Local geometrised Rankin-Selberg method for GL(n)

Following Laumon [10], to a nonramified $\mathrm{\ell }$-adic local system $E$ of rank $n$ on a curve $X$ one associates a complex of $\mathrm{\ell }$-adic sheaves $_n{\cal K}_E$onthemodulistackofrank$n$vectorbundleson$X$ 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.