Euler factorization of global integrals

We consider integrals of cuspforms f on reductive groups G defined over numberfields k against restrictions ι∗E of Eisenstein series E on “larger” reductive groups G̃ over k via imbeddings ι : G → G̃. We give hypotheses sufficient to assure that such global integrals have Euler products. At good primes, the local factors are shown to be rational functions in the corresponding parameters q−s from the Eisenstein series and in the Satake parameters q−si coming from the spherical representations locally generated by the cuspform. The denominators of the Euler factors at good primes are estimated in terms of “anomalous” intertwining operators, computable via orbit filtrations on test functions. The standard intertwining operators (attached to elements of the (spherical) Weyl group) among these unramified principal series yield symmetries among the anomalous intertwining operators, thereby both sharpening the orbit-filtration estimate on the denominator and implying corresponding symmetry in it. Finally, we note a very simple dimension-counting heuristic for fulfillment of our hypotheses, thereby giving a simple test to exclude configurations ι : G→ G̃, E, as candidates for Euler product factorization. Some simple examples illustrate the application of these ideas.