Asymptotic spherical analysis on the Heisenberg group

Jacques Faraut¹

¹ Institut de Mathématiques de Jussieu Université Pierre et Marie Curie 175 rue du Chevaleret 75013 Paris, France

Colloquium Mathematicum, Tome 118 (2010) no. 1, pp. 233-258

Résumé

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.

Détail

DOI : 10.4064/cm118-1-13

Keywords: asymptotics spherical functions large dimensions related spherical functions olshanski spherical pairs paper consider inductive limits gelfand pairs associated heisenberg group group times acts multiplicity cal space polynomials p mathbb space times complex matrices group acts heisenberg group times mathbb result carcano pair ltimes gelfand pair main results paper asymptotics spherical functions related pair large analysis involves asymptotics shifted schur functions