The translator of a mathematical work faces a task that is at once fascinating and frustrating. He has the opportunity of reading closely the work of a master mathematician. He has the duty of retaining as far as possible the flavor and spirit of the original, at the same time rendering it into a readable and idiomatic form of the language into which the translation is made. All of this is challenging. At the same time, the translator should never forget that he is not a creator, but only a mirror. His own viewpoints, his own preferences, should never lead him into altering the original, even with the best intentions. Only an occasional translator's note is permitted. The undersigned is grateful for the opportunity of translating Professor Kirillov's fine book on group representations, and hopes that it will bring to the English-reading mathematical public as much instruction and interest as it has brought to the translator. Deviations from the Russian text have been rigorously avoided, except for a number of corrections kindly supplied by Professor Kirillov. Misprints and an occasional solecism have been tacitly taken care of. The trans­ lation is in all essential respects faithful to the original Russian. The translator records his gratitude to Linda Sax, who typed the entire translation, to Laura Larsson, who prepared the bibliography (considerably modified from the original), and to Betty Underhill, who rendered essential assistance.

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First Part. Preliminary Facts.- § 1. Sets, Categories, Topology.- 1.1. Sets.- 1.2. Categories and Functors.- 1.3. The Elements of Topology.- § 2. Groups and Homogeneous Spaces.- 2.1. Transformation Groups and Abstract Groups.- 2.2. Homogeneous Spaces.- 2.3. Principal Types of Groups.- 2.4. Extensions of Groups.- 2.5. Cohomology of Groups.- 2.6. Topological Groups and Homogeneous Spaces.- § 3. Rings and Module.- 3.1. Rings.- 3.2. Skew Fields.- 3.3. Modules over Rings.- 3.4. Linear Spaces.- 3.5. Algebra.- § 4. Elements of Functional Analysis.- 4.1. Linear Topological Spaces.- 4.2. Banach Algebras.- 4.3. C-Algebras.- 4.4. Commutative Operator Algebras.- 4.5. Continuous Sums of Hilbert Spaces and von Neumann Algebras.- § 5. Analysis on Manifolds.- 5.1. Manifolds.- 5.2. Vector Fields.- 5.3. Differential Forms.- 5.4. Bundles.- § 6. Lie Groups and Lie Algebras..- 6.1. Lie Group.- 6.2. Lie Algebras.- 6.3. The Connection between Lie Groups and Lie Algebras.- 6.4. The Exponential Mapping.- Second Part. Basic Concepts and Methods of the Theory of Representations.- § 7. Representations of Groups.- 7.1. Linear Representations.- 7.2. Representations of Topological Groups in Linear Topological Space.- 7.3. Unitary Representations.- § 8. Decomposition of Representations.- 8.1. Decomposition of Finite Representation.- 8.2. Irreducible Representation.- 8.3. Completely Reducible Representations.- 8.4. Decomposition of Unitary Representations.- § 9. Invariant Integration.- 9.1. Means and Invariant Measures.- 9.2. Applications to Compact Groups.- 9.3. Applications to Noncompact Groups.- § 10. Group Algebras.- 10.1. The Group Ring ofa Finite Group.- 10.2. Group Algebras of Topological Groups.- 10.3. Application of Group C-Algebras.- 10.4. Group Algebras of Lie Groups.- 10.5. Representations of Lie Groups and their Group Algebras.- § 11. Characters..- 11.1. Characters of Finite-Dimensional Representations.- 11.2. Characters of Infinite-Dimensional Representations.- 11.3. Infinitesimal Characters.- § 12. Fourier Transforms and Duality.- 12.1. Commutative Groups.- 12.2. Compact Groups.- 12.3. Ring Groups and Duality for Finite Groups.- 12.4. Other Result.- § 13. Induced Representations.- 13.1. Induced Representations of Finite Groups.- 13.2. Unitary Induced Representations of Locally Compact Groups.- 13.3. Representations of Group Extensions.- 13.4. Induced Representations of Lie Groups and their Generalizations.- 13.5. Intertwining Operators and Duality.- 13.6. Characters of Induced Representations.- § 14. Projective Representation.- 14.1. Projective Groups and Projective Representations.- 14.2. Schur's Theory.- 14.3. Projective Representations of Lie Group.- § 15. The Method of Orbits.- 15.1. The Co-Adjoint Representation of a Lie Group.- 15.2. Homogeneous Symplectic Manifold.- 15.3. Construction of an Irreducible Unitary Representation by an Orbit.- 15.4. The Method of Orbits and Quantization of Hamiltonian Mechanical System.- 15.5. Functorial Properties of the Correspondence between Orbits and Representation.- 15.6. The Universal Formula for Characters and Plancherel Measures.- 15.7. Infinitesimal Characters and Orbits.- Third Part. Various Examples.- § 16. Finite Groups.- 16.1. Harmonic Analysis on the Three-Dimensional Cube.- 16.1. Harmonic Analysis on the Three-Dimensional Cube.- 16.3. Representations of the Group SL(2,Fq).- 16.4. Vector Fields on Spheres.- § 17. Compact Groups.- 17.1. Harmonic Analysis on the Sphere.- 17.2. Representations of the Classical Compact Lie Groups.- 17.3. Spinor Representations of the Orthogonal Group.- § 18. Lie Groups and Lie Algebras.- 18.1. Representations of a Simple Three-Dimensional Lie Algebra.- 18.2. The Weyl Algebra and Decomposition of Tensor Products.- 18.3. The Structure of the Enveloping Algebra $$ U\left( \mathfrak{g} \right) $$ for $$ \mathfrak{g} = \mathfrak{s}\mathfrak{l}\left( {2,C} \right) $$.- 18.4. Spinor Representations of the Symplectic Group.- 18.5. Representations of Triangular Matrix Groups.- § 19. Examples of Wild Lie Groups.- A Short Historical Sketch and a Guide to the Literature.