Invariant Representations of $\mathrm {GSp}(2)$ under Tensor Product with a Quadratic Character

Let $F$ be a number field or a $p$-adic field. The author introduces in Chapter 2 of this work two reductive rank one $F$-groups, $\mathbf{H_1}$, $\mathbf{H_2}$, which are twisted endoscopic groups of $\mathrm{GSp}(2)$ with respect to a fixed quadratic character $\varepsilon$ of the idele class group of $F$ if $F$ is global, $F^\times$ if $F$ is local. When $F$ is global, Langlands functoriality predicts that there exists a canonical lifting of the automorphic representations of $\mathbf{H_1}$, $\mathbf{H_2}$ to those of $\mathrm{GSp}(2)$. In Chapter 4, the author establishes this lifting in terms of the Satake parameters which parameterize the automorphic representations. By means of this lifting he provides a classification of the discrete spectrum automorphic representations of $\mathrm{GSp}(2)$ which are invariant under tensor product with $\varepsilon$.