CONTENTS
Preface
Index Theory for Wiener-Hopf Operators on Convex Cones A. Alldridge
1. Wiener-Hopf Operators on the Half Line
1.1. Problem and Classical Solution
1.2. C -Algebraic Reformulation
2. Composition Series for the Wiener-Hopf Algebra
2.1. Multivariate Wiener-Hop! Operators
2.2. Groupoid C -Algebras
2.3. An Application of Groupoid C"-Algebras
2.4. The Wiener-Hop! Algebra as a Groupoid C"-Algebra
2.5. The Composition Series
3. Index Theory for the Wiener-Hopf Algebra
3.1. Analytical Index
3.2. A Connecting Bibundle
4. The Case of Polyhedral Cones
4.1. Cellular Differential and Index Maps
4.2. Atiyah-Hirzebruch Spectral Sequence
References
A Variant of the Frobenius Reciprocity for Restricted Representations on Nilpotent Lie Groups A. Baklouti, H. Fujiwara and 1. Ludwig
1. Introduction
2. Backgrounds
2.1. Some notations and basic facts
2.2. The orbit theory
2.3. Induced and restricted representations
2.4. Coexponential bases
2.5. Fmbenius vectors
3. The main result
4. Proof of the main result
5. Examples
5.1. On threadlike nilpotent Lie groups
5.2. An eight dimensional nilpotent Lie group
Acknowledgements
References
Mehler Hemigroups and Embedding of Discrete Skew Convolution Semigroups on Simply Connected Nilpotent Lie Groups P. Becker-Kern and W. Hazod
1. Introduction
2. Results on finite-dimensional vector spaces
3. Hemigroup embedding
4. Space-time enlargement and Mehler hemigroups
5. Representations by generalized Lie-Trotter formulas
References
Transforms, Polynomials and Integrable Models Associated with Reflection Groups C. F. Dunkl
1. Introduction
2. Intertwining operator and an integral transform
3. Group-theoretic aspects of V and K
4. Intertwining operator for the group B2
5. Quantum-mechanical models
6. Nonsymmetric Jack polynomials
Acknowledgments
References
Positive and Negative Definite Functions on a Hypergroup and Its Dual H. Heyer
Preface
1. Introduction
2. Bounded positive definite functions
2.1. The dual space of a hypergroup
2.2. The Bochner-Weil and Levy theorems
3. Strongly positive definite functions in the commutative case
3.1. Dual space and dual hypergroup
3.2. The Bochner and Levy theorems
4. Strongly negative definite functions in the commutative case
4.1. The Schoenberg correspondence
4.2. Local Sturm-Liouville hypergroups
4.3. Levy-Khintchine decomposition
5. Conditionally positive definite functions
5.1. Cocycles and co boundaries
5.2. Accompanying arrays
5.3. Infinitely divisible positive definite functions
References
Symbolic Dynamics for the Geodesic Flow on Locally Symmetric Orbifolds of Rank One 1. Hilgert and A. D. Pohl
1. Introduction
2. Symbolic dynamics
3. Cusp expansions
3.1. Associated transfer operators
3.2. The case of the modular surface
4. Outlook
References
Towards Projective Representations and Spin Characters of Finite and Infinite Complex Reflection Groups T. Hirai, E. Hirai and A. Hora
1. Wreath product groups and complex reflection groups
1.1. Wreath products 6 n {T) and their canonical subgroups
1.2. Complex reflection groups and their inductive limits
2. Characters of factor representations of a group
2.1. Some generalities for characters of factor representations
2.2. Chamcter formula for wreath products 6 00 (T) and its canonical subgroups
3. Generality for projective representations
3.1. Schur multiplier H2(G,e X ) and representation groups
3.2. Case of 6 n , ~n and 6
4. Representation groups of complex reflection groups
4.1. Schur multiplier and representation groups of Weyl groups
4.2. Representation groups of generalized symmetric groups
4.3. Representation groups of G(m,p, n) = 6 n (Z ... )S(p)
5. Structure of R(G(m,l,nΒ») and R(G(m,l,ooΒ»)
6. Properties of spin characters of generalized symIlletric groups G(m,l,n) and G(m,l,oo)
References
An Irreducible Homogeneous Non-Selfdual Cone of Arbitrary Rank Linearly Isomorphic to the Dual Cone H. Ishi and T. Nomura
1. Introduction
2. Description of the cone
3. Dual cone
References
Extensions of Commutative Hypergroups S. Kawakami
1. Introduction
2. Preliminaries
3. Splitting extensions of hypergroups
4. Non-splitting extensions of hypergroups
5. Cohomological approach to the extension problem on hypergroups
References
Real Hardy Space for Jacobi Analysis and Its Applications T. Kawazoe
1. Introduction
2. Notations
3. A key relation
4. Real Hardy spaces
5. Littlewood-Paley g-function
6. Lusin area function
References
A Remark on Random Groups of the Triangular Model T. Kondo
1. Introduction
2. CAT(O) spaces
References
Asymptotic Behaviour of Quantum Markov Processes B. K iimmerer
1. From Graphs to Markov Processes
1.1. An Example
1.2. Road-Coloured Graphs and Markov Processes
1.3. Synchronizing Words
1.4. Outlook
2. Markov Processes from Road-Coloured Graphs in Operator Algebras
2.1. Markov Processes from Road-Coloured Graphs
2.2. Algebraization
2.3. Quantization
3. Synchronising Words and Asymptotic Completeness
3.1. Asymptotic Completeness
3.2. Synchronizing Words for Countable State Spaces
4. Synchronizing Words and Preparation of States
4.1. Preparation of States
4.2. Two-Sided Quantum Markov Processes
4.3. The Micro-Maser and Preparation of Quantum States
Acknowledgement
References
On a-amenability of Commutative Hypergroups R. Lasser
1. Introduction
2. Preliminaries and a-amenability
3. Convergence to cp-invariance
4. Examples and comments
References
Moments of Characteristic Polynomials of a Random Matrix Associated with Compact Symmetric Spaces S. Matsumoto
1 Introduction
2 Dyson's circular ensembles
2.1 Definition of circular ,a-ensembles
2.1.1 COE ({3 = 1)
2.1.2 CUE (f3 = 2)
2.1.3 CSE ((3 = 4)
2.1.4 Quaternion expression for CSE matrices
2.2 Jack polynomials
2.2.1 Partitions
2.2.2 Definition of Jack polynomials
2.3 Characteristic polynomial average
3 Jacobi ensembles
3.1 Definition of Jacobi ensembles
3.2 Heckman and Opdam's Jacobi polynomials
3.3 Characteristic polynomial average
4 Closing remarks
References
Semi-bounded Unitary Representations of Infinite-Dimensional Lie Groups K.-H. Neeb
Introduction
1. Semi-equicontinuous convex sets
2. Momentum sets of smooth unitary representations
3. Bounded representations
4. The abelian case
References
A Note on the Capelli Identities for Symmetric Pairs of Hermitian Type K. Nishiyama and A. Wachi
1. Introduction
2. Case C
2.1. Formulas for the Weil representation
2.2. Capelli identity for Case <C (1)
2.3. Capelli identity for Case C (2)
2.4. Proof of the Capelli identity for Case C (I)
2.4.1. Proof of Proposition 2.1
2.4.2. Proof of Proposition 2.2
2.5. Proof of the Capelli identity for Case C (2)
3. Case lR.
3.1. Formulas for the Weil representation
3.2. Capelli identity for Case lR
4. Case lHI
4.1. Formulas for the Weil representation
4.2. Capelli identity for Ca8e lHl
Acknowledgment
Appendix A. Formula for c~
References
Convolution Algebras for Multivariable Bessel Functions M. Rosler
1. Introduction
2. Bessel functions on matrix cones
3. Radial analysis on matrix spaces, integral formulas, and Hankel transforms
4. Hypergroups
5. Bessel hypergroups on matrix cones
6. Wishart-distributions and probabilistic aspects
7. Bessel functions associated with root system Bq
8. Hypergroups on the Weyl chamber
Acknowledgments
References
Irreducible Decompositions of Unitary Representations and Extremal Decompositions of Positive-Definite Functions on Groups H. Shimomura
1. INTRODUCTION
2. IRREDUCIBLE DECOMPOSITIONS OF UNITARY REPRESENTATIONS OF INFINITE DIMENSIONAL GROUPS
2.1 Presentation of the basic results.
2.2 Representations of the group of diffeomorphisms on M.
2.3 Specific examples of realization.
3. IRREDUCIBLE DECOMPOSITIONS OF UNITARY REPRESENTATIONS AND EXTEREMAL DECOMPOSITIONS OF PDF
3.1 Presentation of the problem.
3.2 Main results
3.3 Characters on the infinite permutation group, and their disintegrations.
REFERENCES
Ensembles of Hermitian Random Matrices Associated to Symmetric Spaces M. Stolz
1. Introduction
2. A Wigner-type theorem
2.1. A proof of Theorem 1.1
2.2. Extensions
3. Invariant ensembles and large deviations
Acknowledgement
References
Limit Theorems for Radial Random Walks on Homogeneous Spaces with Growing Dimensions M. Voit
1. Introduction
2. Radial limit theorems on jRP for p ---* 00
3. Radial limit theorems on the matrix spaces Mp,q for p --+ 00
4. A central limit theorem for hyperbolic spaces and Jacobi convolutions on [0, oc[
References