Local geometrised Rankin-Selberg method for GL(n)

Details Local geometrised Rankin-Selberg method for GL(n) Local geometrised Rankin-Selberg method for GL(n)

1999-05-05

arxiv.org/abs/math/9905033v2

Mathematics

Algebraic Geometry

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

Scientific paper

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 $_n{\cal K}_E$ 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 $_n{\cal K}_{E_1}\otimes {_n{\cal K}_{E_2}}$ 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.

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