Harmonic analysis on two and three-step nilmanifolds
Let \(G\) be a connected, simply connected nilpotent Lie group of step 1, 2 or 3, which contains a discrete cocompact subgroup \(\Gamma\). Let \(\tau=\mathrm{ind}^G_\Gamma1\) be the quasi-regular representation of \(G\). It is known that \(\tau\) splits into a discrete direct sum of a countable number of irreducible unitary representations \(\pi\) with finite multiplicities \(m(\pi)\); \(\tau\simeq \sum_{\pi\in\mathcal{S}(G:\Gamma)}m(\pi)\pi\) where \(\mathcal{S}(G:\Gamma)\) is the spectrum. The authors study the two main problems of this formula. Given a nilpotent Lie group \(G\) with discrete cocompact subgroup \(\Gamma\), determine the spectrum \(\mathcal{S}(G:\Gamma)\) and the multiplicity function \(\pi\mapsto m(\pi)\), and construct an explicit intertwining operator between \(\tau\) and its decomposition into irreducibles. Suppose now that \(G\) is 1,2, or 3-step nilpotent. The paper gives an orbital description of the answer on the first question about the spectrum and the multiplicity function. Furthermore, the authors describe an intertwining operator which does not take into account the multiplicities of the decomposition of \(\tau\). They also give a global intertwining operator for \(\tau\) and its disintegration, for three-step connected, simply connected nilpotent Lie groups. Their operator is given by convergent infinite sums for elements of a dense subspace of the space of \(\tau\).
scientific article; zbMATH DE number 6371146
scientific article