A New Tower of Rankin-Selberg Integrals

The notion of a tower of Rankin-Selberg integrals was introduced in [G-R]. To recall this notion, let G be a reductive group defined over a global field F . Let G denote the L group of G. Let ρ denote a finite dimensional irreducible representation of G. Given an irreducible generic cuspidal representation of G(A), we let L(π, ρ, s) denote the partial L function associated with π and ρ. Here s is a complex variable and A denotes the adele ring associated with F . If ρ acts on the vector space V , we denote by C[V ] the symmetric algebra attached to the vector space V . Let C[V ] G denote the G invariant polynomials inside the symmetric algebra. As far as we know all examples of L functions represented by a Rankin-Selberg integral are associated with representations ρ such that C[V ] G is a free algebra. A list of all such groups, representations and the degrees of the generators of the invariant polynomials are given in [K]. The basic observation in [G-R] is that there is some relation between the Eisenstein series one uses to construct the Rankin-Selberg integral and the number of generators of the invariant polynomials and their degrees. This relation is far from being clear and it is mainly based on observation of all known constructions of such integrals. To summarize in an unprecise manner, the relations are: 1) If ρ1 and ρ2 have the same number of generators with the same degrees, then in some cases the Rankin-Selberg integrals which represent the corresponding two L functions, use the same Eisenstein series.