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

Auditors\u27 Third Party Liability: An Ill-Considered Extension of the Law

Numerous legal commentators have recently advocated an extension of auditors\u27 professional liability to third-party users of financial statements, with the dual purposes of inducing improved disclosure in financial statements and obtaining restitution for investors injured by misleading statements, and some judicial decisions have adopted such an expanded liability. This comment evaluates the probable effectiveness and effects of such expanded liability, concluding that the expansion is unlikely to obtain either of its stated objectives and that such an expansion is likely to create pressures upon the cost and availability of audits which will be injurious to both investors and users of capital and will impede the resource-allocating capability of the economy