Strong exponent bounds for the local Rankin-Selberg convolution

Document Type: Special Issue of BIMS in Honor of Professor Freydoon Shahidi

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Abstract

Let $F$ be a non-Archimedean locally compact field‎. ‎Let $\sigma$ and $\tau$ be finite-dimensional representations of the Weil-Deligne group of $F$‎. ‎We give strong upper and lower bounds for the Artin and Swan exponents of $\sigma\otimes\tau$ in terms of those of $\sigma$ and $\tau$‎. ‎We give a different lower bound in terms of $\sigma\otimes\check\sigma$ and $\tau\otimes\check\tau$‎. ‎Using the Langlands correspondence‎, ‎we obtain the bounds for Rankin-Selberg exponents‎.

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