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Fourier transform on graded Lie algebras

Tamanna Chatterjee

TL;DR

The paper addresses extending Lusztig's Fourier transform on graded Lie algebras to positive characteristic, embedding the problem in the modular Springer framework via $G_0$-equivariant derived categories of $\mathfrak{g}_n$ and parity sheaves. Its approach combines a detailed analysis of parity sheaves, cuspidal pairs, and the graded induction/restriction machinery with a careful study of the Fourier-Sato transform on $\mathfrak{g}_n$ and its dual $\mathfrak{g}_{-n}$. Under Assumption on the characteristic and two Mautner conjectures, it proves that the Fourier-Sato transform sends cuspidal pairs to cuspidal pairs and parity complexes to parity complexes, and that restriction preserves parity, while analyzing the behavior of induction/restriction in the graded setting. These results pave the way toward a modular, graded version of the Springer correspondence and a potential 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 $χ:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $χ(\mathbb{C}^\times)$. 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.

Fourier transform on graded Lie algebras

TL;DR

The paper addresses extending Lusztig's Fourier transform on graded Lie algebras to positive characteristic, embedding the problem in the modular Springer framework via -equivariant derived categories of and parity sheaves. Its approach combines a detailed analysis of parity sheaves, cuspidal pairs, and the graded induction/restriction machinery with a careful study of the Fourier-Sato transform on and its dual . Under Assumption on the characteristic and two Mautner conjectures, it proves that the Fourier-Sato transform sends cuspidal pairs to cuspidal pairs and parity complexes to parity complexes, and that restriction preserves parity, while analyzing the behavior of induction/restriction in the graded setting. These results pave the way toward a modular, graded version of the Springer correspondence and a potential block decomposition in positive characteristic.

Abstract

In this paper we study the Fourier transform on graded Lie algebras. Let be a complex, connected, reductive, algebraic group, and be a fixed cocharacter that defines a grading on , the Lie algebra of . Let be the centralizer of . Here under some assumptions on the field and also assuming two conjectures for the group , 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.
Paper Structure (12 sections, 14 theorems, 70 equations)

This paper contains 12 sections, 14 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Induction and restriction
  4. The sets $\mathscr{I}(G,\Bbbk)$, $\mathscr{I}(\mathfrak{g}_n,\Bbbk)$ and cuspidal pairs
  5. Parity sheaves
  6. Restriction
  7. Good double cosets
  8. Bad double cosets
  9. Composition of $\mathop{\mathrm{Ind}}\nolimits^{\mathfrak{g}}_{\mathfrak{p}}$ and $\mathop{\mathrm{Res}}\nolimits^{\mathfrak{g}}_{\mathfrak{p}}$
  10. Fourier Sato transform
  11. Fourier transform on vector spaces
  12. Fourier transform on graded Lie algebras

Key Result

Theorem 1.1

$\mathop{\mathrm{Res}}\nolimits^{\mathfrak{g}}_{\mathfrak{p}}$ sends parity complexes to parity complexes when the characteristic of $\Bbbk$ satisfies Assumption assume.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.5: Ch, Theorem 29
  • Theorem 2.6
  • Theorem 2.7: Ch, Theorem. 26
  • Conjecture 2.8
  • Conjecture 2.9
  • ...and 19 more