An Analysis on Harmonic Function For Measuring Hypergroups

The modern approach to harmonic analysis on a Lie group treats the representations of the group as the central objects of study, while characters are treated as important but auxiliary objects associated to representations. This is in direct contrast to the modern approach to harmonic analysis on a finite group, which treats the determination and study of the characters of the group as the primary problem, and considers representations as important but auxiliary objects associated to characters. For finite groups, the reason for this is three fold; 1) the determination of the irreducible characters is a vastly simpler problem than the determination of the irreducible representations 2) almost all of the standard problems of harmonic analysis may be answered solely by means of the character theory and 3) historically, the theory of characters has preceded that of the theory of representations. This suggests the following interesting question - might it be possible to develop harmonic analysis on a Lie group as essentially a theory of characters, and thereby finesse the present difficulties and technicalities in modern representation theory? [This is not an entirely new idea- in fact Harish Chandra's pivotal work on the existence of discrete series for non-compact semi-simple groups takes this view.] In this paper we show how such an approach may be initiated, and applied to various classes of groups. The first problem is of course - how does one define a character of a group if one does not know what a representation is? Our answer to this question is a variant of the one Ftobenius would have given- the characters of a finite group, say, are exactly those functions on the set of conjugacy ·classes of the group which respect the natural algebraic structure of this set. This algebraic structure has a probabilistic nature and is called an abelian hypergroup. We therefore replace the problem of non-commutative harmonic analysis on the group G with the problem of abelian harmonic analysis on the hypergroup of conjugacy classes, which we call the class hypergroup of G. This allows us to sketch a straight forward algorithm for the construction of the character table of any given finite group. It also provides a general framework for the study of harmonic analysis on an arbitrary group. When G is a Lie group, we show that one may expect an intimate relationship between the class hypergroup of G and the hypergroup of adjoint orbits. This provides an explanation for the general effectiveness of Kirillov theory in harmonic analysis since, formally at least, the dual object of the hypergroup of adjoint orbits is the hypergroup of co-adjoint orbits, although we also see that in general Kirillov theory will provide at best a 'linear approximation' to the unitary dual of G on account of the global difference between the class hypergroup and the hypergroup of adjoint orbits. We therefore propose a program for the determination and study of the unitary dual of a Lie It is well known that Riemannian symmetric spaces can be represented as homogeneous spaces G/K of Lie groups G. Recently, harmonic functions on Riemannian symmetric spaces have been studied in C-H. Chu, A.T-M. Lau (2010) via convolution semigroups of measures on G, where the harmonic spherical functions identify as functions on the double coset space G//K. Since double coset spaces are special examples of hypergroups on which the Borel measures have a convolution structure, it is natural to consider harmonic functions on the wider class of hypergroups where convolution can be exploited. Our objective is to develop a basic theory of harmonic functions on hypergroups which is applicable to a large class of examples besides symmetric spaces, for instance, the group orbit spaces, spaces of conjugacy classes and dual spaces of compact groups. Given a Borel measure  on a hypergroup H, a Borel function f on H is called  - harmonic if it satisfies the convolution equation We first introduce the concept of a nilpotent hypergroup and focus our attention on them. Commutative hypergroups are nilpotent and include, for example, double coset spaces G//K of Gelfand pairs (G, K), polynomial hypergroups and. hypergroups arising from free groups, discrete semigroups, quantum groups and various differential equations. We show that a nilpotent hypergroup admits an invariant measure and the Liouville theorem holds for these hypergroups. Further, we show that positive harmonic functions on metrisable nilpotent hypergroups are integrals of exponential functions. For arbitrary hypergroups with an invariant measure, we derive a Harnack inequality for positive harmonic functions and prove a Liouville theorem for compact hypergroups. All the above results should translate into interesting applications to concrete examples of hypergroups. As an example of such application, consider the double coset space G//K of a Gelfand pair (G, K). It is a commutative hypergroup and therefore Liouville theorem holds. It follows that, given an adapted radial probability measure  on G. the bounded -harmonic spherical functions on G must be constant. In the case of and K the subgroup of rotations, this crucial fact was the key in Y. Benyamini et al. (2002) to showing that the bounded - harmonic functions on the unit disc are exactly the bounded harmonic functions of the Laplacian. In fact, the latter result should hold for more general Lie groups other than, as noted in Y. Benyamini et al. (2002), In this section, we derive a Harnack inequality for positive harmonic functions on a hypergroup G and, if G is nilpotent and metrisable, we determine these functions completely.

To describe positive, possibly unbounded,-harmonic functions on a hypergroup G, one can make use of Choquet’s representation theory on cones of Radon

measures in. Letbe a subcone of. A measureis called extremal inif everywithis a positive multiple of. Letbe the set of extremal measures in. For a Radon measureand a real Borel function / on G, we define a measure byfor each Borel set If G admits a right Haar measureand if / is a positive solution of the equation then, as in, the measure satisfies the equation Therefore one can determine all positive solutions of once one describes every element in the cone To achieve this, we consider the larger cone w hich is closed in the weak* topology. Hence, if G is separable and metrisable, then is weak* complete andis a Borel set. By Choquet’s representation theory, each has an integral representation where, for the measureis supported on the Borel set. If G is nilpotent, we describe and hence, completely. Lemma. Let cr be a positive non-degenerate Radon measure on a hypergroup G. Let f be a non-negative continuous a-harmonic function on G. Then either f is

Proof. Let / be nonzero somewhere. We show. Suppose otherwise, we deduce a contradiction. The open set is nonempty and hencefor someby non-degeneracy of. Hence which is impossible. This proves For each, the left translateis not identically zero since for some implies on some open neighbourhood V of jc and where Repeating the previous arguments to the -harmonic function gives Let G be a hypergroup which has a right Haar measure. Denote bythe modular function of G. Letbe an absolutely continuous probability measure on G, with compact support. By taking an infinite series involving repeated convolutions ofwith itself, one can find a strictly positive lower semicontinuous functionon G such that implies and . whereis a probability measure on G. We note that Theorem. Let G be a hypergroup which admits a right Haar measure co and letbe an -absolutely continuous non-degenerate probability' measure on G, with compact support. Then for each compact subset K of G, there is a constantsuch that for all positive continuous-harmonic functions f on G. continuous-harmonic function / on G, the function defined by is positive continuous-harmonic and its value at e is 1. therefore we have, by the above remark, for all.

A hypergroup (X,*) is called commutative if (M(X),+,*) is a commutative algebra, and hermitian if the involutionis the identity map. Its easy to prove that every her- mitian hypergroup is commutative. A locally bounded measurable function is called a semicharaeter if and for all. Every bounded semicharacter is called a character. If the character is not locally null then it must be continuous. The dualof X is just the set of continuous characters with the compact-open topology in which casemust be locally compact. In this paper we will be concerned with continuous characters on hypergroups. A locally bounded measurable functionis said to be positive definite if for all choice of and . The following two lemmas are in fact, an adaption of whatever done for semigroups. We will not repeat the proof, wherever the proof for semigroups can be applied to the hypergroups with necessary modification.

One should be observe that, a functionis negative definite if and only ifis positive definite for each t > 0. While this result holds for all semigroups it is not clear how to prove the 'only if' part for hypergroups since the usual technique do not apply(the 'if' part always holds provided thatis locally lower bounded). The problem is that except usually not equal so that other methods have to be used to overcome this. A locally bounded measurable function q is called a quadratic form if for alland additive if for all. In the case X is hermitian, that when X carries the identity involution, then every quadratic form is an additive function and every negative definite function is real. A locally bounded measurable functionis said to be negative definite ifand for all choice of and all that satisfy A key result in the study of negative definite functions on hypergroups is the following Levy-Khinchin represent at ion for allwhere q is a nonnegative quadratic form on X and. Both q and the integral part belong to the set of negative definite function on X and the pairis uniquely determined by with q being given by

If G is a connected, simply-connected nilpotent Lie group, then the exponential map is a diffeomorphism fromto G. This suggests there should be a strong connection betweenand, if these objects really exist; in fact Kirillov theory leads us to predict that. The typical conjugacy class is non-compact however, so while it does carry a G-invariant measure, such a measure will not generally be a probability measure. Furthermore the convolution of two such measures may easily not exist, at least in the usual sense. Nevertheless we have the following result. expThenif and only if Since the proof is pertinent here, we recall the main idea. Working in the free Lie algebra generated by X and Y (and the corresponding algebra of formal power series) and letting X • Y = [X,Y]y X * Y = ln(exp X exp Y) and, one can show that there exists formal power series A(X, Y) and B(X, Y) such that Furthermore the assignmentis invertible in the sense that one can write X and Y as similar formal power series in Z and W. These considerations prove thatbut they actually prove more, namely that if we define by and by , then we can find a bijection such that This is nothing but a formal proof of our conjectureIf G was a finite nilpotent group with a Lie algebra 0 and an exponential mapwhich was both a bijection and satisfied the Baker-Campbell-Hausdorff formula, then the above argument would allow us immediately to deduce thatAre there such groups? Yes there axe - if G is a finite (p — .1) step nilpotent p-group (for some prime p) then Howe has shown that there is an abelian group 0 with the structure of a (p — 1) step nilpotent Lie algebra and a bijectionsatisfying the Campbell-Hausdorff formula. We have thus proved.

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