## Lectures on the Orbit MethodIsaac Newton encrypted his discoveries in analysis in the form of an anagram, which deciphers to the sentence ``It is worthwhile to solve differential equations''. Accordingly, one can express the main idea behind the Orbit Method by saying "It is worthwhile to study coadjoint orbits". The orbit method was introduced by the author, A. A. Kirillov, in the 1960s and remains a useful and powerful tool in areas such as Lie theory, group representations, integrable systems, complex and symplectic geometry, and mathematical physics. This book describes the essence of the orbit method for non-experts and gives the first systematic, detailed, and self-contained exposition of the method. It starts with a convenient ``User's Guide'' and contains numerous examples. It can be used as a text for a graduate course, as well as a handbook for non-experts and a reference book for research mathematicians and mathematical physicists. |

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### Contents

Geometry of Coadjoint Orbits | 1 |

Representations and Orbits of the Heisenberg Group | 31 |

The Orbit Method for Nilpotent Lie Groups | 71 |

Solvable Lie Groups | 109 |

Compact Lie Groups | 135 |

Miscellaneous | 179 |

### Common terms and phrases

1-dimensional abelian action of G adjoint algebra g Appendix bundle called canonical coadjoint action coadjoint orbits cocycle coefficients cohomology compact complex condition connected Lie group consider coordinate system corresponding defined definition denote diffeomorphism differential direct sum dual elements equation equivalence classes Example exponential fiber formula Fourier transform functor G-invariant G-manifold G-orbits given Heisenberg group Hence Hermitian Hilbert space homogeneous homomorphism induced representation infinite-dimensional Infinitesimal characters integral invariant isomorphic Lemma Let G Lie algebra Lie group G linear Matn(R matrix maximal measure morphisms multiplication nilpotent Lie groups non-zero notation orbit method Poisson polarization polynomial Proof Proposition realization representation theory root system Rule satisfying scalar self-adjoint operator semisimple simply connected smooth manifold Stab(F structure subalgebra subgroup submanifold subset subspace symplectic manifold tensor Theorem topological space unirreps unitary representation User's Guide Vect(M vector fields vector space Weyl