Rankin-Selberg convolutions for GL( n )×GL( n ) and GL( n )×GL( n −1) for principal series representations

Let k be a local field. Let I v and I_v^' be smooth principal series representations of GL n (k) and GL n -−1 (k), respectively. The Rankin-Selberg integrals yield a continuous bilinear map I_v× I_v^'→ℂ with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map I_v× I_v^'→ℂ with the same invariance property, and show that these integrals equal the Rankin-Selberg integrals up to an explicit constant. Similar results are also obtained for Rankin-Selberg integrals for GL n (k) × GL n (k).