Fourier Transforms Of Invariant Functions On Finite Reductive Lie Algebras (lecture Notes In Mathematics, Vol. 1859)

The study of Fourier transforms of invariant functions on finite reductive Lie algebras has been initiated by 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.

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.

"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."--Jacket 1. Introduction....Pages 1-4 2. Connected Reductive Groups and Their Lie Algebras....Pages 5-31 3. Deligne-Lusztig Induction....Pages 33-43 4. Local Systems and Perverse Sheaves....Pages 45-60 5. Geometrical Induction....Pages 61-113 6. Deligne-Lusztig Induction and Fourier Transforms....Pages 115-149 7. Fourier Transforms of the Characteristic Functions of the Adjoint Orbits....Pages 151-158 References....Pages 159-162 Index....Pages 163-165