Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

Let $p,q$ positive integers. The groups U_p(\b C) and U_p(\b C)\times U_q(\b C) act on the Heisenberg group H_{p,q}:=M_{p,q}(\b C)\times \b R canonically as groups of automorphisms where M_{p,q}(\b C) is the vector space of all complex $p\times q$-matrices. The associated orbit spaces may be identified with \Pi_q\times \b R and \Xi_q\times \b R respectively with the cone $\Pi_q$ of positive semidefinite matrices and the Weyl chamber \Xi_q={x\in\b R^q: x_1\ge...\ge x_q\ge 0}. In this paper we compute the associated convolutions on \Pi_q\times \b R and \Xi_q\times \b R explicitly depending on $p$. Moreover, we extend these convolutions by analytic continuation to series of convolution structures for arbitrary parameters $p\ge 2q-1$. This leads for $q\ge 2$ to continuous series of noncommutative hypergroups on \Pi_q\times \b R and commutative hypergroups on \Xi_q\times \b R. In the latter case, we describe the dual space in terms of multivariate Laguerre and Bessel functions on $\Pi_q$ and $\Xi_q$. In particular, we give a non-positive product formula for these Laguerre functions on $\Xi_q$. The paper extends the known case $q=1$ due to Koornwinder, Trimeche, and others as well as the group case with integers $p$ due to Faraut, Benson, Jenkins, Ratcliff, and others. Moreover, it is closely related to product formulas for multivariate Bessel and other hypergeometric functions of R\"osler

Bessel convolutions on matrix cones

Bessel-type convolution algebras of measures on the matrix cones of positive semidefinite q × q-matrices over R,C,H were introduced recently by Rösler. These convolutions depend on a continuous parameter, generate commutative hypergroups and have Bessel functions of matrix argument as characters. In this paper, we study the algebraic structure of these hypergroups. In particular, the subhypergroups, quotients, and automorphisms are classified. The algebraic properties are partially related to properties of random walks on these matrix Bessel hypergroups. In particular, known properties of Wishart distributions, which form Gaussian convolution semigroups on these hypergroups, are put into a new light. Moreover, limit theorems for random walks are presented. In particular, we obtain strong laws of large numbers and a central limit theorem with Wishart distributions as limits

Triple Systems of Hecke Type and Hypergroups

This is the proceedings of the 2nd Japanese-German Symposium on Infinite Dimensional Harmonic Analysis held from September 20th to September 24th 1999 at the Department of Mathematics of Kyoto University.この論文集は, 1999年9月20日から9月24日の日程で京都大学理学研究科数学教室において開催された第2回日独セミナー「無限次元調和解祈」の成果をもとに編集されたものである.編集 : ハーバート・ハイヤー, 平井 武, 尾畑 信明Editors: Herbert Heyer, Takeshi Hirai, Nobuaki Obata #e

Product formulas for a two-parameter family of Heckman-Opdam hypergeometric functions of type BC

In this paper we present explicit product formulas for a continuous two-parameter family of Heckman-Opdam hypergeometric functions of type BC on Weyl chambers $C_q\subset \mathbb R^q$ of type $B$. These formulas are related to continuous one-parameter families of probability-preserving convolution structures on $C_q\times\mathbb R$. These convolutions on $C_q\times\mathbb R$ are constructed via product formulas for the spherical functions of the symmetric spaces $U(p,q)/ (U(p)\times SU(q))$ and associated double coset convolutions on $C_q\times\mathbb T$ with the torus $\mathbb T$. We shall obtain positive product formulas for a restricted parameter set only, while the associated convolutions are always norm-decreasing. Our paper is related to recent positive product formulas of R\"osler for three series of Heckman-Opdam hypergeometric functions of type BC as well as to classical product formulas for Jacobi functions of Koornwinder and Trimeche for rank $q=1$

Categories of hypermagmas, hypergroups, and related hyperstructures

In order to diagnose the cause of some defects in the category of canonical hypergroups, we investigate several categories of hyperstructures that generalize hypergroups. By allowing hyperoperations with possibly empty products, one obtains categories with desirable features such as completeness and cocompleteness, free functors, regularity, and closed monoidal structures. We show by counterexamples that such constructions cannot be carried out within the category of canonical hypergroups. This suggests that (commutative) unital, reversible hypermagmas -- which we call mosaics -- form a worthwhile generalization of (canonical) hypergroups from the categorical perspective. Notably, mosaics contain pointed simple matroids as a subcategory, and projective geometries as a full subcategory.Comment: 48 pages, 3 figure

Hyperstructures and Idempotent Semistructures

Much of this thesis concerns hypergroups, multirings, and hyperfields. These are analogous to abelian groups, rings, and fields, but have a multivalued addition operation. M. Krasner introduced the notion of a valued hyperfield; The prototypical example is $K/(1+\mathfrak{m}_K^n)$ where $K$ is a local field. P. Deligne introduced a category of triples whose objects have the form $\mathrm{Tr}_n(K)=(\mathcal{O}_K/\mathfrak{m}_K^n,\mathfrak{m}_K/\mathfrak{m}_K^{n+1},\epsilon)$ where $\epsilon:\mathfrak{m}_K/\mathfrak{m}_K^{n+1}\rightarrow \mathcal{O}_K/\mathfrak{m}_K^n$ is the obvious map. In this thesis I relate the category of discretely valued hyperfields to Deligne's category of triples. An extension of a local field is arithmetically profinite if the upper ramification subgroups are open. Given such an extension $L/K$, J.P. Wintenberger defined the norm field $X_K(L)$ as the inverse limit of the finite subextensions of $L/K$ along the norm maps. Wintenberger has defined an addition operation on $X_K(L)$, and shown that $X_K(L)$ is a local field of finite characteristic. Using Deligne's triples, I have given a new proof of Wintenberger's characterization of its Galois group. The semifield $\mathbb{Z}_{\max}$ is defined as $\{0\}\cup\{u^k\mid k\in\mathbb{Z}\}$ with addition given by $u^m+u^n=u^{\max(m,n)}$. An extension of $\mathbb{Z}_{\max}$ is a semifield containing $\mathbb{Z}_{\max}$. The extension is finite if $S$ is finitely generated as a $\mathbb{Z}_{\max}$-semimodule. In this thesis I classify the finite extensions of $\mathbb{Z}_\mathrm{max}$. There are two previously known methods for constructing a hypergroup from a totally ordered set. In this thesis I generalize these to a family of constructions parametrized by hypergroups $H$ satisfying $x-x=H$ for all $x\in H$. We say a hyperfield $K$ is selective if $1+1-1-1=1-1$ and for all $x,y\in K$ one has either $x\in x+y$ or $y=x+y$. In this thesis, I show that a selective hyperfield is characterized by a totally ordered group $\Gamma$, a hyperfield $H$ satisfying $1-1=H$, and an extension $\phi\in\mathrm{Ext}^1(\Gamma,H^\times)$. We say a triple of elements $(x,y,z)$ of an idempotent semiring is a corner triple if $x+y=y+z=x+z$. We say an idempotent semiring is regular if whenever $(x,y,a)$ and $(z,w,a)$ are corner triples, there exists $b$ such that $(x,z,b)$ and $(y,w,b)$ are also corner triples. I prove in this thesis that the category of regular idempotent semirings is a reflective subcategory of the category of multirings