Infinite Dimensional Harmonic Analysis IV

The following sections are included:

• PREFACE

• CONTENTS

We consider Wiener–Hopf operators with continuous symbol, defined on the L² space of a convex cone (w.r.t. Lebesgue measure), and the C*-algebra of bounded operators generated by them.

In the (classical) case of a Wiener–Hopf operator on the half line, the property of being Fredholm can be characterised in terms of the symbol, and if Fremdholmness obtains, the numerical index can be computed in terms of topological data and the symbol. We show how to extend this theory to a large class of cones characterised by two topologial conditions.

For the special case of polyhedral cones, we find an interesting connection between the homological cellular differential and the index maps associated with the corresponding C*-algebra.

This work was partly jointly conducted with Troels Johansen of the University of Kiel.

Let G be a nilpotent connected and simply connected Lie group, K an analytic subgroup of G and π a unitary and irreducible representation of G. We study in this paper a variant of the Frobenius reciprocity for the restricted representation π_|K of π on K. It consists in proving that generically, the multiplicity of any isotopic component involved in the central canonical disintegration of π_|K coincides with the dimension of a certain space of generalized tempered distributions which are semi-invariant under the action of a subgroup of K. This problem was considered and partially solved in our previous work ¹.

It is shown how discrete skew convolution semigroups of probability measures on a simply connected nilpotent Lie group can be embedded into Lipschitz continuous semistable hemigroups by means of their generating functionals. These hemigroups are the distributions of increments of additive semi-selfsimilar processes. Considering these on an enlarged space-time group, we obtain Mehler hemigroups corresponding to periodically stationary processes of Ornstein-Uhlenbeck type, driven by certain additive processes with periodically stationary increments. The background driving processes are further represented by generalized Lie-Trotter formulas for convolutions, corresponding to a random integral approach known for finite-dimensional vector spaces.

The article surveys differential-difference operators and related special functions associated to root systems. There are applications in integral transforms, integrable many-body quantum mechanical models, nonsymmetric Jack polynomials and representations of the symmetric group. An intertwining operator associated with the group B₂ is described in detail.

This is an expository article on (conditionally) positive definite functions within the framework of hypergroups, with applications to the central limit problem for infinitesimal arrays.

We present a strategy for a geometric construction of cross sections for the geodesic flow on locally symmetric orbifolds of rank one. We work it out in detail for Γ\H, where H is the upper half plane and Γ = PΓ₀(p), p prime. Its associated discrete dynamical system naturally induces a symbolic dynamics on ℝ. The transfer operator produced from this symbolic dynamics has a particularly simple structure.

Our main aim is to study the characters of projective factor representations of infinite complex reflection groups G(m, p, ∞) = lim_n→∞ G(m, p, n) containing infinite generalized symmetric groups G(m, 1, ∞) = 𝔖_∞ (Z_m), which is similar to our study on wreath products 𝔖_∞ (T) = D_∞(T) ⋊ 𝔖_∞ of the infinite symmetric group 𝔖_∞ with any finite or compact groups T. To begin with, here we briefly review and extend some results on representation groups and spin characters of finite Weyl groups and also of generalized symmetric groups G(m, 1, n) = 𝔖_n (Z_m). Then we show some basic results for G(m, p, ∞), the inductive limits of complex reflection groups G(m, p, n).

In this note we present an irreducible homogeneous non-selfdual open convex cone of arbitrary rank r (r ≥ 3) that is linearly isomorphic to the dual cone.

The purpose of this paper is to investigate the extension problem for the category of commutative hypergroups. In fact, sufficiently many extensions can be established in general and all extensions can be determined in some concrete cases. Among them splitting extensions can be characterized. Moreover, the duality of extensions will be discussed.

Let α ≥ β ≥ -1/2 and Δ(x) = (2shx)^2α+1(2chx)^2β+1 the weight function on ℝ₊ = [0, ∞). Let L¹(Δ) denote the space of even integrable functions on ℝ with respect to Δ(x)dx. We put ɸ_t(x) = t^-1Δ(x)^-1Δ(x/t)ɸ(x/t) and define the radial maximal operator M_ɸ as usual. We introduce a real Hardy space H¹(Δ) as the set of all even locally integrable functions f on ℝ whose radial maximal function M_ɸ(f) belongs to L¹(Δ). We shall obtain a relation between H¹(Δ) and H¹(ℝ). As an application of H¹(Δ), we shall consider (H¹(Δ), L¹(Δ)) boundedness of integral operators associated to the Poisson kernel for Jacobi analysis; the Poisson maximal operator M_P, the Littlewood-Paley g-function, and the Lusin area function S. They are bounded on L^p(Δ) for p > 1, but not true for p = 1. We shall prove that M_P, g, and a modified S₊ are bounded form H¹(Δ) to L¹(Δ).

We give an exposition of a fixed-point property of random groups of the triangular model obtained in the joint work with Izeki and Nayatani.⁹

Classical Markov processes can be constructed from road-coloured graphs. We discuss an operator algebraic extension of this construction. It turns out that a road-coloured graph has a synchronizing word if and only if the associated Markov dynamics is asymptotically complete. This allows to extend the idea of a synchronizing word to the operator algebraic context. Finally, an application to the micro-maser experiment is discussed.

This text is an extended version of the talk given at the Conference on Infinite-Dimensional Harmonic Analysis, September 10-14, 2007, at the University of Tokyo.

We study the concept of α-amenability of commutative hypergroups K. We establish several characterizations of α-amenability by combining results of Bloom, Heyer, and Filbir where we add the Glicksberg-Reiter property. In addition, as examples compact hypergroups and discrete polynomial hypergroups on ℕ₀ are discussed.

The following sections are included:

• Introduction

• Dyson's circular ensembles
  • Definition of circular β-ensembles
    • COE (β = 1)
    • CUE (β = 2)
    • CSE (β = 4)
    • Quaternion expression for CSE matrices
  • Jack polynomials
    • Partitions
    • Definition of Jack polynomials
  • Characteristic polynomial average

• Jacobi ensembles
  • Definition of Jacobi ensembles
  • Heckman and Opdam's Jacobi polynomials
  • Characteristic polynomial average

• Closing remarks

• Acknowledgements

• References

In this note we introduce the concept of a semi-bounded unitary representations of an infinite-dimensional Lie group G. Semi-boundedness is defined in terms of the corresponding momentum set in the dual 𝔤' of the Lie algebra 𝔤 of G. After dealing with some functional analytic issues concerning certain weak-*-locally compact subsets of dual spaces, called semi-equicontinuous, we characterize unitary representations which are bounded in the sense that their momentum set is equicontinuous, we characterize semi-bounded representations of locally convex spaces in terms of spectral measures, and we also describe a method to compute momentum sets of unitary representations of reproducing kernel Hilbert spaces of holomorphic functions.

We get several identities of differential operators in determinantal form. These identities are non-commutative versions of the formula of Cauchy-Binet or Laplace expansions of determinants, and if we take principal symbols, they are reduced to such classical formulas. These identities are naturally arising from the generators of the rings of invariant differential operators over symmetric spaces, and have strong resemblance to the classical Capelli identities. Thus we call those identities the Capelli identities for symmetric pairs.

In this article we give an overview of recent results on hypergroup algebras which are associated with Bessel functions on cones of positive semidefinite matrices and a class of Dunkl-type Bessel functions for root systems of type B. The Bessel hypergroups on matrix cones are obtained from a discrete series with a group-theoretic background by interpolation with respect to a dimension parameter. Passing to the matrix spectra leads to Bessel functions of Dunkl type and three continuous series of hypergroup algebras matching the Dunkl transform in the Weyl-group invariant case.

This paper concerns with two kind of decompositions of unitary representations or positive-definite functions on groups. The first one is a specific example of irredeucible decompositions of unitary representations of infinite-dimensional groups which was already obtained in [7], and which is regarded as an infinite-version descibed in [11]. In the second part, we consider positive-definite functions (PDF) ɸ on the usual locally compact groups. ɸ has an extremal decomposition through an irreducible decomposition of the unitary representation corresponoding to ɸ (by the GNS construction method). However there are other ways to get extremal decompositions, for example via the Choqet theorem. So it is interesting to find conditions which disdinguish the natural particular one from other decompositions. We describe a necessary and sufficient condition for the above problem as well as an interesting negative example related to [4, 10].

Classical physics-inspired random matrix theory had its focus on ensembles of hermitian, real symmetric, or real quaternionic matrices, which may be viewed as probability measures on the tangent spaces to the classical symmetric spaces of class A, AI or AII. Condensed matter physics provides some motivation to extend this theory to the full list of classical symmetric spaces. This contribution discusses the basic strategies to randomize matrices in this broader setting, and reviews our results in Refs. 1 (joint work with Katrin Hofmann-Credner) and 2 (joint work with Peter Eichelsbacher) on the empirical eigenvalue measure for matrix size tending to infinity.

Let X_p = G_p/K_p be homogeneous spaces with compact subgroups K_p of locally compact groups G_p with some dimension parameter p such that the double coset spaces G_p//K_p can be identified with some fixed locally compact space X. For the projections T_p : X_p → X, and for a given probability measure ν ϵ M¹(X) there exist unique "radial", i.e. K_p-invariant measures ν_p ϵ M¹(X_p) with T_p(ν_p) = ν as well as associated radial random walks on the homogeneous spaces X_p. We generally ask for limit theorems for the random variables on X for n,p → ∞. In particular we give a survey about existing results for the Euclidean spaces X_p = ℝ^p with K_p = SO(_p) and X = [0, ∞[ as well as to some matrix extension of this rank one setting. Moreover, we derive a new central limit theorem for the hyperbolic spaces X_p of dimensions p over the skew fields 𝔽 = ℝ, ℂ, ℍ.