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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