Global geometrization of local smooth integral models in the Hitchin fibration for $\mathrm{GL}_n$: The Bass case

Abstract

In previous joint work, we proposed a new method to study local orbital integrals for $\mathrm{GL}_n$ (where $n=3$ or in the Bass case), by employing a smoothening method of a certain scheme defined over a henselian ring. In this paper, we geometrize this local smoothening method within the framework of the global Hitchin fibration for $\mathrm{GL}_n$ in the Bass case. Consequently, we provide a closed formula for the $\ell$-adic cohomology of the compactified Jacobian of a spectral curve over a finite field with double singularities whose local rings are integral domains, provided that the normalization of a spectral curve is isomorphic to $\mathbb{P}^1$.