On the symmetric square: definitions and lemmas

We define the symmetric square lifting for admissible and automorphic representations, from the group $H = {H_0} = {\text{SL}}(2)$, to the group $G = {\text{PGL}}(3)$, and derive its basic properties. This lifting is defined by means of Shintani character relations. The definition is suggested by the computation of orbital integrals (stable and unstable) in our On the symmetric square: Orbital integrals, Math. Ann. 279 (1987), 173-193. It is compatible with dual group homomorphisms ${\lambda _0}:\widehat{H} \to \widehat{G}$ and ${\lambda _1}:{\widehat{H}_1} \to \widehat{G}$, where ${H_1} = {\text{PGL}}(2)$. The lifting is proven for induced, trivial and special representations, and both spherical functions and orthogonality relations of characters are studied.