viii Contents
Chapter 2. Representations and Orbits of the Heisenberg Group 31
§1. Heisenberg Lie algebra and Heisenberg Lie group 32
1.1. Some realizations 32
1.2. Universal enveloping algebra U(h) 35
1.3. The Heisenberg Lie algebra as a contraction 37
§2. Canonical commutation relations 39
2.1. Creation and annihilation operators 39
2.2. Two-sided ideals in U(h) 41
2.3. H. Weyl reformulation of CCR 41
2.4. The standard realization of CCR 43
2.5. Other realizations of CCR 45
2.6. Uniqueness theorem 49
§3. Representation theory for the Heisenberg group 57
3.1. The unitary dual H 57
3.2. The generalized characters of H 59
3.3. The infinitesimal characters of H 60
3.4. The tensor product of unirreps 60
§4. Coadjoint orbits of the Heisenberg group 61
4.1. Description of coadjoint orbits 61
4.2. Symplectic forms on orbits and the Poisson structure
on
h∗
62
4.3. Projections of coadjoint orbits 63
§5. Orbits and representations 63
5.1. Restriction-induction principle and construction of
unirreps 64
5.2. Other rules of the User’s Guide 68
§6. Polarizations 68
6.1. Real polarizations 68
6.2. Complex polarizations 69
6.3. Discrete polarizations 69
Chapter 3. The Orbit Method for Nilpotent Lie Groups 71
§1. Generalities on nilpotent Lie groups 71
§2. Comments on the User’s Guide 73
2.1. The unitary dual 73
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