Nilpotent Orbits in Representation Theory

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

DMS-0201826 Principal Investigator: Eric Sommers [esommers@math.umass.edu] Abstract: The principal investigator will study several questions in representation theory which are related to nilpotent orbits in Lie algebras. More specifically, the investigator will study a new duality map that he defined (and that has recently been extended by Achar). He will continue to study the Lusztig bijection. He will try to complete the determination of which nilpotent orbits in the exceptional Lie algebras have normal closure and prove results (in all Lie algebras) about the structure of functions on covers of nilpotent orbits. Representation theory is a branch of modern algebra that is concerned with understanding symmetries. A central idea is that complicated algebraic or geometric structures can be represented by a certain set of matrices (arrays of numbers), which are easier to understand than the original structure. For example, the symmetries of the square can be represented (in one possible way) by a certain set of eight two-by-two matrices. This view of representation theory has many applications in chemistry and physics. The investigator's work will contribute to understanding the representation theory of Lie algebras and Lie groups.