The Plancherel formula for real semisimple Lie groups.
(La formule de Plancherel des groupes de Lie semi-simples réels.)

*(French)*Zbl 0759.22017
Representations of Lie groups: analysis on homogeneous spaces and representations of Lie groups, Proc. Symp., Kyoto/Jap. and Hiroshima/Jap. 1986, Adv. Stud. Pure Math. 14, 289-336 (1988).

[For the entire collection see Zbl 0694.00014.]

Let \(G\) be a locally compact unimodular group of type I and \(\widehat G\) the set of equivalence classes of irreducible unitary representations of \(G\). Then there exists a measure \(dm(T)\) called Plancherel measure on \(\widehat G\), supported on a subset \(\widehat G_ r\), such that the Plancherel inversion formula \(\phi({\mathbf{1}})=\int_{\widehat G_ r}\text{tr }T(\phi)dm(T)\) holds for all sufficiently well behaved functions on \(G\).

For the case when \(G\) is a connected semisimple real Lie group with finite center Harish-Chandra has given a parametrization of the set \(\widehat G_ r\) and a formula for the Plancherel measure in these parameters [Harish-Chandra, Ann. Math., II. Ser. 104, 117-201 (1976; Zbl 0331.22007)].

In this paper the authors give a proof of Harish-Chandra’s Plancherel formula which fits into the setting of the orbit method and which is also valid for arbitrary connected semisimple groups. There are two major difficulties which are not present in the classical case of simply connected nilpotent Lie groups:

(1) Since the exponential mapping need not be surjective, to obtain an appropriate version of Kirillov’s character formula, one has to study the character distribution of an irreducible representation at elements which are not contained in the image of the exponential function. Here the authors apply Harish-Chandra’s method of descent which permits us to reduce the study of an invariant distribution on \(G\) in a neighborhood of a semisimple element \(s\) to the study of an invariant distribution in a 0-neighborhood in the Lie algebra of the centralizer of \(s\).

(2) Not every coadjoint orbit corresponds to an irreducible unitary representation. There are certain integrality conditions which have to be satisfied. Therefore the different parts of the Plancherel inversion formula are not simply Fourier integrals (as it is the case for nilpotent groups) nor Poisson sums (as it is the case for compact groups) but a mixture of both.

Let \(G\) be a locally compact unimodular group of type I and \(\widehat G\) the set of equivalence classes of irreducible unitary representations of \(G\). Then there exists a measure \(dm(T)\) called Plancherel measure on \(\widehat G\), supported on a subset \(\widehat G_ r\), such that the Plancherel inversion formula \(\phi({\mathbf{1}})=\int_{\widehat G_ r}\text{tr }T(\phi)dm(T)\) holds for all sufficiently well behaved functions on \(G\).

For the case when \(G\) is a connected semisimple real Lie group with finite center Harish-Chandra has given a parametrization of the set \(\widehat G_ r\) and a formula for the Plancherel measure in these parameters [Harish-Chandra, Ann. Math., II. Ser. 104, 117-201 (1976; Zbl 0331.22007)].

In this paper the authors give a proof of Harish-Chandra’s Plancherel formula which fits into the setting of the orbit method and which is also valid for arbitrary connected semisimple groups. There are two major difficulties which are not present in the classical case of simply connected nilpotent Lie groups:

(1) Since the exponential mapping need not be surjective, to obtain an appropriate version of Kirillov’s character formula, one has to study the character distribution of an irreducible representation at elements which are not contained in the image of the exponential function. Here the authors apply Harish-Chandra’s method of descent which permits us to reduce the study of an invariant distribution on \(G\) in a neighborhood of a semisimple element \(s\) to the study of an invariant distribution in a 0-neighborhood in the Lie algebra of the centralizer of \(s\).

(2) Not every coadjoint orbit corresponds to an irreducible unitary representation. There are certain integrality conditions which have to be satisfied. Therefore the different parts of the Plancherel inversion formula are not simply Fourier integrals (as it is the case for nilpotent groups) nor Poisson sums (as it is the case for compact groups) but a mixture of both.

Reviewer: K.-H.Neeb (Darmstadt)

##### MSC:

22E46 | Semisimple Lie groups and their representations |

22E30 | Analysis on real and complex Lie groups |