Orthogonality relations for k-hypermonogenic automorphic forms

In this paper we deal with monogenic and k-hypermonogenic automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. Monogenic automorphic forms are exactly the 0-hypermonogenic automorphic forms. In the first part we establish an explicit relation to Maaß wave forms. In the second part we introduce Clifford algebra valued k-hypermonogenic cusp forms. We construct k-hypermonogenic Poincaré series which provide non-trivial examples of such functions. Then we establish a decomposition theorem of the spaces of k-hypermonogenic automorphic forms in terms of a direct orthogonal sum of the spaces of k-hypermonogenic Eisenstein series and that of k-hypermonogenic cusp forms. Orthogonality is understood in the sense of the Petersson inner product.