Dissolving a cusp form in the presence of multiplicities
Let \(\Gamma\) be a discrete cofinite but not cocompact group of orientation preserving motions of the hyperbolic plane \(\mathbb{H}\). Denote by \(N(x)\) the number of linearly independent \(\Gamma\)-automorphic Maaß cusp forms with eigenvalue less than \(x>0\). The group \(\Gamma\) is called essentially cuspidal if \(N(x)/x\) converges to \(\frac{1}{4\pi}\) area \((\Gamma\setminus \mathbb{H})\) as \(x\to \infty\) (Weyl's law). For a long time it was supposed that all groups \(\Gamma\) of the above type were essentially cuspidal. This conjecture has neither been proven nor disproven, but now several results indicate that the conjecture is unlikely to be true. The reason is that Maaß cusp form eigenvalues may dissolve into the continuous spectrum under a perturbation of \(\Gamma\). The proofs of these dissolution results depend on a multiplicity hypothesis requiring essentially that the eigenvalue to be dissolved is simple. Since no example of a group \(\Gamma\) with simple spectrum is known this hypothesis really is crucial. The aim of the paper under review is to relax the multiplicity conjecture. The authors show that an eigenspace of arbitrary dimension contains a Maaß cusp form that is dissolved under perturbation provided the dissolving functional does not vanish for some Maaß form belonging to the eigenspace in question. This can be used to establish the dissolution of Maaß cusp form in several cases which were considered in the literature.
scientific article; zbMATH DE number 1042483
scientific article