[Paper Review] Sur les caractères des groupes réductifs finis : applications aux groupes spéciaux linéaires et unitaires

Cédric Bonnafé|arXiv (Cornell University)|Apr 5, 2005

Advanced Algebra and Geometry3 citations

TL;DR

This paper develops a theoretical algorithm for computing the character table of finite reductive groups of type A with non-connected center, such as special linear and unitary groups, when the field size is sufficiently large. It establishes a parametrization of irreducible characters and character sheaves, proves that characteristic functions of character sheaves are Fourier transforms of irreducible characters (verifying Lusztig's conjecture), and provides a systematic framework for computing the full character table using Lusztig's theory of character sheaves and Gelfand-Graev characters in the non-connected center setting.

ABSTRACT

In the first chapters, this paper contains a survey on the theory of ordinary characters of finite reductive groups with non-connected centre. The last chapters are devoted to the proof of Lusztig's conjecture on characteristic functions of character sheaves for all finite reductive groups of type A, split or not.

Motivation & Objective

• To provide a theoretical algorithm for computing the character table of finite reductive groups of type A with non-connected center, such as SL(n,q) and SU(n,q).
• To extend the theory of character sheaves to groups with non-connected center, particularly focusing on those whose support intersects the regular unipotent class.
• To establish a parametrization of irreducible characters and character sheaves in the case of large finite field size.
• To verify Lusztig's conjecture that characteristic functions of character sheaves are Fourier transforms of irreducible characters in this setting.

Proposed method

• Utilizes Lusztig's theory of character sheaves, focusing on F-stable perverse sheaves with support meeting the regular unipotent class.
• Applies the theory of Gelfand-Graev characters and semisimple characters to analyze the representation theory of groups with non-connected center.
• Employs the induction functor of Lusztig to compute the character table in the basis of irreducible characters.
• Uses the Fourier transform of irreducible characters to construct characteristic functions of character sheaves, verifying Lusztig's conjecture.
• Relies on the parametrization of irreducible characters via Lusztig's method, adapted to the non-connected center case.
• Applies results from Deligne-Lusztig theory and the geometry of unipotent and semisimple classes to derive the character table.

Experimental results

Research questions

• RQ1 How can the character table of finite special linear and unitary groups be computed when the center is not connected?
• RQ2 What is the role of character sheaves whose support intersects the regular unipotent class in the representation theory of finite reductive groups with non-connected center?
• RQ3 To what extent do the characteristic functions of character sheaves coincide with the Fourier transforms of irreducible characters in this setting?
• RQ4 Can a complete parametrization of irreducible characters and character sheaves be achieved for finite reductive groups of type A with non-connected center?
• RQ5 What modifications are needed in Lusztig's theory of character sheaves when the center of the reductive group is not connected?

Key findings

• For finite reductive groups of type A with non-connected center and sufficiently large field size, a complete parametrization of irreducible characters is obtained.
• The characteristic functions of character sheaves are shown to be Fourier transforms of irreducible characters, confirming Lusztig's conjecture in this case.
• A theoretical algorithm for computing the full character table of such groups is established, based on the parametrization of irreducible characters and character sheaves.
• The theory of Gelfand-Graev and semisimple characters is extended to the non-connected center setting, providing key tools for the parametrization.
• The support of character sheaves intersecting the regular unipotent class is identified as the natural analogue of semisimple characters in this context.
• The induction functor of Lusztig is explicitly computed in the basis of irreducible characters, enabling the construction of the character table.