# On Weyl's asymptotics and remainder term for the orthogonal and unitary groups

## TL;DR

This paper investigates the spectral asymptotics of orthogonal and unitary groups, showing that existing estimates are not sharp in positive Ricci curvature cases and providing sharp examples with implications for geodesic length spectra.

## Contribution

It demonstrates the non-sharpness of classical spectral asymptotics for certain Lie groups and provides sharp, quantitative examples in positive Ricci curvature settings.

## Key findings

- Classical asymptotics are not sharp for ${f SO}(N)$, ${f SU}(N)$, ${f U}(N)$, and ${f Spin}(N)$.
- Sharp examples are constructed in positive Ricci curvature cases.
- Results have implications for the study of closed geodesics and length spectra.

## Abstract

We examine the asymptotics of the spectral counting function of a compact Riemannian manifold by V.G.~Avakumovic \cite{Avakumovic} and L.~H\"ormander \cite{Hormander-eigen} and show that for the scale of orthogonal and unitary groups ${\bf SO}(N)$, ${\bf SU}(N)$, ${\bf U}(N)$ and ${\bf Spin}(N)$ it is not sharp. While for negative sectional curvature improvements are possible and known, {\it cf.} e.g., J.J.~Duistermaat $\&$ V.~Guillemin \cite{Duist-Guill}, here, we give sharp and contrasting examples in the positive Ricci curvature case [non-negative for ${\bf U}(N)$]. Furthermore here the improvements are sharp and quantitative relating to the dimension and {\it rank} of the group. We discuss the implications of these results on the closely related problem of closed geodesics and the length spectrum.