arbitrary field. This category and its subcategory of perverse sheaves play an important role in Springer theory and the theory of character sheaves. We show that the composition of the Fourier--Sato transform on the Lie algebra followed by restriction to the nilpotent cone restricts to an autoequivalence of the derived category of the nilpotent cone. In the case of $GL_n$, we show that this autoequivalence can be regarded as a geometric version of Ringel duality for the Schur algebra.