Hypercomplex Representations of the Affine Group

In this paper, we consider representations of the affine group that are induced by hypercomplex characters specifically, double and dual characters instead of complex characters. The double and dual number systems provide a rich algebraic structure that allows for the development of generalized representations, extending beyond the complex Hilbert spaces. We derive the hypercomplex representations by starting from certain subgroups of the affine group. Then, we investigate how these representations act on vector spaces equipped with indefinite inner products, naturally leading to the framework of Krein spaces.