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Direct Systems of Spherical Functions and Representations

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 $ϕ(x) = <e, π(x)e>$ where $e$ is a $K_\infty$--fixed unit vector for $π$. Specifically, we look at representations $π_\infty = \varinjlim π_n$ of $G_\infty$ where $π_n$ is $K_n$--spherical, so the spherical representations $π_n$ and the corresponding spherical functions $ϕ_n$ are related by $ϕ_n(x) = <e_n, π_n(x)e_n>$ where $e_n$ is a $K_n$--fixed unit vector for $π_n$, and we consider the possibility of constructing a $K_\infty$--spherical function $ϕ_\infty = \lim ϕ_n$. 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 $ϕ(x)ϕ(y) = \lim_{n\to \infty} \int_{K_n}ϕ(xky)dk$ of Faraut and Olshanskii, which is their definition of spherical functions.