Special Bessel models with the local Maass relation and non-tempered automorphic forms on orthogonal groups
I will provide some general class of automorphic forms or representations on a general orthogonal group, having a non-tempered non-archimedean local component. We call them non-tempered automorphic forms or representations. It is a fundamental problem to find non-tempered cusp forms, which are nothing but counterexamples to the Ramanujan conjecture. The general class above includes the cusp forms given by the Oda-Rallis-Schiffmann lifting to O(2,m) and non-holomorphic lifting to O(1,8n+1) recently given by the joint work with Yingkun Li and Ameya Pitale. Such general class is given by means of the notion of the special Bessel model and the local Maass relation.