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

Abstract

We examine the asymptotics of the spectral counting function of a compact Riemannian manifold by V.G.~Avakumovic \cite{Avakumovic} and L.~Hörmander \cite{Hormander-eigen} and show that for the scale of orthogonal and unitary groups $\mathbf{SO}(N)$, $\mathbf{SU}(N)$, $\mathbf{U}(N)$ and $\mathbf{Spin}(N)$ 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 $\mathbf{U}(N)$]. 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.