Spectral zeta function of the sub-Laplacian on two step nilmanifolds

Abstract

We study the heat kernel trace and the spectral zeta function of an intrinsic sub-Laplace operator ΔL∖Gsub on a two step compact nilmanifold L∖G. Here G is an arbitrary nilpotent Lie group of step 2 and we assume the existence of a lattice L⊂G. We essentially use the well-known heat kernel expressions of the sub-Laplacian on G due to Beals, Gaveau and Greiner. In contrast to the spectral zeta function of the Laplacian on L∖G which can have infinitely many simple poles it turns out that in case of the sub-Laplacian only one simple pole occurs. Its residue divided by the volume of L∖G is independent of L and can be expressed by the Lie group structure of G. By standard arguments this result is equivalent to a specific asymptotic behaviour of the heat kernel trace of ΔL∖Gsub as time tends to zero. As an example we explicitly calculate the spectrum of the sub-Laplacian ΔL∖Gsub in case of the six-dimensional free nilpotent Lie group G and a standard lattice L⊂G by using a decomposition of ΔL∖Gsub into a family of elliptic operators.