Heckman-Opdam Hypergeometric Functions Research Articles

Weighted Bergman kernels for nearly holomorphic functions on bounded symmetric domains

We identify the standard weighted Bergman kernels of spaces of nearly holomorphic functions, in the sense of Shimura, on bounded symmetric domains. This also yields a description of the analogous kernels for spaces of “invariantly-polyanalytic” functions — a generalization of the ordinary polyanalytic functions on the ball which seems to be the most appropriate one from the point of view of holomorphic invariance. In both cases, the kernels turn out to be given by certain spherical functions, or equivalently Heckman-Opdam hypergeometric functions, and a conjecture relating some of these to a Faraut-Koranyi hypergeometric function is formulated based on the study of low rank situations. Finally, analogous results are established also for compact Hermitian symmetric spaces, where explicit formulas in terms of multivariable Jacobi polynomials are given.

Freezing limits for Calogero–Moser–Sutherland particle models

AbstractOne‐dimensional interacting particle models of Calogero–Moser–Sutherland type with N particles can be regarded as diffusion processes on suitable subsets of like Weyl chambers and alcoves with second‐order differential operators as generators of the transition semigroups, where these operators are singular on the boundaries of the state spaces. The most relevant examples are multivariate Bessel processes and Heckman–Opdam processes in a compact and noncompact setting where in all cases, these processes are related to special functions associated with root systems. More precisely, the transition probabilities can be described with the aid of multivariate Bessel functions, Jack and Heckman–Opdam Jacobi polynomials, and Heckman–Opdam hypergeometric functions, respectively. These models, in particular, form dynamic eigenvalue evolutions of the classical random matrix models like β‐Hermite, β‐ Laguerre, and β‐Jacobi, that is, MANOVA, ensembles. In particular, Dyson's Brownian motions and multivariate Jacobi processes are included. In all cases, the processes depend on so‐called coupling parameters. We review several freezing limit theorems for these diffusions where, for fixed N, one or several of the coupling parameters tend to ∞. In many cases, the limits will be N‐dimensional normal distributions and, in the process case, Gauss processes. However, in some cases, normal distributions on half spaces and distributions related to some other ensembles appear as limits. In all cases, the limits are connected with the zeros of the classical one‐dimensional orthogonal polynomials of order N.

The Dunkl-Laplace transform and Macdonald’s hypergeometric series

We continue a program generalizing classical results from the analysis on symmetric cones to the Dunkl setting for root systems of typeAA. In particular, we prove a Dunkl-Laplace transform identity for Heckman-Opdam hypergeometric functions of typeAAand more generally, for the associated Opdam-Cherednik kernel. This is achieved by analytic continuation from a Laplace transform identity for non-symmetric Jack polynomials which was stated, for the symmetric case, as a key conjecture by I.G. Macdonald [arXiv:1309.4568v1]. Our proof for the Jack polynomials is based on Dunkl operator techniques and the raising operator of Knop and Sahi. Moreover, we use these results to establish Laplace transform identities between hypergeometric series in terms of Jack polynomials. Finally, we conclude with a Post-Widder inversion formula for the Dunkl-Laplace transform.

Limits and fluctuations of p-adic random matrix products

We show that singular numbers (also known as elementary divisors, invariant factors or Smith normal forms) of products and corners of random matrices over $${\mathbb {Q}}_p$$ are governed by the Hall–Littlewood polynomials, in a structurally identical manner to the known relations between singular values of complex random matrices and Heckman–Opdam hypergeometric functions. This implies that the singular numbers of a product of corners of Haar-distributed elements of $$\mathrm {GL}_N({\mathbb {Z}}_p)$$ form a discrete-time Markov chain distributed as a Hall–Littlewood process, with the number of matrices in the product playing the role of time. We give an exact sampling algorithm for the Hall–Littlewood processes which arise by relating them to an interacting particle system similar to PushTASEP. By analyzing the asymptotic behavior of this particle system, we show that the singular numbers of such products obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions. In the limit of large matrix size, we also show that the analogues of the Lyapunov exponents for matrix products have universal limits within this class of $$\mathrm {GL}_N({\mathbb {Z}}_p)$$ corners.

Hypergeometric Functions of Type BC and Standard Multiplicities

Abstract We study the Heckman–Opdam hypergeometric functions associated with a root system of type $BC$ and a multiplicity function that is allowed to assume some nonpositive values (a standard multiplicity function). For such functions, we obtain positivity properties and sharp estimates that imply a characterization of the bounded hypergeometric functions. As an application, our results extend the known properties of Harish-Chandra’s spherical functions on Riemannian symmetric spaces of the non-compact type $G/K$ to spherical functions over homogeneous vector bundles on $G/K$, which are associated with certain small $K$-types.

Majorization and Spherical Functions

Abstract In this paper, we generalize a result of Cuttler, Greene, Skandera, and Sra that characterizes the majorization order on Young diagrams in terms of nonnegative specializations of Schur polynomials. More precisely, we introduce a generalized notion of majorization associated to an arbitrary crystallographic root system $\Phi $ and show that it admits a natural characterization in terms of the values of spherical functions on any Riemannian symmetric space with restricted root system $\Phi .$ We also conjecture a further generalization of this theorem in terms of Heckman–Opdam hypergeometric functions.

Sonine Formulas and Intertwining Operators in Dunkl Theory

Abstract Let $V_k$ denote Dunkl’s intertwining operator associated with some root system $R$ and multiplicity $k$. For two multiplicities $k, k^{\prime }$ on $R$, we study the intertwiner $V_{k^{\prime },k} = V_{k^{\prime }}\circ V_k^{-1}$ between Dunkl operators with multiplicities $k$ and $k^{\prime }.$ It has been a long-standing conjecture that $V_{k^{\prime },k}$ is positive if $k^{\prime } \geq k \geq 0.$ We disprove this conjecture by constructing counterexamples for root system $B_n$. This matter is closely related to the existence of Sonine-type integral representations between Dunkl kernels and Bessel functions with different multiplicities. In our examples, such Sonine formulas do not exist. As a consequence, we obtain necessary conditions on Sonine formulas for Heckman–Opdam hypergeometric functions of type $BC_n$ and conditions for positive branching coefficients between multivariable Jacobi polynomials.

A Laplace-type representation for some generalized spherical functions of type $BC$

In [M. R\"osler and M. Voit. Integral Representation and Uniform Limits for Some Heckman-Opdam Hypergeometric Functions of type BC, Transactions of the American Mathematical Society, Vol. 368, No. 8, 6005-6032, 2016.], R\"osler and Voit give a formula for generalized spherical functions of type $BC$ in terms of the spherical functions of type A. We use this formula to describe precisely the support of the associated generalized Abel transform. Furthermore, we derive a similar formula for the generalized spherical functions in the rational Dunkl setting. The support of the intertwining operator V is also deduced. We also show, as a consequence, that a Laplace-type expression exists for the generalized spherical functions both in the trigonometric Dunkl setting and in the rational Dunkl setting.

Paley–Wiener theorem for line bundles over compact symmetric spaces and new estimates for the Heckman–Opdam hypergeometric functions

AbstractPaley–Wiener type theorems describe the image of a given space of functions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous space. In this article we proved a Paley–Wiener theorem for smooth sections f of homogeneous line bundles on a compact Riemannian symmetric space . It characterizes f with small support in terms of holomorphic extendability and exponential growth of their χ‐spherical Fourier transforms, where χ is a character of K. An important tool in our proof is a generalization of Opdam's estimate for the hypergeometric functions associated to multiplicity functions that are not necessarily positive. At the same time the radius of the domain where this estimate is valid is increased. This is done in an appendix.

Superintegrability of d-Dimensional Conformal Blocks.

We observe that conformal blocks of scalar four-point functions in a d-dimensional conformal field theory can be mapped to eigenfunctions of a two-particle hyperbolic Calogero-Sutherland Hamiltonian. The latter describes two coupled Pöschl-Teller particles. Their interaction, whose strength depends smoothly on the dimension d, is known to be superintegrable. Our observation enables us to exploit the rich mathematical literature on Calogero-Sutherland models in deriving various results for conformal field theory. These include an explicit construction of conformal blocks in terms of Heckman-Opdam hypergeometric functions. We conclude with a short outlook, in particular, on the consequences of integrability for the theory of conformal blocks.

A new integral formula for Heckman–Opdam hypergeometric functions

We provide Harish-Chandra type formulas for the multivariate Bessel functions and Heckman–Opdam hypergeometric functions as representation-valued integrals over dressing orbits. Our expression is the quasi-classical limit of the realization of Macdonald polynomials as traces of intertwiners of quantum groups given by Etingof and Kirillov Jr. in [10]. Integration over the Liouville tori of the Gelfand–Tsetlin integrable system and adjunction for higher Calogero–Moser Hamiltonians recovers and gives a new proof of the integral realization over Gelfand–Tsetlin polytopes which appeared in the recent work [4] of Borodin and Gorin on the β-Jacobi corners ensemble.

Integral representation and uniform limits for some Heckman-Opdam hypergeometric functions of type \textsl{BC}

The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various limit transitions known for such hypergeometric functions. In the present paper, we use an explicit form of the Harish-Chandra integral representation as well as an interpolated variant in order to obtain two limit results, each of them for three continuous classes of hypergeometric functions of type BC which extend the group cases over the fields R , C , H . \mathbb R, \mathbb C, \mathbb H. These limits are distinguished from the known results by explicit and uniform error bounds. The first limit realizes the approximation of the spherical functions of infinite dimensional Grassmannians of fixed rank; here hypergeometric functions of type A appear as limits. The second limit is a contraction limit towards Bessel functions of Dunkl type.

Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications

A series expansion for Heckman–Opdam hypergeometric functions φλ is obtained for all λ∈aC⁎. As a consequence, estimates for φλ away from the walls of a Weyl chamber are established. We also characterize the bounded hypergeometric functions and thus prove an analogue of the celebrated theorem of Helgason and Johnson on the bounded spherical functions on a Riemannian symmetric space of the noncompact type. The Lp-theory for the hypergeometric Fourier transform is developed for 0<p<2. In particular, an inversion formula is proved when 1⩽p<2.

Positive convolution structure for a class of Heckman–Opdam hypergeometric functions of type BC

In this paper, we derive explicit product formulas and positive convolution structures for three continuous classes of Heckman–Opdam hypergeometric functions of type BC. For specific discrete series of multiplicities these hypergeometric functions occur as the spherical functions of non-compact Grassmann manifolds G / K over one of the skew fields F = R , C , H . We write the product formula of these spherical functions in an explicit form which allows analytic continuation with respect to the parameters. In each of the three cases, we obtain a series of hypergroup algebras which include the commutative convolution algebras of K-biinvariant functions on G as special cases. The characters are given by the associated hypergeometric functions.

Bessel functions for root systems via the trigonometric setting

By taking an appropriate limit, we obtain the Bessel functions related to root systems as limit of Heckman-Opdam hypergeometric functions . A more general class of Bessel functions is also investigated, which we shall call the \Theta-Bessel functions. Explicit formuals for the \Theta-Bessel functions are obtained when the multiplicity functions are even and positive integer-valued. This class encloses the Bessel functions on the tangent space at the origin of non-compact causal symmetric spaces, were an integral representation for these special functions is shown.