Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Infinite Dimensional Multiplicity Free Spaces III: Matrix Coefficients
  and Regular Functions   In earlier papers we studied direct limits $(G,K) = \varinjlim (G_n,K_n)$ of
two types of Gelfand pairs. The first type was that in which the $G_n/K_n$ are
compact Riemannian symmetric spaces. The second type was that in which $G_n =
N_n\rtimes K_n$ with $N_n$ nilpotent, in other words pairs $(G_n,K_n)$ for
which $G_n/K_n$ is a commutative nilmanifold. In each we worked out a method
inspired by the Frobenius--Schur Orthogonality Relations to define isometric
injections $\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)$ for $m
\geqq n$ and prove that the left regular representation of $G$ on the Hilbert
space direct limit $L^2(G/K) := \varinjlim L^2(G_n/K_n)$ is multiplicity--free.
This left open questions concerning the nature of the elements of $L^2(G/K)$.
Here we define spaces $\cA(G_n/K_n)$ of regular functions on $G_n/K_n$ and
injections $\nu_{m,n} : \cA(G_n/K_n) \to \cA(G_m/K_m)$ for $m \geqq n$ related
to restriction by $\nu_{m,n}(f)|_{G_n/K_n} = f$. Thus the direct limit
$\cA(G/K):= \varinjlim \{\cA(G_n/K_n), \nu_{m,n}\}$ sits as a particular
$G$--submodule of the much larger inverse limit $\varprojlim \{\cA(G_n/K_n),
\text{restriction}\}$. Further, we define a pre Hilbert space structure on
$\cA(G/K)$ derived from that of $L^2(G/K)$. This allows an interpretation of
$L^2(G/K)$ as the Hilbert space completion of the concretely defined function
space $\cA(G/K)$, and also defines a $G$--invariant inner product on $\cA(G/K)$
for which the left regular representation of $G$ is multiplicity--free.

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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: q-bio
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The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
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The categories of this paper are: adap-org
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The categories of this paper are: hep-lat
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The categories of this paper are: atom-ph
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The categories of this paper are: econ
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The categories of this paper are: chem-ph
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The categories of this paper are: eess