Archimedean zeta integrals on $GL_n \times GL_m$ and $SO_{2n+1} \times GL_m$

TL;DR

This paper evaluates archimedean zeta integrals for automorphic L-functions on specific group products, expressing them explicitly via Mellin transforms, Gamma functions, and Barnes integrals, advancing the understanding of their analytic properties.

Contribution

It provides explicit formulas for archimedean zeta integrals on $GL_n imes GL_m$ and $SO_{2n+1} imes GL_m$, connecting Mellin transforms with special functions.

Findings

Explicit Mellin transform expressions in terms of Gamma functions.

Barnes integral representations for zeta integrals.

Enhanced understanding of automorphic L-functions at archimedean places.

Abstract

In this paper, we evaluate archimedean zeta integrals for automorphic $L$-functions on $GL_n \times GL_{n-1+\ell}$ and on $ SO_{2n+1} \times GL_{n+\ell}$, for $\ell=-1$, $0$, and $1$. In each of these cases, the zeta integrals in question may be expressed as Mellin transforms of products of class one Whittaker functions. Here, we obtain explicit expressions for these Mellin transforms in terms of Gamma functions and Barnes integrals.