Spin L-functions for GSO10 and GSO12

In this paper we consider two Rankin-Selberg integrals which were discovered by David Ginzburg, and announced in [G-H1]. These integrals are defined on a split form of GSO2n (see below for precise definition), and involve a generic cuspidal automorphic representation of this group. We content ourselves with showing that both integrals unfold to Eulerian integrals involving Whittaker functions, and computing the contributions from the unramified places. In each case we get a product of two partial Langlands L functions, at least one of which is a “Spin” L-function. Recall that a Langlands L function requires two pieces of data. The first is an automorphic representation π defined on some group G, from which we obtain a family, indexed by all but finitely many places of our global field, of semisimple conjugacy classes in a certain complex Lie group G. The second is a finite dimensional representation r of that complex Lie group. Recall also that the special orthogonal group SO2n is not simply connected, but possesses a simply connected double cover, known as the spin group. This group possesses two fundamental representations, usually called the half-spin representations, which do not factor through the projection. By a Spin L function we mean a Langlands L function in which the role of r is played by either of these two representations. In this paper we consider an integral on GSO10 and a similar one on GSO12. In both cases we unfold the integral and compute the contribution from the unramified places (being Archimedean is treated as a form of ramification), obtaining a product of partial Langlands L functions. These are: in the GSO10 case L(3s1 − 2s2, π, Spin)L(3s1 + 2s2 − 2, π, Spin) and in the GSO12 case L(5s2 − 2, π ⊗ χ2, St)L(4s1 − 3 2 , π ⊗ χ1, Spin).