Direct Systems of Spherical Functions and Representations

TL;DR

This paper investigates spherical functions and representations on infinite-dimensional symmetric spaces, establishing conditions for their convergence and functional equations, with implications for harmonic analysis and the structure of these spaces.

Contribution

It characterizes when spherical functions on infinite-dimensional symmetric spaces can be constructed as limits of finite-dimensional ones, linking this to the space's finite rank and proving related functional equations.

Findings

Convergence of fixed vectors characterizes finite rank symmetric spaces.

Spherical functions satisfy a specific limit functional equation.

Finite rank spaces are equivalent to Grassmann manifolds of finite dimension.

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.