Asymptotics and zeta functions on compact nilmanifolds

TL;DR

This paper derives asymptotic formulas for spectral kernels and zeta functions on compact nilmanifolds, focusing on sub-Laplacians and Rockland operators, revealing simplified short-time asymptotics with a single dominant term.

Contribution

It provides new asymptotic formulas for spectral kernels and zeta functions on nilmanifolds, especially for sub-Laplacians and graded cases, with a focus on the diagonal asymptotics.

Findings

Short-time asymptotics contain only one non-trivial term.

Asymptotic formulas are obtained for spectral kernels and zeta functions.

Results apply to stratified and graded nilpotent Lie groups.

Abstract

In this paper, we obtain asymptotic formulae on nilmanifolds $\Gamma \backslash G$, wher $G$ is any stratified (or even graded) nilpotent Lie group equipped with a co-compact discrete subgroup $\Gamma$. We study especially the asymptotics related to the sub-Laplacians naturally coming from the stratified structure of the group $G$ (and more generally any positive Rockland operators when $G$ is graded). We show that the short-time asymptotic on the diagonal of the kernels of spectral multipliers contains only a single non-trivial term. We also study the associated zeta functions.