Shintani functions and automorphic \(L\)-functions for \(GL(n)\)
This paper develops an idea due to T. Shintani. Suppose \(k\) is a global field; we let \(G=GL_n/k\) and \(G_0=GL_{n-1}/k\). We can regard \(G_0\) as a subgroup of \(G\), embedded in the `upper left corner'. The authors consider a square-integrable automorphic form \(f\) on \(G_0(k_A)\) and a cuspidal automorphic form \(F\) on \(G(k_A)\). The authors first prove, following Shintani's idea of a uniqueness result, that the integral \[ \int_{G_0(k) \setminus G_0(k_A)} f(x)F(xg)dx \] is a product of local factors. They determine the local factors which arise. They show further that a certain family of Rankin-Selberg convolutions (depending on two complex parameters) is of the form of an \(L\)-function times \[ \int_{G_0(k) \setminus G_0(k_A)} f(x)F(xg)|\text{det}(x) |^s dx. \] Combining these two results they show that the Rankin-Selberg convolution is essentially the standard \(L\)-function of the contragredient representation to that of \(F\) times that of the tensor product of those of \(f\) and \(F\). Moving to \(GL_{n+1}\) they can also determine a similar integral to the one above involving \(F\) and an Eisenstein series induced from \(f\) (again in two complex variables). This has to be regularised to give it a sense, but it turns out to be a very interesting product of \(L\)-functions.
scientific article; zbMATH DE number 927275
scientific article