Colin J Bushnell

Strong exponent bounds for the local Rankin-Selberg convolution

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