Cuspidal representations of Lie algebras and modules finitely generated over Cartan subalgebra

• Grant number: 22/05915-6
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• Start date: August 22, 2022
• End date: June 21, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Iryna Kashuba
• Grantee: Eduardo Monteiro Mendonça
• Supervisor: Olivier Mathieu
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Institution abroad: Université Claude Bernard Lyon 1, France
• Associated to the scholarship: 20/14313-4 - Cuspidal Representations of Lie Algebras, BP.DR

Abstract

The goal of this project is to study combinatoric aspects of irreducible cuspidal representations of a simple lie algebra g and their relation with g-modules that are finitely generated by the universal envelop algebra of h, a fixed Cartan sub algebra of g. Cuspidal representation was classified by Olivier Mathieu, that also gave a formula for their weight spaces dimensions. We intend to refine his description with a combinatoric point of view, and that way obtain a formula of such dimensions only with positive coefficients. Furthermore, we have as second goal to study the weighting functor, the functor that map a g-module of finite type over U(h) to a weight module. By studying tensor modules, we hope to construct a inverse functor to the weighting functor. (AU)