Representation of Lie Groups and Special Functions

Representation of Lie Groups and Special Functions Recent Advances by N. Ja. Vilenkint formerly of The Correspondence Pedagogical Institute, Moscow, Russia and A.U. Klimyk Institute for Theoretical Physics, Ukrainian Academy of Sciences, Kiev, Ukraine KLUWER ACADEMIC PUBLISHERS DORDRECHT/ BOSTON / LONDON

Preface xiii Chapter 1: /i-harmonic Polynomials, /i-hankel Transform, and Coxeter Groups 1 1.1. Coxeter Groups 1 1.1.1. Dihedral groups 1 1.1.2. Generating elements and defining relations 2 1.1.3. Coxeter groups 3 1.1.4. Coxeter matrices. The classification of irreducible Coxeter groups 5 1.1.5. Invariants of Coxeter groups 7 1.1.6. Invariant bilinear forms 9 1.1.7. Irreducible representations 11 1.1.8. Representations on polynomials 14 1.1.9. Representations on a group algebra 14 1.1.10. Polynomials p g (t) 16 1.2. The h Laplacian and ft-harmonic Polynomials 18 1.2.1. The /i-laplacian 18 1.2.2. ^-Harmonic polynomials 21 1.2.3. Differential-difference operators TJ 25 1.2.4. The operators I? 27 1.2.5. Averaging operator 30 1.2.6. The minimum principle 32 1.2.7. Polynomials related to representations 34 1.2.8. Examples of/i-harmonic polynomials 38 1.3. The Poisson Kernel for h-harmonic Functions 44 1.3.1. /i-exact 1-forms 44 1.3.2. The intertwining operator 47 1.3.3. Kernels K r (x,y) 49 1.3.4. The space? (R n, h 2 d/j.). 51 1.3.5. The bilinear form on polynomials 53 1.3.6. The operator exp (-A ft /2) 54 1.3.7. Properties of AT r (x, y) and K(x,y) 57 1.3.8. The Poisson kernel 58 1.4. /i-hankel Transform 61 1.4.1. Definition 61 1.4.2. R e s t r i c t i o n of / i - H a n k e l t r a n s f o r m o n t o t h e s p h e r e... 63

1.4.3. /i-bessel functions 64 1.4.4. / i - H a n k e l t r a n s f o r m a n d classical s p e c i a l f u n c t i o n s... 6 4 Chapter 2: Symmetric Polynomials and Symmetric Functions 67 2.1. Simplest Symmetric Polynomials and Symmetric Functions 67 2.1.1. Partitions and their orderings 67 2.1.2. The ring of symmetric functions. Monomial symmetric polynomials and functions... 69 2.1.3. Elementary symmetric functions 72 2.1.4. Complete symmetric functions 73 2.1.5. Power-sum symmetric functions 74 2.1.6. Schur functions 77 2.2. The Scalar Product on A and Skew Schur Functions... 80 2.2.1. The scalar product on A 80 2.2.2. Matrices of transitions 83 2.2.3. Skew Schur functions 86 2.2.4 Summation formulas containing Schur functions.... 89 2.3. Hall-Littlewood Polynomials and Functions 92 2.3.1. Definition 92 2.3.2. The functions q\ and S\ 95 2.3.3. The scalar product on A(Q(f)) 96 2.3.4. Skew Hall-Littlewood polynomials 103 2.4. Jack Symmetric Polynomials and Functions 105 2.4.1. Definition 105 2.4.2. Symmetric functions <7 n (x; a) 109 2.4.3. Differential operator D(a) 112 2.4.4. Duality relation 115 2.4.5. Skew Jack symmetric functions 117 2.4.6. Expression for J M in terms of J^-i 119 2.4.7. Expression for J\{1,..., l;or).. 121 2.4.8. Expressions for c\(a), r\(a), and jx(ct) 123 2.4.9. Expression for J\/n 126 2.4.10. Jack polynomials and zonal polynomials 126 2.5 Generalized Binomial Coefficients and Jack Polynomials. 127 2.5.1. Generalized binomial coefficients 127 2.5.2. The main theorem 131 2.5.3. E x p r e s s i o n s for g e n e r a l i z e d b i n o m i a l coefficients.... 1 3 5 2.5.4. S p e c i a l c a s e s of g e n e r a l i z e d b i n o m i a l coefficients.... 1 3 7 2.5.5. Relations for Jack polynomials 137

2.5.6. Estimate of Jack polynomials 139 2.5.7. Jack polynomials of two variables 141 2.6. Macdonald Symmetric Polynomials and Functions.... 143 2.6.1. The space A(F) 143 2.6.2. The operator D 145 2.6.3. Macdonald symmetric functions and polynomials... 148 2.6.4. Duality relation 150 2.6.5. Skew Macdonald symmetric functions 152 2.7. Macdonald's Orthogonal Polynomials Associated with Root Systems 154 2.7.1. Root systems 154 2.7.2. Classification of irreducible root systems 157 2.7.3. Admissible pairs of irreducible root systems 164 2.7.4. The group algebra A 166 2.7.5. Scalar products on A 168 2.7.6. The operator E 172 2.7.7. Orthogonal polynomials associated with root systems. 178 2.7.8. Special cases of polynomials P\ 179 Chapter 3: Hypergeometric Functions Related to Jack Polynomials 185 3.1. Hypergeometric Functions Related to Jack Polynomials. 185 3.1.1. Definition 185 3.1.2. Differential equations for 2F[ d) 187 3.1.3. Integral representation of 2F[ d) 192 3.1.4. The integral relation for Jack polynomials 196 3.1.5. Properties of hypergeometric functions 197 3.1.6. Symmetric orthogonal polynomials associated to Jack polynomials 200 3.2. Hypergeometric Functions of Two Variables 206 3.2.1. Expressions in terms of the functions i F\ and 2^1 206 3.2.2. The Appell function F 4 209 3.2.3. Expression for 2F[ d) in terms of F 4 211 3.2.4. Generalized Laplace transform 213 3.2.5. Generalized Laguerre polynomials related to Jack polynomials 216 3.2.6. Hankel transform 221 3.3. Hypergeometric Functions Associated to Root Systems. 222 3.3.1. Introduction 222

3.3.2. Zonal spherical functions 224 3.3.3. Hypergeometric functions associated to root systems.. 227 3.3.4. Symmetric Jacobi polynomials associated to root systems 237 3.3.5. Relations between Jack polynomials and Jacobi polynomials associated to the root system A n -\ - 239 3.3.6. Jacobi polynomials and hypergeometric functions associated to the root system BC n 245 3.3.7. Relation between Jacobi polynomials associated to Jack polynomials and Jacobi polynomials associated to the root system BC n 250 3.4. Basic Hypergeometric Functions Related to Schur Polynomials 252 3.4.1. Definition 252 3.4.2. E x p r e s s i o n s for t h e V a n d e r m o n d e d e t e r m i n a n t.... 2 5 4 3.4.3. Determinental formulas for rv'l+i and r^ 256 3.4.4. Summation formulas 259 3.4.5. Integral representation 261 3.4.6. Transformation properties of 2Vi 263 Chapter 4: Clebsch Gordan Coefficients and Racah Coefficients of Finite Dimensional Representations 265 4.1. Finite Dimensional Representations of Semisimple Lie Groups and Algebras 265 4.1.1. Semisimple Lie groups and algebras 265 4.1.2. Finite dimensional representations 269 4.1.3. Finite dimensional representations of semisimple Lie algebras 270 4.1.4. Properties of a Weyl group 273 4.2. Tensor products of finite dimensional representations... 276 4.2.1. Expressions for representation multiplicities in terms of weight multiplicities 276 4.2.2. F o r m u l a s for d e c o m p o s i t i o n of t e n s o r p r o d u c t s.... 2 7 8 4.2.3. Ranges of disposition of highest weights in decompositions of tensor products 280 4.2.4 Upper bound for multiplicities of representations in tensor products 282 4.2.5 The theorem on shifts of highest weights 283 4.2.6. Expressions for n,- 288

4.3. Clebsch-Gordan Coefficients of Compact Groups 289 4.3.1. Definition 289 4.3.2. CGC's and matrix elements of representations 291 4.3.3. Problems of uniqueness for CGC's 293 4.3.4. Permutation symmetry of CGC's 296 4.4. Clebsch-Gordan Coefficients and Scalar Factors 299 4.4.1. Subgroup chains and corresponding orthonormal bases. 300 4.4.2. Definition of scalar factors 301 4.4.3. Orthogonality relations for scalar factors 305 4.4.4. Permutation symmetries of scalar factors 306 4.5. Racah Coefficients 309 4.5.1. Definition 309 4.5.2. Special cases of RC's 312 4.5.3. Permutation symmetries 313 4.5.4. RC's and characters of representations 315 4.5.5. The addition theorem and the Biedenharn-EUiott identity. ; 316 Chapter 5: Clebsch-Gordan Coefficients of the group U(n) and Related Generalizations of Hypergeometric Functions 317 5.1. Clebsch-Gordan Coefficients of the Group U(n) and the Denominator Function 317 5.1.1. CGC's of the tensor product T m g) T( P 0 ) 317 5.1.2. CGC's with multiplicities 319 5.1.3. CGC's with multiplicities and scalar factors 323 5.1.4. The denominator function 326 5.1.5. A n o t h e r definition of t h e d e n o m i n a t o r f u n c t i o n.... 3 2 8 5.1.6. The path sum formula 330 5.2. The algebra of Boson Operators and Clebsch-Gordan Coefficients of the Group U(n) 333 5.2.1. Creation and annihilation operators 333 5.2.2. The algebra of creation and annihilation operators... 336 5.2.3. Boson and dual boson polynomials 338 5.2.4. Properties of boson polynomials 342 5.2.5. Construction of boson polynomials 343 5.2.6. Symmetry relation for scalar factors of the tensor product T m (g) T( Pt o) 349 5.2.7. M a t r i x e l e m e n t s of t h e o p e r a t o r T m (5f n _i(7r/2)).... 3 5 0 5.2.8. RC's and scalar factors 353

5.3. Hypergeometric Series Well-Poised in U(n) 359 5.3.1. Generalized hypergeometric series related to the group U(n) 359 5.3.2. Summation formulas for well-poised series 361 5.3.3. An analogue of the Whipple formula 366 5.3.4. Corollaries of t h e g e n e r a l i z e d W h i p p l e i d e n t i t y.... 3 6 9 5.3.5. The recurrence relation for Wq"\z) 370 5.3.6. Integral relations for F (n) 371 5.4. Polynomials Related to Hypergeometric Series Well-Poised in U(n). 373 5.4.1. Functions G q n) 5.4.2. Symmetries of functions G q n) 373; 5.4.3. The functions ^G^ 378 5.4.4. The functions G^n) 381 5.4.5. The functions G q n) 5.5. Basic Hypergeometric Series Well-Poised in U{n) and Their Properties 385 5.5.1. Basic hypergeometric functions well-poised in U(n).. 385 5.5.2. Summation formulas 388 5.5.3. 5 - A n a l o g u e of t h e g e n e r a l i z e d W h i p p l e f o r m u l a.... 3 9 1 Chapter 6: Gel'fand Hypergeometric Functions 393 6.1. General Hypergeometric Series 393 6.1.1. Introduction 393 6.1.2. Horn hypergeometric series 395 6.1.3. Gel'fand general hypergeometric series 398 6.1.4. General hypergeometric series associated with subspaces 399 6.1.5. General hypergeometric series with common convergence domain 402 6.2. Gel'fand General Hypergeometric Functions 404 6.2.1. General hypergeometric systems of equations 404 6.2.2. Spaces of general hypergeometric functions 406 6.2.3. General hypergeometric functions associated with subspaces 407 6.2.4. Generalized hypergeometric functions 410 6.3. Gel'fand q Hypergeometric Series and (A, I)) Hypergeometric Series 412 6.3.1. Horn g-hypergeometric series 412 376 383

6.3.2. General g-hypergeometric series 413 6.3.3. (V,X>)-Hypergeometric series 415 6.3.4. Difference analogues of hypergeometric functions... 417 6.4. Hypergeometric Functions on Real Grassmannians.... 419 6.4.1. Real Grassmannians 419 6.4.2. The Radon transform 420 6.4.3. Hypergeometric functions on Grassmannians 423 6.4.4. Hypergeometric systems of equations on Grassmannian. 424 6.5. Hypergeometric Functions and Hypergeometric Series on Complex Grassmannians "^. 428 6.5.1. Hypergeometric systems of equations and hypergeometric functions on V 428 6.5.2. General hypergeometric functions on C?3,6(C)..... 432 6.5.3. General hypergeometric series on Zjt n (C) 436 6.5.4. Reduction relations 438 6.5.5. Hypergeometric functions on strata 442 6.6. Hypergeometric Functions on Strata of Grassmannian G3,6(C).... - 444 6.6.1. Strata of Grassmannian <J3,6(C) 444 6.6.2. General hypergeometric functions in neighborhoods of one-orbit strata 446 6.6.3. Bases of spaces of hypergeometric functions on nondegenerate strata 448 6.6.4. Hypergeometric functions on strata of type A 449 6.6.5. Hypergeometric functions on strata of type B 453 6.6.6. Hypergeometric functions on Grassmannian (^^(C).. 459 6.6.7. Hypergeometric functions on strata of type C... 459 Bibliography 463 Supplementary Bibliography 484 Bibliography Notes 488 Subject Index 494