THE SPACE OF INVARIANT FUNCTIONS ON A FINITE LIE-ALGEBRA

We show that the operations of Fourier transform and duality on the sp ace of adjoint-invariant functions on a finite Lie algebra commute wit h each other. This result is applied to give formulae for the Fourier transform of a ''Brauer function''-i.e. one whose value at X depends o nly on the semisimple part X(s) of X and for the dual of the convoluti on of any function with the Steinberg function. Geometric applications include the evaluation of the characters of the Springer representati ons of Weyl groups and the study of the equivariant cohomology of loca l systems on G/T, where T is a maximal torus of the underlying reducti ve group G.