Order estimates for irreducible projections in \(L^ 2\) of a nilmanifold
Let \(\pi\) be an irreducible representation occuring in \(L^ 2(\Gamma \setminus N)\) where N is a connected, simply connected, nilpotent Lie group and \(\Gamma\) a discrete, cocompact subgroup of N. It is known that the projection onto any irreducible \(\pi\)-summand in \(L^ 2(\Gamma \setminus N)\) is given by convolution against a distribution \(D_{\pi}.\) The main theorem is an estimate of the order of \(D_{\pi}\) in terms of ''canonical objects'' associated to elements of the Kirillov orbit of \(\pi\). The author states that the main ''standard'' examples for irreducibles in \(L^ 2(\Gamma \setminus N)\) are covered by the theorem and the earlier paper by \textit{L. Corwin}, F. P. Greenleaf and \textit{R. Penney} [Commun. Pure Appl. Math. 30, 355-372 (1977; Zbl 0343.22009)].
scientific article; zbMATH DE number 3908723