Fourier transforms on invariant functions on finite reductive Lie algebras
Let \(k\) be an algebraic closure of a finite field \({\mathbb F}_q\) with \(q\) a power of a prime \(p\). Let \(G\) be a connected reductive algebraic group over \(k\) which is defined over \({\mathbb F}_q\) and let \(\mathfrak g\) be the Lie algebra of \(G\). Denote by \(F: G\to G\) and \(F: {\mathfrak g}\to{\mathfrak g}\) the Frobenius endomorphisms corresponding to the \({\mathbb F}_q\)-structures. Let \(G^F=\{ g\in G: F(g)=g\}\), \({\mathfrak g}^F=\{X\in{\mathfrak g}: F(X)=X\}\). Then \(G^F\) is a finite group of Lie type and \({\mathfrak g}^F\) is a finite reductive Lie algebra. The group \(G^F\) acts on \({\mathfrak g}^F\) via the adjoint representation. This book is devoted to the study of Fourier transforms of adjoint invariant functions on \({\mathfrak g}^F\) in the sense of \textit{T.~A.~Springer} [Invent. Math. 36, 173--207 (1976; Zbl 0374.20054)]. The table of contents in the book is as follows. 1. Introduction. 2. Connected reductive groups and their Lie algebras. 3. Deligne-Lusztig induction. 4. Local systems and perverse sheaves. 5. Geometrical induction. 6. Deligne-Lusztig induction and Fourier transforms. 7. Fourier transforms of the characteristic functions of the adjoint orbits. 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 and he proves this formula in many cases. The trigonometric sums on finite reductive Lie algebras are the Fourier transforms of the characteristic functions of the \(G^F\)-orbits of \({\mathfrak g}^F\). As an application the computation of the values of the trigonometric sums on \({\mathfrak g}^F\) is shown to reduce to the computation of the generalized Green functions and to the computation of the Lusztig constants which are known in many cases.
scientific article; zbMATH DE number 2133293