Harmonic analysis on reduced homogeneous manifolds and unitary representation theory.

*(English. Japanese original)*Zbl 0895.22006
Am. Math. Soc., Transl., 2. Ser. 183, 1-50 (1998); translation from Sūgaku 46, No. 2, 124-143 (1994).

This is an expository paper concerning recent advances on the relationship between harmonic analysis on homogeneous manifolds \(G/H\) of reductive type and unitary representations of real reductive Lie groups \(G\), where \(H\) is a reductive subgroup of \(G\). The paper is divided into six sections; they are 1. Homogeneous manifolds of reductive type, 2. Unitary representations of real reductive Lie groups, 3. Noncommutative harmonic analysis on Riemannian homogeneous manifolds, 4. Discrete series with values in vector bundles over symmetric spaces, 5. Harmonic analysis on para-Hermitian symmetric spaces, and 6. Ramification principle of representations and its application to discrete series.

A little more precisely, Section 1 introduces the geometry of homogeneous manifolds of reductive type needed in this paper. As preparation, in the following sections harmonic analysis on homogeneous manifolds possessing invariant complex, (pseudo-)Riemannian, symmetric, para-Hermitian and other geometric structures are examined. Section 2 is related to Schmid’s and Wong’s recent results about the construction of representations on the space of Dolbeault cohomology of reductive-type homogeneous manifolds with complex structure, and stresses the relationship with the additive group of Zuckerman’s derived operators and Vogan’s idea of unitary representations, Section 3 expounds the relationship between the point spectrum of the Laplacian of a noncompact Riemannian manifold and the Blattner conjecture on discrete series. Section 4 describes the construction of discrete series of reductive-type homogeneous manifolds having the structure of principal bundles over symmetric spaces. The last section states the author’s recent results on the reduced decomposability of restriction of reduced unitary representations to subgroups, i.e., the ramification principle of representations (“symmetry breaking” in physics), which is one of the basic problems of unitary representations.

A little more precisely, Section 1 introduces the geometry of homogeneous manifolds of reductive type needed in this paper. As preparation, in the following sections harmonic analysis on homogeneous manifolds possessing invariant complex, (pseudo-)Riemannian, symmetric, para-Hermitian and other geometric structures are examined. Section 2 is related to Schmid’s and Wong’s recent results about the construction of representations on the space of Dolbeault cohomology of reductive-type homogeneous manifolds with complex structure, and stresses the relationship with the additive group of Zuckerman’s derived operators and Vogan’s idea of unitary representations, Section 3 expounds the relationship between the point spectrum of the Laplacian of a noncompact Riemannian manifold and the Blattner conjecture on discrete series. Section 4 describes the construction of discrete series of reductive-type homogeneous manifolds having the structure of principal bundles over symmetric spaces. The last section states the author’s recent results on the reduced decomposability of restriction of reduced unitary representations to subgroups, i.e., the ramification principle of representations (“symmetry breaking” in physics), which is one of the basic problems of unitary representations.

Reviewer: Ji Sheng Na (MR 95k:22011)

##### MSC:

22E30 | Analysis on real and complex Lie groups |

43A85 | Harmonic analysis on homogeneous spaces |

22D10 | Unitary representations of locally compact groups |

22E47 | Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) |