The Fourier transforms of invariant functions on finite reductive Lie algebras are due to T.A. Springer (1976) in connection with the geometry of nilpotent orbits. In this book the author studies Fourier transforms using Deligne-Lusztig induction and the Lie algebra version of Lusztig's character sheaves theory. He conjectures a commutation formula between Deligne-Lusztig induction and Fourier transforms that he proves in many cases. As an application the computation of the values of the trigonometric sums (on reductive Lie algebras) is shown to reduce to the computation of the generalized Green functions and to the computation of some fourth roots of unity.

Includes supplementary material: sn.pub/extras

Inhalt
Preface.- Introduction.- Connected Reductive Groups and their Lie Algebras.- Deligne-Lusztig Induction.- Local Systems and Perverse Shaeves.- Geometrical Induction.- Deligne-Lusztig Induction and Fourier Transforms.- Fourier Transforms of the Characteristic Functions of the Adjoint Orbits.- References.- Index.