We explicitly construct non-holomorphic cusp forms on the orthogonal group of signature (1 , 8 n+ 1) for an arbitrary natural number n as liftings from Maass cusp forms of level one. In our previous works  and  the fundamental tool to show the automorphy of the lifting was the converse theorem by Maass. In this paper, we use the Fourier expansion of the theta lifts by Borcherds  instead. We also study cuspidal representations generated by such cusp forms and show that they are irreducible and that all of their non-archimedean local components are non-tempered while the archimedean component is tempered, if the Maass cusp forms are Hecke eigenforms. Our non-archimedean local theory relates Sugano’s local theory  to non-tempered automorphic forms or representations of a general orthogonal group in a transparent manner.
ASJC Scopus subject areas
- 数学 (全般)