Exotic one-parameter semigroups of endomorphisms of a symmetric cone

We construct an exotic one-parameter semigroup of endomorphims of a symmetric cone $C$, whose generator is not the sum of a Lie group generator and an endomorphism of $C$. The question is motivated by the theory of affine processes on symmetric cones, which plays an important role in mathematical finance. On the other hand, theoretical question that we solve in this paper seems to have been implicitly open even much longer then this motivation suggests.