Fourier transform on graded Lie algebras

TL;DR

This paper investigates the behavior of the Fourier transform on graded Lie algebras, showing it preserves parity complexes under certain conditions, and explores primitive pairs' role in block decomposition in positive characteristic.

Contribution

It establishes that the Fourier transform sends parity complexes to parity complexes on graded Lie algebras under specific assumptions and examines the role of primitive pairs in this context.

Findings

Fourier transform preserves parity complexes under certain assumptions.

Primitive pairs are key to understanding block decomposition in graded Lie algebras.

The study advances towards proving block decomposition in positive characteristic.

Abstract

In this paper we study the Fourier transform on graded Lie algebras. Let $G$ be a complex, connected, reductive, algebraic group, and $\chi:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $\chi(\mathbb{C}^\times)$. Here under some assumptions on the field $\Bbbk$ and also assuming two conjectures for the group $G$, we prove that the Fourier transform sends parity complexes to parity complexes. Primitive pairs have played an important role in Lusztig's paper \cite{Lu} to prove a block decomposition in the graded setting. A long term goal of this project is to prove a similar block decomposition in positive characteristic. In this paper we have tried to understand the primitive pair and its relation with the Fourier transform.