The space of invariant functions on a finite Lie algebra

Abstract

We show that the operations of Fourier transform and duality on the space of adjoint-invariant functions on a finite Lie algebra commute with each other. This result is applied to give formulae for the Fourier transform of a “Brauer function”—i.e. one whose value at X X depends only on the semisimple part X s X_s of X X and for the dual of the convolution of any function with the Steinberg function. Geometric applications include the evaluation of the characters of the Springer representations of Weyl groups and the study of the equivariant cohomology of local systems on G / T G/T , where T T is a maximal torus of the underlying reductive group G G .