Direct Systems of Spherical Functions and Representations
Abstract
Spherical representations and functions are the building blocks for harmonic analysis on riemannian symmetric spaces. In this paper we consider spherical functions and spherical representations related to certain infinite dimensional symmetric spaces $G_\infty/K_\infty = \varinjlim G_n/K_n$. We use the representation theoretic construction $\phi (x) = <e, \pi(x)e>$ where $e$ is a $K_\infty$fixed unit vector for $\pi$. Specifically, we look at representations $\pi_\infty = \varinjlim \pi_n$ of $G_\infty$ where $\pi_n$ is $K_n$spherical, so the spherical representations $\pi_n$ and the corresponding spherical functions $\phi_n$ are related by $\phi_n(x) = <e_n, \pi_n(x)e_n>$ where $e_n$ is a $K_n$fixed unit vector for $\pi_n$, and we consider the possibility of constructing a $K_\infty$spherical function $\phi_\infty = \lim \phi_n$. We settle that matter by proving the equivalence of condtions (i) $\{e_n\}$ converges to a nonzero $K_\infty$fixed vector $e$, and (ii) $G_\infty/K_\infty$ has finite symmetric space rank (equivalently, it is the Grassmann manifold of $p$planes in $\F^\infty$ where $p < \infty$ and $\F$ is $\R$, $\C$ or $\H$). In that finite rank case we also prove the functional equation $\phi(x)\phi(y) = \lim_{n\to \infty} \int_{K_n}\phi(xky)dk$ of Faraut and Olshanskii, which is their definition of spherical functions.
 Publication:

arXiv eprints
 Pub Date:
 October 2011
 arXiv:
 arXiv:1110.0655
 Bibcode:
 2011arXiv1110.0655D
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Differential Geometry;
 Mathematics  Functional Analysis;
 43A35;
 53C35;
 22E46
 EPrint:
 17 pages. New material added on the finite rank cases