Spectral zeta function of the sub-Laplacian on two step nilmanifolds (Q764912)

Let \(G\) be a nilpotent Lie group of step 2 and let \(L\subset G\) be a cocompact lattice. Let \(M=L\backslash G\) be the associated two step compact nilmanifold. Let \(\Delta_M^{sub}\) be the intrinsic sub-Laplace operator. The authors examine the heat kernel trace and the spectral zeta function of this operator. The heat trace has a meromorphic extension to the complex plane with but a single simple pole. When the residue is normalized by dividing by the volume of \(M\), the resulting invariant only depends on the Lie group structure of \(G\). The authors calculate the spectrum of \(\Delta_M^{sub}\) for the standard lattice in the case of the six-dimensional free nilpotent Lie group.