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 $[E:F]\le n$.

The author would like to express his gratitude to an anonymous referee who pointed out that in Definition 4.1 where the regularized mirabolic integral is defined, the summation over all parabolic subgroups is divergent. The author is currently working on the issue. Meanwhile, the author welcomes any ideas and suggestions on possible improvements via email