The unramified computation of a Shimura integral for $\mathrm{SL}(2)\times \mathrm{GL}(2)$

Abstract

In this note, we revisit the Rankin-Selberg integral of Shimura type for generic representations of ${\mathrm{SL}}_2\times {\mathrm{GL}}_2$, constructed by Ginzburg, Rallis, and Soudry. We give a different and more ``intrinsic'' proof of the unramified computation. In contrast to their proof we avoid local functional equation for the general linear groups but use the Casselman-Shalika formulas for unramified Whittaker functions for ${\mathrm{SL}}_2$ and ${\mathrm{GL}}_2$.

arXiv comment12 pages