Iwahori-Matsumoto Involution and Linear Koszul Duality

We use linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras, to give a geometric realization of the Iwahori-Matsumoto involution of affine Hecke algebras. More generally, we prove that linear Koszul duality is compatible with convolution in a general context related to convolution algebras.