The Penney-Fujiwara Plancherel formula for nilpotent Lie groups
Let \(G= \exp{\mathfrak g}\) be a connected, simply connected nilpotent Lie group and \(H= \exp{\mathfrak h}\) be an analytic subgroup with unitary character \(\chi: H\to \mathbb{C}\). We consider the monomial representation \(\tau= \text{ind}^G_H\chi\) of \(G\). Using their intertwining operator between \(\tau\) and their irreducible decomposition [J. Lie Theory 9, 157-191 (1999; Zbl 0921.22006)], the authors describe the Penney's abstract Plancherel formula explicitly in terms of the orbit method [cf. \textit{R. Penney}, J. Funct. Anal. 18, 177-190 (1975; Zbl 0305.22016); cf. also the reviewer's paper in Pac. J. Math. 127, 329-351 (1987; Zbl 0588.22008)]. This is a direct extension of the reviewer's result to the case, where \(\tau\) has infinite multiplicities in its irreducible decompositions. The paper also contains a short proof of a Corwin-Greenleaf's result on the algebra \(D_\tau(G/H)\) of invariant differential operators on the line bundle constructed from the character \(\chi\) on the base space \(G/H\). Suppose that \(\tau\) has finite multiplicities in its irreducible decomposition. Let \(f\in{\mathfrak g}^*\) such that \(\chi(\exp X)= e^{if(X)}\) \((\forall X\in{\mathfrak h})\), and let \(\Gamma_f= f+{\mathfrak h}^\perp\) in \({\mathfrak g}^*\). Now, we can state the result in question as follows: suppose that there exists a common polarization \({\mathfrak b}\) for generic \(\ell\in \Gamma_f\) and that \({\mathfrak b}\) is normalized by \({\mathfrak h}\), then the algebra \(D_\tau(G/H)\) is isomorphic to the algebra \(\mathbb{C}[\Gamma_f]^H\) of \(H\)-invariant polynomial functions on \(\Gamma_f\).
scientific article; zbMATH DE number 1502025