Non-elliptic Laplace equations on nilpotent Lie groups


Let $N$ be a connected, simply connected nilpotent Lie group which admits a left invariant complex structure. A non-degenerate, $J$ invariant, alternating two form $\phi$ on $\mathcal{N}$ is called pseudo-Kählerian if $\phi([X,Y],W) + \phi([Y,W],X) + \phi([W,X],Y) = 0$. $\phi$ defines a pseudo-Käherlian structure on the complex manifold $N$. We analyze the complex Laplacian defined from the non-positive form $\phi$ acting on either $L^2(N)$, $L^2(\Gamma/N)$ or certain line bundles over these spaces. Under additional hypotheses, we completely describe the discrete and continuous spectrum of these operators as well as give a technique which is capable, in principle, of computing the exponential and inverse of these operators. As applications we generalize some results of Folland-Greiner-Stein and obtain some counterexamples to a non-elliptic Hodge theory.


Richard C. Penney