On the occurrence of admissible representations in the real Howe correspondence in stable range

Abstract

Let \(({\rm G, G'}) \subset {\rm Sp}({\rm W})\) be an irreducible real reductive dual pair of type I in stable range, with G the smaller member. In this note, we prove that all irreducible genuine representations of \(\tilde{\rm G}\) occur in the Howe correspondence. The proof uses structural information about the groups forming a reductive dual pair and estimates of matrix coefficients.