Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\phi (x) = <e, \pi(x)e>$ 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.

26
The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
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The categories of this paper are: stat
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The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
22
The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: eess
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The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom