Local geometrised Rankin-Selberg method for GL(n)

Abstract

Following Laumon [10], to a nonramified $\ell$-adic local system $E$ of rank $n$ on a curve $X$ one associates a complex of $\ell$-adic sheaves ${}_n{\mathcal{K}}_E$ on the moduli stack of rank $n$ vector bundles on $X$ with a section, which is cuspidal and satisfies Hecke property for $E$. This is a geometric counterpart of the well-known construction due to Shalika [17] and Piatetski-Shapiro [16]. We express the cohomology of the tensor product ${}_n{\mathcal{K}}_{E_1}\otimes {}_n{\mathcal{K}}_{E_2}$ in terms of cohomology of the symmetric powers of $X$. This may be considered as a geometric interpretation of the local part of the classical Rankin-Selberg method for GL(n) in the framework of the geometric Langlands program.

36 pages, LaTeX2e, new results are added, final version to appear in Duke Mathematical Journal published by Duke University Press