ON THE SYMMETRICAL SQUARE - DEFINITIONS AND LEMMAS

We define the symmetric square lifting for admissible and automorphic representations, from the group H = H-0 = SL(2), to the group G = PGL( 3), and derive its basic properties. This lifting is defined by means of Shintani character relations. The definition is suggested by the co mputation of orbital integrals (stable and unstable) in our On the sym metric square. Orbital integrals, Math. Ann. 279 (1987), 173-193. It i s compatible with dual group homomorphisms lambda-0: H --> G and lambd a-1: H-1 --> G, where H-1 = PGL(2). The lifting is proven for induced, trivial and special representations, and both spherical functions and orthogonality relations of characters are studied.