The Adjoint L-function of Su2,1

To the memory of my grandfather, Harold H. Hensold, Jr. In these notes we give a construction for a certain L-function attached to a globally generic automorphic representation of the quasi-split unitary group in 3 variables associated to a quadratic extension E/F of number fields. Recall that the finite Galois form of the L-group of this group is a semidirect product of GL3(C) and Gal(E/F ). The representation we consider has the property that when restricted to GL3(C) it is the adjoint representation of this group. For this reason, we refer to the associated L function as the adjoint L function. In fact, ther are two representations of GL3(C)⋊Gal(E/F ) with the above property– related to one another by twistng by the unique nontrivial one dimensional representation of Gal(E/F ). We pin down precisely which on we are talking about in section 1.1 below. Let us mention that a small modification of this construction gives the other. The construction is a slight modification of that given in [3].