Cross-sections and an explicit formula for the Plancherel measure on a nilpotent Lie group
This paper gives a simple, calculable formula for the Plancherel measure \(\mu\) for a nilpotent Lie group with Lie algebra \({\mathfrak g}\). From the Dixmier-Kirillov-Pukanszky theory, it is known that \(\mu\) is the quotient of Lebesgue measure by the canonical coadjoint orbit measures. Further, \(\mu\) can be realized on a subspace W of \({\mathfrak g}^ *\) and \(d\mu =| R(z)| dz\) for some rational function R on W. The problem is to determine R(z). The formula is given as a product of two straight-forward determinants, one of these arising in Pukanszky's formula for canonical measures. The main idea for the proof is that of a cross-section, which associates with elements of \({\mathfrak g}^ *\) a rationally varying weak Mal'cev basis fitting in ``nicely'' with the irreducible representations associated with the functionals. In practice, it is easy to write down cross- sections, and the determinants are immediately specified by the cross- section. The explicitness of the formula can be used to explain the curious fact (c.f. the work of O. A. Nielsen) that R(z) is actually polynomial for dim \(G\leq 6\). The author believes that the cross-section approach should prove useful in dealing with other problems in harmonic analysis on nilpotent Lie groups.
scientific article; zbMATH DE number 4019690
scientific article