Local geometrised Rankin-Selberg method for GL(n)

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Following Laumon [10], to a nonramified ℓ-adic local system E of rank n on a curve X one associates a complex of ℓ-adic sheaves ₙ Kₑ 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 Kₑ₁⊗ ₙ Kₑ₂ 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.