Asymptotic spherical analysis on the Heisenberg group

Volume 118 / 2010

Jacques Faraut Colloquium Mathematicum 118 (2010), 233-258 MSC: 43A90, 22E27, 33C50. DOI: 10.4064/cm118-1-13


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 $K=U(n)\times U(p)$ acts multiplicity free on ${\cal P}(V)$, the space of polynomials on $V=M(n,p;{\mathbb C})$, the space of $n\times p$ complex matrices. The group $K$ acts also on the Heisenberg group $H=V\times {\mathbb R}$. By a result of Carcano, the pair $(G,K)$ with $G=K\ltimes H$ 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.


  • Jacques FarautInstitut de Mathématiques de Jussieu
    Université Pierre et Marie Curie
    175 rue du Chevaleret
    75013 Paris, France

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