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 $\phi (x) =$ 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.Comment: 17 pages. New material added on the finite rank case

M\u27Naghten: Right or Wrong for Florida in the 1980s? It Flunks the Test

The members of the jury listened solemnly as the judge instructed them on the law. The defendant was charged with murder in the first degree