Stevens' Approach to Kloosterman Sums on Split Groups

Abstract

We develop a uniform framework for Stevens' approach to Kloosterman sums on split groups over $p$-adic fields. The main result reduces the bounding of Kloosterman sums to that for the so-called Plücker-constrained sets, which are certain subsets of affine spaces constrained by conditions giving by the Plücker coordinates. We then give a nontrivial bound for the Kloosterman sums on the exceptional group $G_2$. As an application, we obtain the regularity of Bessel distributions associated with generic representations on $G_2$.