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 $G_{\mathrm{\infty }}\mathrm{/}{K}_{\mathrm{\infty }}=\mathrm{}{G}_n\mathrm{/}{K}_n$. We use the representation theoretic construction $\varphi (x)=<e,\pi (x)e>$ where $e$ is a $K_{\mathrm{\infty }}$–fixed unit vector for $\pi$. Specifically, we look at representations ${\pi }_{\mathrm{\infty }}=\mathrm{}{\pi }_n$ of $G_{\mathrm{\infty }}$ where ${\pi }_n$ is $K_n$–spherical, so the spherical representations ${\pi }_n$ and the corresponding spherical functions ${\varphi }_n$ are related by $\phi_n(x) = <e_n, \pi_n(x)e_n>$where$e_n$isa$K_n$--fixedunitvectorfor$\pi_n$, and we consider the possibility of constructing a $K_{\mathrm{\infty }}$–spherical function ${\varphi }_{\mathrm{\infty }}=\lim {\varphi }_n$. We settle that matter by proving the equivalence of condtions (i) $\{e_n\}$ converges to a nonzero $K_{\mathrm{\infty }}$–fixed vector $e$, and (ii) $G_{\mathrm{\infty }}\mathrm{/}{K}_{\mathrm{\infty }}$ has finite symmetric space rank (equivalently, it is the Grassmann manifold of $p$–planes in ${\text{F}}^{\mathrm{\infty }}$ where $p<\mathrm{\infty }$ and $\text{F}$ is $\mathbb{R}$, $\text{C}$ or \H). In that finite rank case we also prove the functional equation $\varphi (x)\varphi (y)={\lim }_{n\to \mathrm{\infty }}{\int }_{K_n}\varphi (xky)dk$ of Faraut and Olshanskii, which is their definition of spherical functions.