The differential form spectrum of quaternionic hyperbolic spaces (Q2387390)

The \(L^{2}\) spectrum of the Hodge-de Rham Laplacian \(\Delta _{l}\) acting on quaternionic hyperbolic spaces is determined by using harmonic analysis and representation theory. It is shown that the unique possible discrete eigenvalue and the lowest continuous eigenvalue of \(\Delta _{l}\) can both be realized by some subspace of hyper-effective differential forms. The study is focused on the case when \(\Delta _{l}\) acts on smooth differential \(l\)-forms of \(G/K\) (symmetric space of noncompact type). It is proven that the discrete \(L^{2}\) spectrum of \(\Delta _{l}\) on \(G/K\) is empty, unless \(G\) and \(K\) have equal complex rank and \(l=\frac{1}{2}\dim _{R}t( G/K) \) in which case it reduces to zero. Moreover, the space of \(L^{2}\) harmonic forms consists of the sum of all discrete series representations of \(G\) having trivial infinitesimal character. If \(G/K\) is a quaternionic hyperbolic space, the eigenvalues \(\alpha _{l}\) (i.e. the spectrum) of \(\Delta _{l}\) are calculated and the lowest value of \(\alpha _{l}\) is realized by some subspace of hyper-effective \(l\)-forms, whose corresponding\ \(K\)-type occurs with multiplicity one. The spectrum of the Bochner Laplacian \(B_{l}\) = \(\nabla ^{\ast }\nabla \) is also calculated and the results are similar those in the case of \(\Delta _{l}.\) However, the author remarks that the spectrum \(\beta _{l}\) of \(B_{l}\) does not `fully' depend on the value of \(l\), in contrast to the real case but similar to the complex case.