ON THE ARCHIMEDEAN THEORY OF RANKIN-SELBERG CONVOLUTIONS FOR 202L+1XGLN

In this paper, we study the local theory over an archimedean field F o f certain Rankin-Selberg convolutions for pairs of generic representat ions (pi, tau) of SO2l+1 (F) and GL(n) (F). The corresponding local in tegrals involve Whittaker functions of pi and sections of the represen tation rho(tau,s) of SO2n (F), induced from tau (X) \det.\s-1/2, viewe d as a representation of the ''Siegel'' parabolic subgroup. The integr als converge absolutely for Re(s) large enough and are shown to have a meromorphic continuation in s to the whole plane, to a continuous bil inear form on pi x rho(tau,s), which satisfies certain equivariance pr operties. These properties determine such bilinear forms in an essenti ally unique way. An important ingredient here is an application of Wal lach's results on asymptotics of matrix coefficients (and variations). Using all this, we compute the corresponding gamma factors which turn to be, by results of Shahidi, the Artin gamma factors.