Plancherel Measure Research Articles

Experiments in calculating the dimension of a typical representation of the symmetric group

We give numerical data in favor of the conjecture of the asymptotic equidistribution of the probabilities of typical representations of the symmetric group and we estimate the entropy of the limiting Plancherel measure.

The plancherel measure for symmetric graphs

Let G be the free product of r copies of the cyclic groupZk.We obtain the Plancherel formula for the commutative O*-algebra of radial convolution operators on l2(G). The Plancherel measure is expressed in terms of the c-function appearing in the expansion of spherical functions on G as linear combinations of exponentials.

Unimodularity and atomic Plancherel measure

Unimodularity and atomic Plancherel measure

Fourier Transforms of Intertwining Operators and Plancherel Measures for GL(n)

Fourier Transforms of Intertwining Operators and Plancherel Measures for GL(n)

The Plancherel measure for polygonal graphs

We compute explicitely the Plancherel measure for groups acting isometrically and simply transitively on polygonal graphs.

Les fonctions sphériques d'un couple de Gelfand symétrique et les chaînes de Markov

After an elementary description of Gelfand pairs, spherical functions and Plancherel measure, some explicit computations on the related Markov chains are performed. Random walks on polyhedra belong to this class of Markov chains; two more examples of chains on graphs are worked out, and the necessary and sufficient condition of transcience of random walks on p-adic numbers with spherical symmetry is given as an application of the techniques of the paper.

Les fonctions sphériques d'un couple de Gelfand symétrique et les chaînes de Markov

After an elementary description of Gelfand pairs, spherical functions and Plancherel measure, some explicit computations on the related Markov chains are performed. Random walks on polyhedra belong to this class of Markov chains; two more examples of chains on graphs are worked out, and the necessary and sufficient condition of transcience of random walks on p-adic numbers with spherical symmetry is given as an application of the techniques of the paper.

Harmonic analysis on reductive Lie groups

This is a survey on harmonic analysis on real reductive Lie groups for the period 1950–1978. Contained is a description of the fundamental series of representations of a reductive group G, the theory of characters, a description of the Plancherel measure, a description of dual spaces with respect to the Fourier transform on G, the classification of irreducible representations of the group G, questions of harmonic analysis on symmetric spaces of noncompact type.

Tensor products of principal series for the De Sitter group

The decomposition of the tensor product of two principal series representations is determined for the simply connected double covering, G = Spin(4, 1), of the DeSitter group. The main result is that this decomposition consists of two pieces, T, and Td, where T, is a continuous direct sum with respect to Plancherel measure on G of representations from the principal series only and Td is a discrete sum of representations from the discrete series of G. The multiplicities of representations occurring in T, and Td are all finite. Introduction. Let G = Spin(4, 1) be the simply connected double covering of the DeSitter group, G = KAN an Iwasawa decomposition of G, M the centralizer of A in K, and P = MAN the associated minimal parabolic subgroup of G. For a E M and T E A, a x T is a representation of P via a X T(man) = a(m)T(a) and a representation of the form 7(a, T) = Indp a x T is called a principal series representation of G. The main goal of this paper is to determine the decomposition of the tensor product of two principal series representations of G into irreducibles. It was shown in [7], by using Mackey's tensor product theorem and the Mackey-Anh reciprocity theorem, that this problem reduces to knowing how to decompose the restriction to MA of almost every principal series representation of G and each discrete series representation of G. For a representation v7 belonging to the principal series of G, the restriction of v7 to MA, (7)MAA was determined by using Mackey's subgroup theorem. However, in that paper, we were not able to determine explicitly (7)MA for a representation 'r belonging to the discrete series of G. This we do in ?3 of this paper by using Lie algebraic methods and the realizations of these representations given by Dixmier in [2]. This paper is organized as follows. In ?? 1 and 2 we summarize the main results concerning the structure and representation theory of G that we shall use. In ?3 we determine (7)MA when v7 is a discrete series representation of G. We also include the results of [7] concerning the decomposition of (@)MA when v7 is a principal series representation of G. In ?4 we show how to decompose the tensor product of two principal series representations of G. The main results are contained in Theorem 4. The basic methodology used in this paper to decompose principal series tensor products originates in the works of G. Mackey [6], N. Anh [1], and F. Williams [11]. Received by the editors April 15, 1980. 1980 Mathematics Subject Classification Primary 22E43, 81C40. 'This research was partially supported by a grant from the National Science Foundation. ? 1981 American Mathematical Society 0002-9947/81 /0000-0207/$04.75 121 This content downloaded from 157.55.39.118 on Fri, 22 Apr 2016 04:32:41 UTC All use subject to http://about.jstor.org/terms

FIA] B − Gruppen und Hypergruppen

The spaceGB of\(\bar B\)-orbits of an [FIA]B− groupG is a commutative hypergroup (=commutative convo in Jewett's terminology). Its space of “hypergroup characters” is a hypergroup itself (see [3]) which can be identified withE (G, B) (the extreme points ofB-invariant positive definite continuous functionsp:G→ℂ withp (e)=1), endowed with the topology of compact convergence. We determine the structure space ofL1 (E(G, B), μPL) whereμPL is the Plancherel measure (which is the “Haar measure” ofE(G, B)) and characterize the convolution algebrasM (E(G, B)) andL1(E(G, B)) via the inverse Fourier-Stieltjes transformation.

Invariant polynomials and zonal spherical functions

In the second and third sections we derive again some results of Nolan Wallach (with some modest simplifications). In the second section we give his proof of a simple lemma (lemma 10) which is the basis for the inversion of the Abel transform. To add a little spice, we cast the lemma in the setting of Euclidean motion groups; this gives us the Paley-Wiener theorem for certain Bessel functions (Jn/2, n ~ 7/). The third sect~n gives Wallach's proof of the inversion formula for the Abel transform and of the zonal Paley-Wiener theorem for rank one semi-simple groups [H2, G]. I have included these because they are not readily available. The fourth section is devoted to finding the Plancherel measure in rank one, class one. The techniques of this paper are quite elementary; the most difficult facts used are Mather's theorem and the parameterization of zonal spherical function [H, ch 10]. (Actually one can give a proof of Mather's theorem in the case used here which is considerably simpler). Aside from this we use only some standard structure theory for semi-simple groups, regular singular point theory for ordinary differential equations, and a contour integral.

TENSOR PRODUCTS OF UNITARY REPRESENTATIONS OF THE THREE-DIMENSIONAL LORENTZ GROUP

The author obtains a decomposition into irreducible representations of the tensor product of any two irreducible unitary representations of the group SO0(1,2). An explicit construction of this decomposition is given, and the corresponding Plancherel measure is found. Bibliography: 13 titles.

Square-integrable representations and the dual topology

We show that the square-integrable factor representations of a connected locally compact group G are precisely the normal representations whose kernels in C ∗(G) are open points of Prim r ( G) (the support of Plancherel measure). Related results hold for certain other groups. We also settle questions of Dixmier, and Duflo and Moore, by giving examples of square-integrable irreducible representations (of totally disconnected groups) which are not open in the reduced dual.

Fourier inversion on Borel subgroups of Chevalley groups: the symplectic group case

In recent papers, the author and J. A. Wolf have developed the Plancherel theory of parabolic subgroups of real reductive Lie groups. This includes describing the irreducible unitary representations, computing the Plancherel measure, and-since parabolic groups are nonunimodular-explicating the (unbounded) Dixmier-Pukanszky operator that appears in the Plancherel formula. The latter has been discovered to be a special kind of pseudodifferential operator. In this paper, the author considers the problem of extending this analysis to parabolic subgroups of semisimple algebraic groups over an arbitrary local field. Thus far he has restricted his attention to Borel subgroups (i.e. minimal parabolics) in Chevalley groups (i.e. split semisimple groups). The results he has obtained are described in this paper for the case of the symplectic group. The final result is (perhaps surprisingly), to a large extent, independent of the local field over which the group is defined. Another interesting feature of the work is the description of the “pseudodifferential” Dixmier-Pukanszky operator in the nonarchimedean situation.

Noncompact unimodular groups with purely atomic Plancherel measures

A noncompact unimodular group whose regular representation decomposes as a direct sum of irreducible subrepresentations was constructed by J. M. G. Fell. In the present paper, groups of this kind are studied, and several new examples are provided to settle open structure problems.

Noncompact Unimodular Groups with Purely Atomic Plancherel Measures

Noncompact Unimodular Groups with Purely Atomic Plancherel Measures

Complementary series for 𝑝-adic groups. I

Let Ω \Omega be a nonarchimedean local field, G the group of Ω \Omega -points of a connected reductive algebraic group defined over Ω \Omega . This paper establishes that to each zero of the Plancherel measure of G one can associate complementary series. Our result is the analogue for p-adic groups of a similar statement, announced separately by Knapp-Stein and Harish-Chandra, for real groups.

On the Plancherel measure for linear Lie groups of rank one

In this paper we find a very explicit, simple form for the Plancherel measure for rank one, linear simple groups, including the normalizing constant.

A Plancherel measure for the discrete Heisenberg group

A Plancherel measure for the discrete Heisenberg group

Nuclearity of some spaces of C∞ vectors in induced representations

We give conditions on irreducible unitary induced representations of inhomogeneous Lie groups ensuring nuclearity of the space of differentiable vectors. We study the consistency of these conditions. Finally we show that our conditions hold almost everywhere with respect to the Plancherel measure for some generalizations of the Poincaré group.