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Freydoon Shahidi : Local Langlands correspondence and the exterior and symmetric square root numbers for GL(n)

We will discuss the notion of Artin root numbers attached to an n-dimensional continuous Frobenius-semisimple complex representation of the Weil-Deligne group and show their equalities with those defined by Langlands-Shahidi method through local Langlands correspondence for GL(n) and the exterior and symmetric square representation of the L-group GL(n,C) of GL(n). The proof is a robust deformation argument using local-global techniques, complemented with suitable asymptotic expansions for partial Bessel functions inspired by certain generalized Shalika germ expansions of Jacquet and Ye. This is a joint work with J. Cogdell and T.-L. Tsai.

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