Theta correspondence I - Semistable range: Construction and irreducibility

The main purpose of this paper is to study theta correspondence from representation theoretic point of view. There are two problems we have in mind. One is the construction of unipotent representations of semisimple Lie group. The other is the parametrization of unitary dual of semisimple Lie group. In the first paper of this series, we define semistable range in the domain of theta correspondence. Roughly speaking, semistable range is a range where one can define certain averaging operator analytically In this paper, we prove that if the averaging operator is not vanishing, then it produces the theta correspondence. This paper pave the way to study theta correspondence using analytic machinery.