An explicit construction of non-tempered cusp forms on $$O(1,8n+1)$$

Abstract

We explicitly construct non-holomorphic cusp forms on the orthogonal group of signature $$(1,8n+1)$$ for an arbitrary natural number n as liftings from Maass cusp forms of level one. In our previous works [31] and [24] 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 [4] 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 [39] to non-tempered automorphic forms or representations of a general orthogonal group in a transparent manner.