On the Mirabolic Trace Formula for $\mathfrak{gl}(n)$

Abstract

In this paper, a Chaudouard type trace formula is established for the Lie algebra $\mathfrak{gl}(n)$, by integrating the Lie algebra analogue of the Selberg kernel function against a mirabolic Eisenstein series on $\mathrm{GL}(n)$. The result is a combination of zeta functions $\zeta_E(s)$ of extensions over the base field $F$ of degree $[E:F]\leq n$.