## Representation Theory of Semisimple Groups: An Overview Based on ExamplesIn this classic work, Anthony W. Knapp offers a survey of representation theory of semisimple Lie groups in a way that reflects the spirit of the subject and corresponds to the natural learning process. This book is a model of exposition and an invaluable resource for both graduate students and researchers. Although theorems are always stated precisely, many illustrative examples or classes of examples are given. To support this unique approach, the author includes for the reader a useful 300-item bibliography and an extensive section of notes. |

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### Contents

REPRESENTATIONS OF SU2 SL2 R | 28 |

C VECTORS AND THE UNIVERSAL | 46 |

4 Girding Subspace | 55 |

REPRESENTATIONS OF COMPACT LIE GROUPS 1 Examples of Root Space Decompositions | 60 |

2 Roots | 65 |

3 Abstract Root Systems and Positivity | 72 |

4 Weyl Group Algebraically | 78 |

5 Weights and Integral Forms | 81 |

8 Behavior on the Singular Set | 371 |

9 Families of Admissible Representations | 374 |

10 Problems | 383 |

INTRODUCTION TO PLANCHEREL FORMULA 1 Constructive Proof for SU2 | 385 |

2 Constructive Proof for SL2 C | 387 |

3 Constructive Proof for SL2 R | 394 |

4 Ingredients of Proof for General Case | 401 |

5 Scheme of Proof for General Case | 404 |

6 Centalizers of Tori | 86 |

7 Theorem of the Highest Weight | 89 |

8 Verma Modules | 93 |

9 Weyl Group Analytically | 100 |

10 Weyl Character Formula | 104 |

11 Problems | 109 |

STRUCTURE THEORY FOR NONCOMPACT GROUPS 1 Cartan Decomposition and the Unitary Trick | 113 |

2 Iwasawa Decomposition | 116 |

3 Regular Elements Weyl Chambers and the Weyl Group | 121 |

4 Other Decompositions | 126 |

5 Parabolic Subgroups | 132 |

6 Integral Formulas | 137 |

7 BorelWeil Theorem | 142 |

8 Problems | 147 |

HOLOMORPHIC DISCRETE SERIES 1 Holomorphic Discrete Series for SU11 | 150 |

2 Classical Bounded Symmetric Domains | 152 |

3 HarishChandra Decomposition | 153 |

4 Holomorphic Discrete Series | 158 |

5 Finiteness of an Integral | 161 |

6 Problems | 164 |

INDUCED REPRESENTATIONS 1 Three Pictures | 167 |

2 Elementary Properties | 169 |

3 Bruhat Theory | 172 |

4 Formal Intertwining Operators | 174 |

5 iindikinKarpelevic Formula | 177 |

6 Estimates on Intertwining Operators Part I | 181 |

7 Analytic Continuation of Intertwining Operators Part I | 183 |

8 Spherical Functions | 185 |

9 FiniteDimensional Representations and the H function | 191 |

10 Estimates on Intertwining Operators Part II | 196 |

11 Tempered Representations and Langlands Quotients | 198 |

12 Problems | 201 |

ADMISSIBLE REPRESENTATIONS 1 Motivation | 203 |

2 Admissible Representations | 205 |

3 Invariant Subspaces | 209 |

4 Framework for Studying Matrix Coefficients | 215 |

5 HarishChandra Homomorphism | 218 |

6 Infinitesimal Character | 223 |

7 Differential Equations Satisfied by Matrix Coefficients | 226 |

8 Asymptotic Expansions and Leading Exponents | 234 |

Subrepresentation Theorem | 238 |

Analytic Continuation of Interwining Operators Part II | 239 |

Control of KFinite ZgcFinite Functions | 242 |

12 Asymptotic Expansions near the Walls | 247 |

Asymptotic Size of Matrix Coefficients | 253 |

Identification of Irreducible Tempered Representations | 258 |

Langlands Classification of Irreducible Admissible Representations | 266 |

16 Problems | 276 |

CONSTRUCTION OF DISCRETE SERIES 1 Infinitesimally Unitary Representations | 281 |

2 A Third Way of Treating Admissible Representations | 282 |

3 Equivalent Definitions of Discrete Series | 284 |

4 Motivation in General and the Construction in SU11 | 287 |

5 FiniteDimensional Spherical Representations | 300 |

6 Duality in the General Case | 303 |

7 Construction of Discrete Series | 309 |

8 Limitations on K Types | 320 |

9 Lemma on Linear Independence | 328 |

10 Problems | 330 |

GLOBAL CHARACTERS 1 Existence | 333 |

2 Character Formulas for SL2 R | 338 |

3 Induced Characters | 347 |

4 Differential Equations | 354 |

5 Analyticity on the Regular Set Overview and Example | 355 |

6 Analyticity on the Regular Set General Case | 360 |

7 Formula on the Regular Set | 368 |

6 Properties of Ff | 407 |

7 Hirais Patching Conditions | 421 |

8 Problems | 425 |

EXHAUSTION OF DISCRETE SERIES 1 Boundedness of Numerators of Characters | 426 |

2 Use of Patching Conditions | 432 |

3 Formula for Discrete Series Characters | 436 |

4 Schwartz Space | 447 |

5 Exhaustion of Discrete Series | 452 |

6 Tempered Distributions | 456 |

7 Limits of Discrete Series | 460 |

8 Discrete Series of M | 467 |

9 Schmids Identity | 473 |

10 Problems | 476 |

PLANCHEREL FORMULA 1 Ideas and Ingredients | 482 |

3 RealRankOne Groups Part II | 485 |

4 Averaged Discrete Series | 494 |

5 Sp2R | 502 |

6 General Case | 511 |

7 Problems | 512 |

IRREDUCIBLE TEMPERED REPRESENTATIONS 1 SL2 R from a More General Point of View | 515 |

2 Eisenstein Integrals | 520 |

3 Asymptotics of Eisenstein Integrals | 526 |

4 The r Functions for Intertwining Operators | 535 |

5 First Irreducibility Results | 540 |

6 Normalization of Intertwining Operators and Reducibility | 543 |

7 Connection with Plancherel Formula when dim A 1 | 547 |

8 HarishChandras Completeness Theorem | 553 |

9 R Group | 560 |

10 Action by Weyl Group on Representations of M | 568 |

11 Multiplicity One Theorem | 577 |

12 Zuckerman Tensoring of Induced Representations | 584 |

13 Generalized Schmid Identities | 587 |

14 Inversion of Generalized Schmid Identities | 595 |

15 Complete Reduction of Induced Representations | 599 |

16 Classification | 606 |

17 Revised Langlands Classification | 614 |

18 Problems | 621 |

MINIMAL K TYPES 1 Definition and Formula | 626 |

2 Inversion Problem | 635 |

3 Connection with Intertwining Operators | 641 |

4 Problems | 647 |

UNITARY REPRESENTATIONS 1 SL2RandSL2C | 650 |

2 Continuity Arguments and Complementary Series | 653 |

3 Criterion for Unitary Representations | 655 |

4 Reduction to Real Infinitesimal Character | 660 |

5 Problems | 665 |

ELEMENTARY THEORY OF LIE GROUPS 1 Lie Algebras | 667 |

2 Structure Theory of Lie Algebras | 668 |

3 Fundamental Group and Covering Spaces | 670 |

4 Topological Groups | 673 |

5 Vector Fields and Submanifolds | 674 |

6 Lie Groups | 679 |

REGULAR SINGULAR POINTS OF PARTIAL DIFFERENTIAL EQUATIONS 1 Summary of Classical OneVariable Theory | 685 |

2 Uniqueness and Analytic Continuation of Solutions in Several Variables | 690 |

3 Analog of Fundamental Matrix | 693 |

4 Regular Singularities | 697 |

5 Systems of Higher Order | 700 |

6 Leading Exponents and the Analog of the Indicial Equation | 703 |

7 Uniqueness of Representation | 710 |

ROOTS AND RESTRICTED ROOTS FOR CLASSICAL | 713 |

NOTES | 719 |

747 | |

763 | |

### Common terms and phrases

abelian admissible representation apply Cartan subalgebra Cartan subgroup Cayley transform Chapter commuting completes the proof conjugate Corollary corresponding decomposition define denote differential discrete series representation dominant eigenvalue element equivalent finite follows function given global character group G Haar measure Harish-Chandra Hence highest weight holomorphic identity imaginary implies induced representation infinitesimal character inner product intertwining operators invariant isomorphism K-finite Langlands Lemma Let G Lie algebra Lie group limit of discrete linear connected linear connected reductive matrix coefficients minimal K type multiple nonzero notation obtain orthogonal parabolic subgroup parameter Plancherel formula polynomial positive roots positive system Problems proof of Theorem Proposition prove quotient rank G real analytic reduced representation of G restricted roots result right side root of g root system satisfies scalar simple roots simply connected subset subspace suppose tion unitary representation Weyl group write Z(gc