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Asymptotic spherical analysis on the Heisenberg group

100%
Faraut J.
Colloquium Mathematicum
|
2010
|
tom 118
|
nr 1
233-258
EN
The asymptotics of spherical functions for large dimensions are related to spherical functions for Olshanski spherical pairs. In this paper we consider inductive limits of Gelfand pairs associated to the Heisenberg group. The group K = U(n) × U(p) acts multiplicity free on 𝓟(V), the space of polynomials on V = M(n,p;ℂ), the space of n × p complex matrices. The group K acts also on the Heisenberg group H = V × ℝ. By a result of Carcano, the pair (G,K) with G = K ⋉ H is a Gelfand pair. The main results of the paper are the asymptotics of the spherical functions related to the pair (G,K) for large n and p. This analysis involves the asymptotics of shifted Schur functions.
2
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The Wigner semi-circle law and the Heisenberg group

63%
Faraut J., Saal L.
Banach Center Publications
|
2007
|
tom 78
|
nr 1
133-143
EN
The Wigner Theorem states that the statistical distribution of the eigenvalues of a random Hermitian matrix converges to the semi-circular law as the dimension goes to infinity. It is possible to establish this result by using harmonic analysis on the Heisenberg group. In fact this convergence corresponds to the topology of the set of spherical functions associated to the action of the unitary group on the Heisenberg group.
3
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Markov-Krein transform

63%
Faraut J., Fourati F.
Colloquium Mathematicum
|
2016
|
tom 144
|
nr 1
137-156
EN
The Markov-Krein transform maps a positive measure on the real line to a probability measure. It is implicitly defined through an identity linking two holomorphic functions. In this paper an explicit formula is given. Its proof is obtained by considering boundary values of holomorhic functions. This transform appears in several classical questions in analysis and probability theory: Markov moment problem, Dirichlet distributions and processes, orbital measures. An asymptotic property for this transform involves Thorin-Bondesson distributions.
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