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Fourier transforms on a semisimple Lie algebra over \(F_ q\). (English) Zbl 0654.20047
Algebraic groups, Proc. Symp., Utrecht/Neth. 1986, Lect. Notes Math. 1271, 177-188 (1987).
[For the entire collection see Zbl 0619.00008.]
Let \({\mathfrak g}\) be the Lie algebra of a reductive connected algebraic group G over k, an algebraic closure of the finite prime field \(F_ p\). In this paper we shall assume that p is large. We assume chosen a non- singular G-invariant symmetric bilinear form \(<.,.>: {\mathfrak g}\times {\mathfrak g}\to k\). If we are given (a) an \(F_ q\)-rational structure on G (hence on \({\mathfrak g})\) with Frobenius map F, such that \(<.,.>\) is defined over \(F_ q\), we can define the Fourier transform of a function \(f: {\mathfrak g}^ F\to \bar Q_{\ell}\) to be the function (b) \(\hat f:\) \({\mathfrak g}^ F\to \bar Q_{\ell}\), \(\hat f(\xi)=\sum_{\xi'\in {\mathfrak g}^ F}\Psi <\xi,\xi'>f(\xi')\), where \(\psi: F_ q\to \bar Q^*_{\ell}\) is a fixed non-trivial character. (c) Let N be the variety of nilpotent elements in \({\mathfrak g}\). The purpose of this paper is to describe those G F-invariant functions \(f: {\mathfrak g}^ F\to \bar Q_{\ell}\) such that both f and \(\hat f\) vanish on \({\mathfrak g}^ F-N^ F\).

MSC:
20G05 Representation theory for linear algebraic groups
17B20 Simple, semisimple, reductive (super)algebras
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
20G40 Linear algebraic groups over finite fields