An introduction to harmonic analysis on the infinite symmetric group

The irreducible unitary highest weight representations $(\pi_\lambda,\mathcal{H}_\lambda)$ of the group $U(\infty)$, which is the countable direct limit of the compact unitary groups $U(n)$, are classified by the orbits of the weights $\lambda \in \mathbb{Z}^{\mathbb{N}}$ under the Weyl group $S_{(\mathbb{N})}$ of finite permutations. Here, we determine those weights $\lambda$ for which the first cohomology space $H^1(U(\infty),\pi_\lambda,\mathcal{H}_\lambda)$ vanishes. For ...

We initiate a Stein's method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov's central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained; the proof gives an error term. The construction of an exchangeable pair needed for applying Stein's method arises from the theory of harmonic functions on Bratelli diagrams. We also find the spectrum of the Markov chain on partitions...

We extend the classical Schur-Weyl duality between representations of the groups $SL(n,\C)$ and $\sN$ to the case of $SL(n,\C)$ and the infinite symmetric group $\sinf$. Our construction is based on a "dynamic," or inductive, scheme of Schur-Weyl dualities. It leads to a new class of representations of the infinite symmetric group, which have not appeared earlier. We describe these representations and, in particular, find their spectral types with respect to the Gelfand-Tsetl...

I survey some recent developments in the theory of zeta functions associated to infinite groups and rings, specifically zeta functions enumerating subgroups and subrings of finite index or finite-dimensional complex representations.

Refection Positivity is a central theme at the crossroads of Lie group representations, euclidean and abstract harmonic analysis, constructive quantum field theory, and stochastic processes. This book provides the first presentation of the representation theoretic aspects of Refection Positivity and discusses its connections to those different fields on a level suitable for doctoral students and researchers in related fields.

In the framework of the supervised learning of a real function defined on a space X , the so called Kriging method stands on a real Gaussian field defined on X. The Euclidean case is well known and has been widely studied. In this paper, we explore the less classical case where X is the non commutative finite group of permutations. In this setting, we propose and study an harmonic analysis of the covariance operators that enables to consider Gaussian processes models and fore...

In 1964 R.Gangolli published a L\'{e}vy-Khintchine type formula which characterised $K$ bi-invariant infinitely divisible probability measures on a symmetric space $G/K$. His main tool was Harish-Chandra's spherical functions which he used to construct a generalisation of the Fourier transform of a measure. In this paper we use generalised spherical functions (or Eisenstein integrals) and extensions of these which we construct using representation theory to obtain such a char...

This article, which is substantially motivated by the previous joint work with J. McKay [8], establishes the analytic analogues of the relations we found free probability has with Witt vectors. Therefore, we first present a novel analytic derivation of an exponential map which relates the free additive convolution on $\mathbb{R}$ with the free multiplicative convolution on either the unit circle or $\mathbb{R}_+^*$, for compactly supported, freely infinitely divisible probabi...

In this paper we review some connections between harmonic analysis and the modern theory of automorphic forms. We indicate in some examples how the study of problems of harmonic analysis brings us to the important objects of the theory of automorphic forms, and conversely. We consider classical groups and their unitary, tempered, automorphic and unramified duals. The most important representations in our paper are the isolated points in these duals.

We study two classes of extension problems, and their interconnections: (i) Extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups $G$; (ii) In case of Lie groups, representations of the associated Lie algebras $La\left(G\right)$ by unbounded skew-Hermitian operators acting in a reproducing kernel Hilbert space (RKHS) $\mathscr{H}_{F}$. Why extensions? In science, experimentalists frequently gather spectral data in cases when...