Representation of Lie Groups and Special Functions

Transcription

1 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

2 Preface xiii Chapter 1: /i-harmonic Polynomials, /i-hankel Transform, and Coxeter Groups Coxeter Groups Dihedral groups Generating elements and defining relations Coxeter groups Coxeter matrices. The classification of irreducible Coxeter groups Invariants of Coxeter groups Invariant bilinear forms Irreducible representations Representations on polynomials Representations on a group algebra Polynomials p g (t) The h Laplacian and ft-harmonic Polynomials The /i-laplacian ^-Harmonic polynomials Differential-difference operators TJ The operators I? Averaging operator The minimum principle Polynomials related to representations Examples of/i-harmonic polynomials The Poisson Kernel for h-harmonic Functions /i-exact 1-forms The intertwining operator Kernels K r (x,y) The space? (R n, h 2 d/j.) The bilinear form on polynomials The operator exp (-A ft /2) Properties of AT r (x, y) and K(x,y) The Poisson kernel /i-hankel Transform Definition 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

3 /i-bessel functions / 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 Chapter 2: Symmetric Polynomials and Symmetric Functions Simplest Symmetric Polynomials and Symmetric Functions Partitions and their orderings The ring of symmetric functions. Monomial symmetric polynomials and functions Elementary symmetric functions Complete symmetric functions Power-sum symmetric functions Schur functions The Scalar Product on A and Skew Schur Functions The scalar product on A Matrices of transitions Skew Schur functions Summation formulas containing Schur functions Hall-Littlewood Polynomials and Functions Definition The functions q\ and S\ The scalar product on A(Q(f)) Skew Hall-Littlewood polynomials Jack Symmetric Polynomials and Functions Definition Symmetric functions <7 n (x; a) Differential operator D(a) Duality relation Skew Jack symmetric functions Expression for J M in terms of J^-i Expression for J\{1,..., l;or) Expressions for c\(a), r\(a), and jx(ct) Expression for J\/n Jack polynomials and zonal polynomials Generalized Binomial Coefficients and Jack Polynomials Generalized binomial coefficients The main theorem 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 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 Relations for Jack polynomials 137

4 Estimate of Jack polynomials Jack polynomials of two variables Macdonald Symmetric Polynomials and Functions The space A(F) The operator D Macdonald symmetric functions and polynomials Duality relation Skew Macdonald symmetric functions Macdonald's Orthogonal Polynomials Associated with Root Systems Root systems Classification of irreducible root systems Admissible pairs of irreducible root systems The group algebra A Scalar products on A The operator E Orthogonal polynomials associated with root systems Special cases of polynomials P\ 179 Chapter 3: Hypergeometric Functions Related to Jack Polynomials Hypergeometric Functions Related to Jack Polynomials Definition Differential equations for 2F[ d) Integral representation of 2F[ d) The integral relation for Jack polynomials Properties of hypergeometric functions Symmetric orthogonal polynomials associated to Jack polynomials Hypergeometric Functions of Two Variables Expressions in terms of the functions i F\ and 2^ The Appell function F Expression for 2F[ d) in terms of F Generalized Laplace transform Generalized Laguerre polynomials related to Jack polynomials Hankel transform Hypergeometric Functions Associated to Root Systems Introduction 222

5 Zonal spherical functions Hypergeometric functions associated to root systems Symmetric Jacobi polynomials associated to root systems Relations between Jack polynomials and Jacobi polynomials associated to the root system A n -\ Jacobi polynomials and hypergeometric functions associated to the root system BC n Relation between Jacobi polynomials associated to Jack polynomials and Jacobi polynomials associated to the root system BC n Basic Hypergeometric Functions Related to Schur Polynomials Definition 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 Determinental formulas for rv'l+i and r^ Summation formulas Integral representation Transformation properties of 2Vi 263 Chapter 4: Clebsch Gordan Coefficients and Racah Coefficients of Finite Dimensional Representations Finite Dimensional Representations of Semisimple Lie Groups and Algebras Semisimple Lie groups and algebras Finite dimensional representations Finite dimensional representations of semisimple Lie algebras Properties of a Weyl group Tensor products of finite dimensional representations Expressions for representation multiplicities in terms of weight multiplicities 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 Ranges of disposition of highest weights in decompositions of tensor products Upper bound for multiplicities of representations in tensor products The theorem on shifts of highest weights Expressions for n,- 288

6 4.3. Clebsch-Gordan Coefficients of Compact Groups Definition CGC's and matrix elements of representations Problems of uniqueness for CGC's Permutation symmetry of CGC's Clebsch-Gordan Coefficients and Scalar Factors Subgroup chains and corresponding orthonormal bases Definition of scalar factors Orthogonality relations for scalar factors Permutation symmetries of scalar factors Racah Coefficients Definition Special cases of RC's Permutation symmetries RC's and characters of representations 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 Clebsch-Gordan Coefficients of the Group U(n) and the Denominator Function CGC's of the tensor product T m g) T( P 0 ) CGC's with multiplicities CGC's with multiplicities and scalar factors The denominator function 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 The path sum formula The algebra of Boson Operators and Clebsch-Gordan Coefficients of the Group U(n) Creation and annihilation operators The algebra of creation and annihilation operators Boson and dual boson polynomials Properties of boson polynomials Construction of boson polynomials Symmetry relation for scalar factors of the tensor product T m (g) T( Pt o) 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)) RC's and scalar factors 353

7 5.3. Hypergeometric Series Well-Poised in U(n) Generalized hypergeometric series related to the group U(n) Summation formulas for well-poised series An analogue of the Whipple formula 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 The recurrence relation for Wq"\z) Integral relations for F (n) Polynomials Related to Hypergeometric Series Well-Poised in U(n) Functions G q n) Symmetries of functions G q n) 373; The functions ^G^ The functions G^n) The functions G q n) 5.5. Basic Hypergeometric Series Well-Poised in U{n) and Their Properties Basic hypergeometric functions well-poised in U(n) Summation formulas 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 Chapter 6: Gel'fand Hypergeometric Functions General Hypergeometric Series Introduction Horn hypergeometric series Gel'fand general hypergeometric series General hypergeometric series associated with subspaces General hypergeometric series with common convergence domain Gel'fand General Hypergeometric Functions General hypergeometric systems of equations Spaces of general hypergeometric functions General hypergeometric functions associated with subspaces Generalized hypergeometric functions Gel'fand q Hypergeometric Series and (A, I)) Hypergeometric Series Horn g-hypergeometric series

8 General g-hypergeometric series (V,X>)-Hypergeometric series Difference analogues of hypergeometric functions Hypergeometric Functions on Real Grassmannians Real Grassmannians The Radon transform Hypergeometric functions on Grassmannians Hypergeometric systems of equations on Grassmannian Hypergeometric Functions and Hypergeometric Series on Complex Grassmannians "^ Hypergeometric systems of equations and hypergeometric functions on V General hypergeometric functions on C?3,6(C) General hypergeometric series on Zjt n (C) Reduction relations Hypergeometric functions on strata Hypergeometric Functions on Strata of Grassmannian G3,6(C) Strata of Grassmannian <J3,6(C) General hypergeometric functions in neighborhoods of one-orbit strata Bases of spaces of hypergeometric functions on nondegenerate strata Hypergeometric functions on strata of type A Hypergeometric functions on strata of type B Hypergeometric functions on Grassmannian (^^(C) Hypergeometric functions on strata of type C Bibliography 463 Supplementary Bibliography 484 Bibliography Notes 488 Subject Index 494