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

We examine the asymptotics of the spectral counting function of a compact Riemannian manifold by Avakumovic (Math Z 65:327–344, [1]) and Hörmander (Acta Math 121:193–218, [15]) and show that for the scale of orthogonal and unitary groups SO(N), SU(N), U(N) and Spin(N) it is not sharp. While for negative sectional curvature improvements are possible and known, cf. e.g., Duistermaat and Guillemin (Invent Math 29:39–79, [7]), here, we give sharp and contrasting examples in the positive Ricci curvature case [non-negative for U(N)]. Furthermore here the improvements are sharp and quantitative relating to the dimension and rank of the group. We discuss the implications of these results on the closely related problem of closed geodesics and the length spectrum