Linear Koszul duality and Fourier transform for convolution algebras

In this paper we prove that the linear Koszul duality isomorphism for convolution algebras in $K$-homology of [MR3] and the Fourier transform isomorphism for convolution algebras in Borel–Moore homology of [EM] are related by the Chern character. So, Koszul duality appears as a categorical upgrade of Fourier transform of constructible sheaves. This result explains the connection between the categorification of the Iwahori–Matsumoto involution for graded affine Hecke algebras in [EM] and for ordinary affine Hecke algebras in [MR3].